sensors Article
Celestial Object Imaging Model and Parameter Optimization for an Optical Navigation Sensor Based on the Well Capacity Adjusting Scheme Hao Wang, Jie Jiang * and Guangjun Zhang Key Laboratory of Precision Opto-mechatronics Technology, Ministry of Education, School of Instrumentation Science and Opto-electronics Engineering, Beihang University, No. 37 Xueyuan Road, Haidian District, Beijing 100191, China;
[email protected] (H.W.);
[email protected] (G.Z.) * Correspondence:
[email protected]; Tel.: +86-10-8233-8497 Academic Editor: Francesco De Leonardis Received: 3 January 2017; Accepted: 19 April 2017; Published: 21 April 2017
Abstract: The simultaneous extraction of optical navigation measurements from a target celestial body and star images is essential for autonomous optical navigation. Generally, a single optical navigation sensor cannot simultaneously image the target celestial body and stars well-exposed because their irradiance difference is generally large. Multi-sensor integration or complex image processing algorithms are commonly utilized to solve the said problem. This study analyzes and demonstrates the feasibility of simultaneously imaging the target celestial body and stars well-exposed within a single exposure through a single field of view (FOV) optical navigation sensor using the well capacity adjusting (WCA) scheme. First, the irradiance characteristics of the celestial body are analyzed. Then, the celestial body edge model and star spot imaging model are established when the WCA scheme is applied. Furthermore, the effect of exposure parameters on the accuracy of star centroiding and edge extraction is analyzed using the proposed model. Optimal exposure parameters are also derived by conducting Monte Carlo simulation to obtain the best performance of the navigation sensor. Finally, laboratorial and night sky experiments are performed to validate the correctness of the proposed model and optimal exposure parameters. Keywords: optical navigation sensor; well capacity adjusting; star centroid estimation; edge extraction; exposure parameter optimization
1. Introduction Optical autonomous navigation is a key technology in deep space exploration. This process is usually accomplished by multi-sensor integration, such as star sensors, navigation cameras, inertial measurement devices, and other equipment. The attitude information of a spacecraft is obtained by a star sensor and inertial measurement element. The navigation camera captures the target celestial image with background stars and extracts the target celestial line-of-sight (LOS) vector according to the current spacecraft attitude. Then, the spacecraft position can be calculated by integrating these optical navigation measurements according to the geometric relationship. An optical navigation system with multi-sensor integration is not only complicated in structure and has high cost and power consumption but also has installation errors between sensors, which further restricts the improvement of navigation accuracy. The best solution for deep space exploration missions is when a single navigation sensor can simultaneously obtain the attitude and LOS vector from the sensor to the centroid of the target celestial body. This approach requires the sensor to image the stars and target celestial body and to extract their navigation measurements simultaneously. However, a large
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gap exists between the irradiance of stars and the target celestial body. For reference, standard image sensors have a dynamic range of 40 dB to 70 dB [1], which is insufficient to ensure that the target celestial body and stars are well-exposed simultaneously. The problem of insufficient dynamic range for image sensors can be generally solved in three ways. The design of optical systems to lower the incident flux of a celestial body is first considered. In Reference [2], the combined Earth-/star sensor for attitude and orbit determination of geostationary satellites is investigated. The combined Earth-/star sensor has two fields of view (FOV) for the observation of the Earth and stars. The two FOVs are combined on the detector through a beamsplitter. The partially transmissive mirror reflects 91% of starlight onto the detector, while transmitting only 9% of the Earth’s brightness. The Earth’s incident flux is further reduced by a filter. This method directly lowers the incident flux of the high irradiance target from the source, which is convenient for the subsequent image processing. A disadvantage of this approach is the complexity of the optical system design, higher weight, and higher costs. The second method enhances the dynamic range by image processing algorithms. In [3–6], multi-exposure fusion techniques are adopted to enhance the dynamic range. A set of different exposure images is obtained. Then, these images are fused into an image where all scenes or areas of interest appear well exposed. The advantage of the multi-exposure technique is that it can enhance the dynamic range without degrading the signal-to-noise ratio (SNR) [7]. The main drawback of exposure fusion is its limitation to static scenes, and any object movement incurs severe ghosting artifacts in the fused result. Given that a spacecraft is always in the motion state, this method is inapplicable in such condition. New image sensor designs are also proposed to attain an extended dynamic range. The photocurrent in logarithmic response image sensors is fed to a resistor with a logarithmic current-voltage characteristic [8–10]. Logarithmic-response image sensors can obtain a wide dynamic range, but it has several disadvantages (i.e., image lag, low SNR, large fixed pattern noise, and poor image quality). This undesired lag effect is most pronounced at low light conditions, and it is caused by a long settling time constant that can exceed the frame time. Another wide dynamic image sensor based on time-to-saturation information was reported in [11–14]. The pixel attains its saturation level and extrapolates the incident light by measuring and recording the time required to attain the saturation state. The light intensity is derived by the information on the time stored in the memory of each pixel and then the final image can be reconstructed. However, each pixel of the detector requires a signal detection circuit, comparator, digital memory, and other components to detect the saturation state and storage time information. This condition results in large pixel sizes and low fill factor, which limit the sensor resolution. The well capacity adjusting (WCA) scheme described by Knight [15] and Sayag [16] and implemented by Decker [17] compresses the sensor’s current versus charge response curve using a lateral overflow gate. This technology is currently widely employed by integrating a lateral overflow integration capacitor in a pixel in complementary metal-oxide-semiconductor (CMOS) detectors [18–20]. The well capacity is monotonically increased once or multiple times to its maximum value during integration. The accumulated photoelectrons of the high irradiance signal are suppressed, but the low irradiance signal is unaffected. The WCA scheme enhances the dynamic range, but at the expense of substantial degradation in SNR. In [21], the navigation camera used a sequence of long and short exposures for optical navigation. A short exposure in which the celestial body is not saturated will permit determination of the celestial body’s location within the image. A long exposure will permit determination of the stars’ location within the image. Setting a long exposure time to ensure that the dim stars satisfy the detection SNR limit is necessary to ensure that the navigation sensor has a reliable attitude determination function. However, a long exposure time can lead to the overexposure of the target celestial body, which results in the expansion of the image shape because the apparent diameter increases and high stray light effect which will overwhelm the natural response to the target stars. Given that the WCA scheme is widely
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utilized in CMOS image sensors, high dynamic range images can be obtained by a single exposure with this technique. This study first analyzes the irradiance characteristics of a celestial body. Then, the celestial body edge model and star spot imaging model are established when the WCA scheme is applied. The effect of exposure parameters on the accuracy of star centroid estimation and edge detection is analyzed based on the proposed model. The exposure parameters are optimized to ensure that the optical navigation measurements satisfy the requirement of navigation accuracy. Comparing with the conventional navigation sensors, this study provides a feasible method for the study of a miniaturized single FOV optical navigation sensor, which is less cost, less weight, simple to design and can obtain attitude information and LOS vector from the sensor to the centroid of the target celestial body simultaneously. This navigation sensor has a strong applicability and can be utilized for a variety of navigation tasks. 2. Irradiance Characteristics of a Celestial Body The study of the irradiance characteristics of a celestial body is the prerequisite for the optimization of the exposure parameters of navigation sensors. The irradiance of a celestial body received by the detector is analyzed in this section. Figure 1 shows the spatial position relationship between the navigation sensor and the observed target. A planet typically emits energy in a manner that reflects solar radiation energy. The irradiance of the sun is assumed to be isotropic because the irradiance is inversely proportional to the square of the distance [22]. Thus, the irradiance received at the target celestial surface is expressed as: IP = Isun (rS /RSP )2
(1)
where Isun is the irradiance of the sun, rS is the radius of the sun, and RSP is the distance between the sun and the target celestial body. The incident energy on the surface of a planet is only partially reflected back into cosmic space, whereas the rest is absorbed by the surface. r P is defined as the radius of the celestial body, A is the Bond albedo that represents the fraction of energy incident on an astronomical body scattered back into space at all wavelengths and phase angles. Thus, the total radiant flux reflected by the surface of the celestial body is expressed as: LR =
π Ar2P rS2 · Isun R2SP
(2)
We consider that navigation sensor observes the planet at a distance of R PC . Only a part of the illuminated area of the celestial body can be observed in most cases. If the surface of a planet is assumed to be homogeneous, then the distribution of the reflected radiant energy only depends on two factors, distance R PC and phase angle ξ which is the angle at the celestial body object between the Sun and the observer. Phase function is the ratio of the reflected radiant flux at phase angle ξ to the radiant ◦ flux at zero phase angle, which is denoted as P(ξ ). When ξ = 0 , P(ξ ) = 1. Thus, the irradiance received by the navigation sensor is expressed in the following form: IC = CP(ξ ) ·
CAr2P rS2 LR = · P(ξ ) · Isun 4πR2PC 4R2PC R2SP
(3)
where C is a constant. The total reflected energy of the celestial body is distributed on a spherical surface. As the distance increases, the radiant flux density decreases, but the total radiant flux remains the same. Therefore, the total radiant flux on the sphere from the center of the celestial body of radius R PC is expressed as: Z Z CP(ξ ) L R L R = IC dS = dS (4) 4πR2PC S
S
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RPC
RSP
Figure Figure1.1.Spatial Spatialposition positionrelationship relationshipbetween betweenthe thesensor sensorand andtarget targetcelestial celestialbody. body.
dS can be inexpressed Figure shows the dS area terms ofasits coordinates as Figure 22shows thatthat the area can be expressed terms of itsin coordinates dS = R2PC sin ξdϕdξ. 2 C is derived as: dS RPC sinC disderived d . Therefore, Therefore, as: 2 2 44πR RPC
44π 22 PC C C = R = R π = Rϕ2= 2π Rξ =π )dS sinξdξ ) sin dξdϕdξ d 0PP(ξ)sin d P P(ξdS ϕ=0 ξ =P0P(ξsin S S
0
0
(5)(5)
0
Inastronomy, astronomy, the theBond Bondalbedo albedo( (A) relatedtotothe thegeometric geometricalbedo albedo( (ρ) by the the expression expression In ) by A ) isisrelated A = ρq [23], where: A q [23], where: Z π Z π I (ξ ) sin ξdξ = 2 P(ξ ) sin ξdξ (6) q=2 0 0 II(0) q 2 sin d 2 P sin d (6) 0 I 0 0 following Thus, the constant C must obey the relationship:
Thus, the constant C must obey the following 4ρ relationship: C= (7) 4A C (7) A received by the sensor can be expressed as: The previously analysis shows that the irradiance The previously analysis shows that the irradiance received by the sensor can be expressed as: ρr2P rS2 IC = 2 2 22 · P(ξ ) · Isun (8) rSSP R rPR I C PC P I (8) sun 2 2 RPC RSP Equation (8) shows that IC is the function of R PC , RSP , and phase angle ξ. Visual magnitude is RSP , and phase Equationquantity (8) shows that I C adopted is the function of RPC angle object. . VisualFor magnitude the relative generally to measure the, irradiance of a celestial example, when the Moon is observed in a geosynchronous orbit, the visual magnitude varies from −2.5 to is the relative quantity generally adopted to measure the irradiance of a celestial object. For example, − 12.74. The irradiance of a celestial body is commonly higher than that of stars. when the Moon is observed in a geosynchronous orbit, the visual magnitude varies from −2.5 to −12.74. The irradiance of a celestial body is commonly higher than that of stars.
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Figure 2. Reflected radiation flux over a sphere.
Figure fluxover overaasphere. sphere. Figure2.2.Reflected Reflected radiation radiation flux
3. Celestial Object Imaging Model based on the Well Capacity Adjusting Scheme
3. Celestial ObjectImaging ImagingModel Modelbased based on on the the Well Well Capacity 3. Celestial Object CapacityAdjusting AdjustingScheme Scheme
The WCA scheme is widely employed in CMOS image sensors. During integration period of the The WCAscheme schemeisis widely widely employed in CMOS image sensors. During integration periodperiod of the of The WCA employed CMOS image sensors. integration WCA scheme, well capacity of a certain pixelin is increased several times to During extend the range of incident WCA scheme, well capacity of a certain pixel is increased several times to extend the range of incident the WCA well range capacity of aiscertain is ratio increased times to extend thetorange signal.scheme, The dynamic which definedpixel as the of the several largest nonsaturating signal the of signal. The dynamic range which is defined as the ratio of the largest nonsaturating signal to the standard deviation of the noise under darkisconditions enhanced WCA scheme. Figure 3a plots incident signal. The dynamic range which defined asisthe ratio ofinthe largest nonsaturating signal standard deviation of the noise under dark conditions is enhanced in WCA scheme. Figure 3a plots thestandard accumulated photoelectrons versus the dark integration time for three different irradiance signals in 3a to the deviation of the noise under conditions is enhanced in WCA scheme. Figure the accumulated photoelectrons the integration time for three different irradiance signalsinin integration mode. The versus accumulated photoelectrons increase linearly with the increase plotsnormal the accumulated photoelectrons versus the integration time for three different irradiance signals normal integration mode. accumulated photoelectrons increase linearly with the aincrease integration time until they The reach the full well capacity. The accumulated pixel canin in in normal integration mode. The accumulated photoelectrons increasephotoelectrons linearly withofthe increase integration time until they reach the full well capacity. The accumulated photoelectrons of a pixel can be expressed as: they reach the full well capacity. The accumulated photoelectrons of a pixel integration time until can be expressed as: be expressed as: I T if I Q / T ( Q I T if I QMAX / T (9) MAX IQ · T otherwise if I ≤ Q MAX /T (9) (9) Q = Q MAX
otherwise QQ otherwise MAX MAX
where I is the photocurrent of the incident signal. The largest photocurrent of nonsaturating where isphotocurrent theisphotocurrent of the signal. The largest photocurrent of nonsaturating QMAXincident / Tsignal. incident signal given by, . where I is Ithe of Ithe The largest photocurrent of nonsaturating incident MAX incident I Q / T incident signal is given by, . MAX /T. MAX signal is given by, I MAX = Q MAX Photoelectrons
Photoelectrons
Photoelectrons Q
Photoelectrons
QMAX
MAX
QMAX
QMAX IH
IH
QS
IM
IM
QS IL
IL (a)
IH
IH T
T
Time
IM
IM
IL
IL (b)
TS
TS
T
Time
T
Time Time (a) (b) Figure 3. (a) Accumulated photoelectrons versus time in normal integration mode; and (b) accumulated photoelectrons versus time using the WCA scheme. Figure 3. Accumulated (a) Accumulated photoelectrons in integration normal integration mode; and (b) Figure 3. (a) photoelectrons versusversus time intime normal mode; and (b) accumulated accumulated photoelectrons versus time using the WCA scheme. photoelectrons versus time using the WCA scheme.
The total integration time is divided into several time segments when the WCA scheme is utilized. The well capacity is adjusted to a higher value at the beginning of each time segment until The total integration time is divided into several time segments when the WCA scheme is itThe increases the full well capacity. The accumulated photoelectrons are awhen piecewise total to integration time is divided into several time segments the linear WCAfunction scheme is utilized. The well capacity is adjusted to a higher value at the beginning of each time segment until with respect to the integration time. Figure 3b shows that the integration time is divided into twountil utilized. The well adjusted to accumulated a higher value at the beginning each time segment it increases to thecapacity full welliscapacity. The photoelectrons are a of piecewise linear function T T T T and . is designated as the adjusting integration time (AIT) at this segments, namely, S S The S accumulated photoelectrons are a piecewise linear function it increases to the fullintegration well capacity. with respect to the time. Figure 3b shows that the integration time is divided into two QMAX time TS . Notably, the full well capacity at time point. Thetowell is adjustedFigure from Q3b with respect the capacity integration shows integration is divided two S to TS and T time. TS . TS is designated asthat the the adjusting integration time (AIT) into at this segments, namely, Q I (e.g., the high irradiance signal ( ) case in the when the accumulated photoelectrons reach segments, namely, TS andisT adjusted − TS . TSfrom is designated adjusting time at this H QS toS the as QMAX at time TS (AIT) fullthe well capacityintegration . Notably, point. The well capacity . The capacity excess photoelectrons will T spill over figure), output photoelectrons arefrom clipped time point. The the well capacity is adjusted QSuntil to the fullTS well Q MAX at time . Notably, S in the when the accumulated photoelectrons reach QS (e.g., the high irradiance signal ( I H ) case
when the accumulated photoelectrons reach QS (e.g., the high irradiance signal (IH ) case in the figure), figure), the output photoelectrons are clipped until time TS . The excess photoelectrons will spill over the output photoelectrons are clipped until time TS . The excess photoelectrons will spill over from the
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lateral overflow gate. However, the accumulated photoelectrons of the low irradiance signal (IL ) are Sensors 2017, 17, 915 6 of 23 unaffected. Therefore, the accumulated photoelectrons of a pixel when using the WCA scheme can be expressed as: from the lateral overflow gate. However, the accumulated photoelectrons of the low irradiance signal Therefore, the accumulated photoelectrons of a pixel when using the WCA ( I L ) are unaffected.
I·T QS + I · ( T − TS ) Q I T MAX
scheme can be expressed as:
Q=
if I ≤ QS /TS if QS /TS < I ≤ ( Q MAX − QS )/( T − tS ) TS if I QS / TS otherwise
(10)
Q QS I T TS if QS / TS I QMAX QS / T tS TS (10) Q of incidentotherwise The largest photocurrent signal when using the WCA scheme is given by, MAX
I 0 MAX = ( Q MAX − QS )/( T − TS ). The standard deviation of the noise under dark conditions is the The dynamic largest photocurrent of incident signal when using the WCA scheme is given by, same. Thus, range is enhanced by a factor: (QMAX QS ) / (T TS ) . The standard deviation of the noise under dark conditions is the same. I MAX
0 Thus, dynamic range is enhanced by a Ifactor: MAX
λ=
=
( Q MAX − QS ) T Q MAX (QT − T TS ) Q
I MAX I S MAX MAX (11) I Q T TS MAX For a signal that does not saturate after MAX integration, Equation (10) can also be expressed as:
(11)
Equation (10) can also be expressed as: For a signal that does not saturate afterintegration, Q Q = IT − ε I − S · ( ITS − QS ) Q T Q IT I SS ITS QS
(
(12) (12)
TS
t≥0 where ε(t) = 1 t 0is the unit step function. As shown in Equation (12), the exposure parameters where t 0 t < 0 is the unit step function. As shown in Equation (12), the exposure parameters 0 t 0 QS and TS explicitly show the effects of output photoelectrons. Therefore, celestial body edge model QS and TS explicitly show the effects of output photoelectrons. Therefore, celestial body edge and star spot imaging model are first established in this study. Then, the influence of exposure model and star spot imaging model are first established in this study. Then, the influence of exposure parameters on the accuracy of the extraction of optical navigation measurements is analyzed. 1
parameters on the accuracy of the extraction of optical navigation measurements is analyzed.
3.1. Celestial Body Edge Model
3.1. Celestial Body Edge Model
The edge of a celestial body can ideally be modeled as a step function on 1D section. Given the The edge of a celestial body can ideally be modeled as a step function on 1D section. Given the effects of the point spread function (PSF) of the optic system, the real edge adjusts to the blurring effect effects of the point spread function (PSF) of the optic system, the real edge adjusts to the blurring and effect is called blurred model. The parameter Gaussian PSF PSF radius σPSFindicates the extent and the is called the edge blurred edge model. The parameter Gaussian radius PSF indicates the of the blurring The celestial body image is assumed to be to anbeideal disk,disk, suchsuch thatthat thethe radial extent of theeffect. blurring effect. The celestial body image is assumed an ideal energy distribution along the direction normal to the edge is isotropic. Therefore, the 2D imaging radial energy distribution along the direction normal to the edge is isotropic. Therefore, the 2D ◦ around the center model can bemodel regarded asregarded having been formed byformed 1D edge rotates imaging can be as having been by model 1D edgethat model that 360 rotates 360° around of the Aof 1D celestial edge body model is established, and the subsequent analyses are based thedisk. center the disk. A body 1D celestial edge model is established, and the subsequent analyses are on the said simplify theoretical analysis and calculation. The blurred edge (Figure 4) can be basedmodel on theto said model the to simplify the theoretical analysis and calculation. The blurred edge (Figure 4) canby be convolving modeled by convolving the edge ideal with step edge withwhich the PSF, is expressed modeled the ideal step the PSF, is which expressed as [24]:as [24]: k x x− ll 1 h ( x)k er erf + 1 + h √ f f ( x ) f= 2PSF 2 2 2σ PSF
(13) (13)
hk
h
l Figure4. 4. Blurred Blurred edge Figure edgemodel. model.
x
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This model has four parameters, namely, background intensity h , edge contrast k , edge PSF . background This lmodel four parameters, namely, intensity h, edgeiscontrast k, edge location and has Gaussian PSF radius The background intensity zero regardless of thel location and Gaussian PSF radius σ . The background intensity is zero regardless of the random noise. Then, PSF random noise. Then, the 1D celestial body edge model is expressed as: the 1D celestial body edge model is expressed as: PQE T x l f x φP η T erf x − l 1 (14) QE 2 er f √2 PSF +1 f (x) = (14) 2 2σPSF Thus, the 1D edge gradient model is derived as: Thus, the 1D edge gradient model is derived as: 2 PQET " x l # f x φP ηQE T exp ( x − 2 (15) 2 l) f 0 ( x ) = √ 2 PSF exp − 2 PSF (15) 2 2σPSF 2πσPSF where T is the integration time, P is the incident flux density of the celestial body on the image where T is the integration time, φP is the incident flux density of the celestial body on the image plane, plane, and QE is the quantum efficiency of the imager sensor. Equation (15) implies that the and ηQE is the quantum efficiency of the imager sensor. Equation (15) implies that the gradient is xl . gradient is maximized the edge maximized at the edge at location x =location l. In Figure 5, 5, the the small small amplitude amplitude blue blue solid solid line line denotes denotes the the intensity intensity distribution distribution with short In Figure with aa short integration time, and the central area is under-saturated. The edge location can be obtained at integration time, and the central area is under-saturated. The edge location can be obtainedhalf at the amplitude intersection point. The large amplitude light blue dashed line denotes the intensity half the amplitude intersection point. The large amplitude light blue dashed line denotes the distribution where the long time time should havehave been. However, intensity distribution where theintegration long integration should been. However,the theactual actual intensity intensity distribution is denoted as the blue solid line because the central area is oversaturated. This scenario scenario distribution is denoted as the blue solid line because the central area is oversaturated. This will result result in in an an extension extension of of the the apparent apparent diameter diameter of of the the celestial celestial body. body. Therefore, real edge edge will Therefore, the the real location cannot be extracted. location cannot be extracted.
Figure 5. 1D profile of the celestial body image in normal integration mode.
The WCA scheme is used to avoid oversaturation of the celestial body. Let the total integration The WCA scheme is used to avoid oversaturation of the celestial body. Let the total integration time be T and AIT be TS . The well capacity is adjusted at time TS from QS to QMAX . In Figure 6, time be T and AIT be TS . The well capacity is adjusted at time TS from QS to Q MAX . In Figure 6, the red TS and the accumulated photoelectrons the red solid line denotes the intensity distribution solid line denotes the intensity distribution at timeatTtime accumulated photoelectrons in the S and the Q . The dark blue solid line denotes the intensity distribution at the in the central area are clipped at central area are clipped at QS . The dark blue solid line denotes the intensity distribution at the end of S the integration. Compared with Figure the central of the celestial image is under-saturated, end of the integration. Compared with5,Figure 5, thearea central area of the body celestial body image is underwhich avoids theavoids extension of the apparent diameter diameter of the celestial However, Figure 6 Figure shows saturated, which the extension of the apparent of the body. celestial body. However, a shallow ring formsring around thearound central the area. Therefore, 1D celestial body edge model 6that shows that aenergy shallow energy forms central area. the Therefore, the 1D celestial body becomes a piecewise when the WCA is applied can be and expressed edge model becomes afunction piecewise function whenscheme the WCA schemeand is applied can be as: expressed as:
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PQE T x l x xc erf 1 2 2 PSF f x φP ηQE T (16) x−l x + T er Tf S √ 1 x ≤ x l c P QE erf 1 x xc 2σPSF QS 2 2 f (x) = (16) 2 PSF φ η x−l P QE ( T − TS ) er f √ +1 x > xc QS + 2σPSF x l 2 2QS where xC is the solution of erf 1 , which denotes the inflection point of the x −2l PSF P2Q SS QE T where xC is the solution of er f √ = − 1, which denotes the inflection point of the φP ηQEbody TS is commonly high, such that T ≫ Q . 2σPSF intensity distribution. The irradiance of a celestial P QE S S intensity distribution. The irradiance of a celestial body is commonly high, such that φP ηQE TS QS . x l is derived. Therefore, Therefore, xCC < l is derived. The 1D scheme is is utilized utilized can can be be derived derived as: as: The 1D edge edge gradient gradient model model when when the the WCA WCA scheme 2 T " x l)2l # PT QE φ η x − ( P QE exp x xx≤ √ C x 2 − C 22 PSF 2exp PSF 2πσPSF 2σPSF 0 f x (17) f (x) = (17) " x l 2 2 # (T−TTS )) P QE φ η ( T x − l ( ) P QE S exp √ C xC exp − 2 2 x xx> 2PSF 22σPSF PSF PSF 2πσ
ll,, the the second second term term of of Equation Equation (17) (17) obtains obtains the the maximum maximum value. Therefore, the true edge when xx= location can be extracted when the WCA scheme is adopted.
Figure Figure 6. 6. 1D 1D profile profile of of the the celestial celestial body body image image when when using using the the WCA WCA scheme. scheme.
3.2. 3.2. Star Star Spot Spot Imaging Imaging Model Model Establishing anaccurate accurate star imaging model is the step to achieving high star Establishing an star spotspot imaging model is the first stepfirst to achieving high star centroiding centroiding accuracy. Stars canpoint be considered point Stellar sourcesrays at can infinity. Stellar rays can be accuracy. Stars can be considered sources at infinity. be approximated to parallel approximated to parallel light rays. Incident stellar lights pass through the optical lens and are light rays. Incident stellar lights pass through the optical lens and are focused at a point in the focal focused at a point in the focal plane. However, the lens is generally slightly defocused to improve the plane. However, the lens is generally slightly defocused to improve the centroiding accuracy of centroiding of a star which spreads toimage severalplane. pixels in theprofile imageof plane. profile of a star spot, accuracy which spreads tospot, several pixels in the The a starThe spot can be adescribed star spotby canthe beGaussian describedPSF, by the Gaussian PSF, whereas the parameter Gaussian PSF radius PSF whereas the parameter Gaussian PSF radius σ indicates the extent PSF
PSF where is high, the spot region where out a star spot spreads is indicates the extent dispersion. of dispersion. WhenofσPSF is high, When the region a star spreads is large. The starout spot imaging in normal integration commonly assumes a 2D Gaussian function 7a) large. Themodel star spot imaging model in mode normal integration mode commonly assumes a 2D(Figure Gaussian and can be expressed as: can be expressed as: function (Figure 7a) and " # φS TηQE ( x − x0 )2 + ( y − y0 )2 Enor ( x, y) = exp − = ΦS ( x, y) T (18) 2 2 2πσPSF 2σPSF
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x x0 y y0 S TQE exp Enor ( x, y ) 2 2 2 PSF 2 PSF 2
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255
255
192
192 Intensity
Intensity
S is the incident flux of the star on the image thetrue truecentroid centroid where (xx0,,yy0) isisthe where of of thethe starstar andand 0 0 " #φS is the incident flux of the star on the image plane. 2 2 2 2 φS ηQE SQE ( x − xx0 ) x+ (y − y0 )y0 0 y ΦS ( x, y)= exp −exp is defined as the energy distribution function at this plane. is defined as the energy distribution function at 2 S x, y 2 2 2πσPSF2 2 2σPSF 2 PSF PSF point. The accumulated photoelectrons of a bright star are suppressed when utilizing the WCA scheme, this point. The accumulated photoelectrons of a bright star are suppressed when utilizing the WCA and the excess photoelectrons are drained via the overflow gate. The star spot imaging model when scheme, and the excess photoelectrons are drained via the overflow gate. The star spot imaging model the WCA scheme is applied (Figure 7b) can be expressed as: when the WCA scheme is applied (Figure 7b) can be expressed as: QS Q EWCA ( x, y) = ΦS ( x, y) T − ε ΦS ( x, y) − S (19) (Φ ( x, y) T − Q ) EWCA ( x, y ) S x, y T S x, y TS S Sx, y TS S QS S (19) T S The celestial body edge model and star spot imaging model when utilizing the WCA scheme The celestial body edge model and star spot imaging model when utilizing the WCA scheme have been established so far. In the subsequent section, the effect of exposure parameters on the have been established so far. In the subsequent section, the effect of exposure parameters on the accuracy of star centroiding and edge detection is analyzed using the proposed models. Then, accuracy of star centroiding and edge detection is analyzed using the proposed models. Then, the the exposure parameters are optimized to obtain the best performance of the navigation sensor exposure parameters are optimized to obtain the best performance of the navigation sensor utilizing utilizing the WCA scheme. the WCA scheme.
128 64 0 5
5 y [p
ixe 0 ls]
0 -5
-5
ls] ixe p [ x
128 64 0 5
5 y [p
ixe 0 ls]
(a)
0 -5
-5
ls] ixe p [ x
(b)
Figure 7. 7. (a) (a) Star Star signal signal intensity intensity distribution distribution utilizing utilizing normal normal integration integration mode; star signal signal Figure mode; and and (b) (b) star intensity distribution using the WCA scheme. intensity distribution using the WCA scheme.
4. Celestial Object Image Feature Extraction Accuracy Performance Utilizing the WCA Scheme 4. Celestial Object Image Feature Extraction Accuracy Performance Utilizing the WCA Scheme The navigation measurements of deep space optical navigation systems are generally combined The navigation measurements of deep space optical navigation systems are generally combined to calculate the LOS direction and spacecraft location [25]. The accuracy of the apparent diameter and to calculate the LOS direction and spacecraft location [25]. The accuracy of the apparent diameter celestial body centroiding is determined by the precision of edge detection. The accuracy of the LOS and celestial body centroiding is determined by the precision of edge detection. The accuracy of the direction of the navigation sensor is determined by the attitude measurement precision, that is, the LOS direction of the navigation sensor is determined by the attitude measurement precision, that is, centroiding accuracy of the background stars. In this section, the effect of exposure parameters on the the centroiding accuracy of the background stars. In this section, the effect of exposure parameters accuracy of star centroiding and edge detection is analyzed using the proposed image model, which on the accuracy of star centroiding and edge detection is analyzed using the proposed image model, provides theoretical support for parameter optimization. which provides theoretical support for parameter optimization. 4.1. Edge the WCA WCA Scheme Scheme 4.1. Edge Detection Detection Accuracy Accuracy Performance Performance Utilizing Utilizing the The edge edge is is the the part part of of the the image image where where brightness brightness changes changes sharply. sharply. The The edge edge points points based based on on The the blurred edge model are given by the maxima of the first image derivative or zero crossing point the blurred edge model are given by the maxima of the first image derivative or zero crossing point of the the second second image image derivative. derivative. Steger of Steger proposed proposed aa subpixel subpixel edge edge extraction extraction algorithm algorithm in in his his doctoral doctoral thesis [26]. The basic principle of the algorithm is to perform the second-order Taylor expansion thesis [26]. The basic principle of the algorithm is to perform the second-order Taylor expansion about about the pixel pixel where where the the local local gradient gradient is is maximized maximized in in the the direction of the the edge edge normal normal and and to the direction of to determine determine the subpixel point of of the the edge edge of of the the the subpixel location location of of the the zero zero crossing crossing point the second second derivative. derivative. In In this this study, study, the celestial body is extracted using this algorithm, which is essentially a fitting interpolation algorithm. celestial body is extracted using this algorithm, which is essentially a fitting interpolation algorithm. Given that the edge detection algorithm is based on image derivative information, it is highly sensitive to noise. Therefore, the image derivatives must be estimated by convolving the image with
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the derivatives of the Gaussian smoothing kernel. Edges appear as bright lines in an image that contains the absolute value of the gradient. The second-order Taylor expansion about the maximized gradient pixel in the direction of the edge normal is expressed as: 1 r 0 ( x ) = r 0 ( x0 ) + r 00 ( x0 ) x + r 000 ( x0 ) x2 2
(20)
where x0 is the pixel center. The subpixel location of the edge where r00 ( x ) = 0 is expressed as: l0 = x 0 −
r00 ( x0 ) r 000 ( x0 )
(21)
The 1D celestial body edge model is a piecewise function when the WCA scheme is applied. The gradient of the edge model is derived by convolving the edge model with the first derivative of the Gaussian smoothing kernel and expressed as:
R∞ r 0 ( x ) = f ( x ) ∗ g0 ( x ) = −∞ f 0 (τ ) g( x − τ )dτ " " 2 # # R xc φP ηQE T x−τ 2 τ−l 1 exp − √ = −∞ √ exp − √ ·√ dτ + 2πσPSF 2πσ 2σPSF 2σ " " 2 # # R ∞ φP ηQE ( T − TS ) τ−l 1 x−τ 2 √ exp − √ ·√ exp − √ dτ xc 2πσPSF 2πσ 2σPSF 2σ " # φP ηQE TS ( x − l )2 TS + er f [Θ( x )] = q exp − T− 2 2 2 2 σPSF + σ2 2 2π (σPSF + σ2 )
(22)
Thus, the second derivative of edge model is derived as: # 2 TS ( x − l ) TS r 00 ( x ) = − q T − exp − + er f Θ ( x )] [ 2 2 2 2 σPSF + σ2 2 2 2π (σPSF + σ2 )(σPSF + σ2 ) " # n o φP ηQE TS σPSF ( x − l )2 exp −Θ2 ( x ) exp − − 2 2 2πσ (σPSF + σ2 ) 2 σPSF + σ2 φP ηQE ( x − l )
where Θ( x ) =
"
(23)
2 ( x − x ) + σ2 ( x − l ) σPSF c c q . The edge location is the zero crossing point of the second 2 2(σPSF + σ2 )σPSF σ
derivative, which indicates that the solution of Equation (23) and r000 ( x )r 0 ( x ) < 0 are required. Given that the analytical solution cannot be calculated, the effect of exposure parameters on the edge detection accuracy is analyzed by numerical simulations. The edge localization error is the function of the Gaussian radius σPSF , well capacity QS , total integration time T, AIT TS , Gaussian smoothing kernel radius σ, and edge location l. For a given optical system, σPSF is constant, σ is a parameter of the edge detection algorithm, and T is determined by the limiting detectable star visual magnitude for the navigation sensor. This study focuses on analyzing the edge detection error caused by QS and TS . Systematic error in edge detection is also introduced by pixelization. However, l is a random variable in practice that is uniformly distributed over a pixel. The root mean square error is defined as the error of edge detection, which is expressed as: δE ( QS , TS ) =
Z
0.5
−0.5
2
δ ( QS , TS , l )dl
2 (24)
σPSF = 0.67 pixels, σ = 0.55 pixels, T = 30 ms are set at this point. Then, the simulated celestial body images with temporal noise and fixed pattern noise are generated. Sources of noises
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which are taken into consideration include photon shot noise, dark current noise, readout noise, which are taken into consideration include photon shot noise, dark current noise, readout noise, quantization noise, dark signal non-uniformity and photon response non-uniformity. The full well quantization noise, dark signal non-uniformity and photon response non-uniformity. The full well consist with the image sensor we utilized. The edge detection error capacity is set to QMAX 15, 000e − capacity is set to Q MAX = 15, 000e consist with the image sensor we utilized. The edge detection simulation results are shown in Figure 8. 8. error simulation results are shown in Figure andAIT AITTTSis is discussed.The Theblack blackline line First,the therelationship relationship between between edge First, edge detection detectionerror error δ E and discussed. E
S
evidentlyaffected affectedby byTST.S .An An interval interval exists wherein δEE isissignificantly significantlysmall. small. indicates that that δE Eis isevidently indicates The relatively long when TS TisS short, which leads to T TT relatively long when is short, which leads Thesecond secondsegment segmentofofthe theintegration integrationtime timeT − S Sis is the of theofcentral regionregion pixels pixels and extension of the apparent diameterdiameter of the celestial to oversaturation the oversaturation the central and extension of the apparent of the body. However, T − T is relatively short when T is excessively long, which leads to a small intensity S S celestial body. However, T TS is relatively short when TS is excessively long, which leads to a contrast betweencontrast the central region and the energy Theenergy algorithm the edge of the small intensity between the central regionring. and the ring.will Theextract algorithm will extract “rings” instead of the actual edge location. Therefore, reaches theminimum number and E initiallyTherefore, the edge of the “rings” instead of the actual edge δlocation. E initially reaches the then increases with the increase in TS . minimum number and then increases with the increase in TS .
Figure8.8.Edge Edgedetection detectionerror error δEE (left) AIT TS T (right). Figure (left) versus versus well well capacity capacity QQ and AIT S Sand S (right).
QSS isisdiscussed. Second, the the relationship relationship between edge discussed. and well well capacity Q Second, edge detection detectionerror error δEE and Setting relatively small small δEE. .TSTS=29.6 29.6ms msisisset setatatthis thispoint. point. Settingthe theAIT AITto toan an appropriate appropriate value value leads to a relatively The red line indicates that δ varies slightly and remains nearly constant at the beginning with the The red line indicates that E E varies slightly and remains nearly constant at the beginning withthe increase in QS . As QS continues to increase, δE increases sharply. The reason for this relationship increase in QS . As QS continues to increase, E increases sharply. The reason for this relationship is provided in Figure 9, which shows the simulated images of the celestial object and the second is provided in Figure 9, which shows the simulated images of the celestial object and the second derivative of the edge model expressed in Equation (23) when the well capacity values are 1000e−, derivative of the edge model expressed in Equation (23) when the well capacity values are 1000 e , 5000e−and 14,000e− . 5000 e and 14,000 e . In Figure 9, the purple color represents the saturated pixels, whereas the cyan color represents the In Figure 9, the purple color represents the saturated pixels, whereas the cyan color represents pixels with zero intensity. The yellow arc represents the true edge of the celestial body. The red cross the pixels with zero intensity. The yellow arc represents the true edge of the celestial body. The red symbol indicates the zero-crossing point of the second derivative, which is located at the true edge cross symbol indicates the zero-crossing point of the second derivative, which is located at the true location, whereas the blue circle symbol indicates the zero-crossing point that deviates from the true edge location, whereas the blue circle symbol indicates the zero-crossing point that deviates from the edge location. Figure 9a shows that the intensity of the energy ring is small when the well capacity QS true edge location. Figure 9a shows that the intensity of the energy ring is small when the well is small and that the zero-crossing points can be extracted at the actual edge location. Figure 9b shows capacity QS is small and that the zero-crossing points can be extracted at the actual edge location. that the intensity of the energy ring increases with the increase in QS and that the zero crossing points QS edge and that the Figure 9b shows the energy intensity of the ringextracts increases with edges, the increase infalse exist at the locationthat of the ring. Theenergy algorithm double and the can be zero crossing existthat at the location of the energy ring. The extracts double edges, rejected. Figurepoints 9c shows the intensity of the energy ring is algorithm higher with a larger QS and that and no the false edge can be shows that the intensity of the energy ring is from higher zero-crossing points arerejected. obtainedFigure at the 9c actual edge location. The extracted edge deviates thewith truea obtained at the actual edge location. The extracted larger QThus, location. largeno QSzero-crossing value resultspoints in falseare edge extraction. S anda that edge deviates from the true location. Thus, a large QS value results in false edge extraction.
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Thus, the well capacity and AIT are the main factors that affect the edge detection error. TS Thus, the well capacity and AIT are the main factors that affect the edge detection error. evidently influences on the accuracy of edge detection. The edge detection error initially reaches the TS evidently influences on the accuracy of edge detection. The edge detection error initially reaches minimum number and then increases with the increase in T ; Q must not be excessively large. the minimum number and then increases with the increase in STS ; QSS must not be excessively large.
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0
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−; Figure of different well capacity on the edge on detection resultsdetection when: (a) Q Figure9. Effect 9. Effect of different well values capacity values the edge results when: S = 1000e − − (b) = 14, . QS 14,000e . (a)QSQ= 1000e ; ;and (b) (c) QS QS5000 e 000e ; and (c) S 5000e
4.2. 4.2.Star StarCentroiding CentroidingAccuracy AccuracyPerformance PerformanceUtilizing Utilizingthe theWCA WCAScheme Scheme Star Starcentroiding centroidingaccuracy accuracyisisthe thebasis basisfor forattitude attitudeaccuracy. accuracy.The Thestar starcentroiding centroidingaccuracy accuracy performance performanceisisanalyzed analyzedwhen whenthe theWCA WCAscheme schemeisisemployed employedtotoensure ensureattitude attitudeaccuracy. accuracy.The Thetotal total centroiding centroidingerror errorisisdecomposed decomposedinto intothe thex-and x-andy-component y-componenterrors. errors.The Theerrors errorsinineach eachcase casecan canbebe proven asas anan example. proventotobebethe thesame. same.Thus, Thus,this thisstudy studyfocuses focusesononanalyzing analyzingthe thex-component x-componenterrors errors example. The the WCA scheme is applied is expressed as:as: Thecentroiding centroidingerror errorofofthe thex-component x-componentδx when when the WCA scheme is applied is expressed x ∑i ∑ j xi Iij i j xi I ij x − x0 x0 ∑i ∑ j Iij i j I ij x , y − QQSS · Φ x , y T − Q − x · ε Φ xi , y j Tx ∑ ∑ ∑i ∑ j Φ S S Sxi , yij TjS Q SS S ixi , y j Tx i xi S iS xji , y j i j S i i j i j TTSS = x0 − x0 Q S Q S φS ηQE T −S∑ · Φ x x, y, yT T ∑ ε Φ j xi ,Sy jxi , − Q− Q yj QEiT j i S TSTS S S i i j j S S S S
δx =
(25) (25)
InInEquation (25), δx isthe function of the Gaussian radius σ radius , wellcapacity Q capacity , total integration QS , total Equation (25), x is the function of the Gaussian PSF PSF , well S time T, AIT T , incident flux of the star on the image plane φ , and actual star location x . T is S S plane S , and actual star 0location T , AIT TS , incident flux of the star on the image integration time determined by the limiting detectable star visual magnitude for the navigation sensor. If x0 moves x0 . T is determined by the limiting detectable star visual magnitude for the navigation sensor. If within a pixel, then δx changes periodically. However, x0 is a random variable that is uniformly x0 moves within a pixel, then x changes periodically. However, x0 is a random variable that is distributed over a pixel within the range [−0.5, 0.5) in practice. The root mean square error is defined distributed over a pixel within the range [−0.5, 0.5) in practice. The root mean square error asuniformly the error of x, as: Z 0.5 2 is defined as the error of x, as: 2 δx,S ( QS , TS ) = δx ( QS , TS , x0 )dx0 o (26) 2 −0.50.5 2 x, S (QS , TS ) x (QS , TS , x0 )dx0 (26) 0.5noise to the simulated After adding temporal noise and fixed pattern star image, the relationship amongAfter δx,S , adding QS , andtemporal TS is analyzed with different magnitudes. Figurestar 10 image, shows the the relationship simulated noise and fixed patternstar noise to the simulated results when theQstar magnitude is 2, 4, 5, and 6. First, the relationship between star centroiding error among x , S , S , and TS is analyzed with different star magnitudes. Figure 10 shows the simulated
results when the star magnitude is 2, 4, 5, and 6. First, the relationship between star centroiding error
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x , S and AIT TS is discussed. Figure 10a shows that the centroiding error of the star magnitude = 2 δslowly AIT TS is with discussed. Figure in 10aTSshows that10b–d the centroiding of the star magnitude =2 . Figure shows thaterror the centroiding error variation the increase x,S and increases slowly increases with the increase in T . Figure 10b–d shows that the centroiding error variation caused by TS can be neglected. Thus, S x , S is less affected by TS . Second, the relationship between caused by TS can be neglected. Thus, δx,S is less affected by TS . Second, the relationship between star star centroiding error x , S and well capacity QS is discussed. Figure 10a–c shows that x , S centroiding error δx,S and well capacity QS is discussed. Figure 10a–c shows that δx,S decreases with decreases increase intheQcentroiding the centroiding of a dim (Figure 10d) S . However,error the increasewith in Qthe . However, variation oferror a dimvariation star (Figure 10d)star caused by well S caused by capacity can be neglected. capacity canwell be neglected. Thetotal totalstar starcentroiding centroidingerror errorisisexpressed expressedas: as: The q 2x2, S 2y2, S δS,cenS , cen= δx,S + δy,S
(27) (27)
The relationship between total star centroiding error and exposure parameters is consistent with The relationship between centroiding parameters consistent with the x-component errors. Thus,total the star centroiding errorerror of a and dimexposure star is unaffected by is the WAC scheme. the errors.ofThus, thestar centroiding a dim star is by the scheme. Thex-component centroiding error a bright decreaseserror withofthe increase inunaffected well capacity, andWAC the AIT effect The centroiding error of a bright star decreases with the increase in well capacity, and the AIT effect can be ignored. can be ignored.
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Figure 10. δx,S versus well capacity QS and AIT TS for different star magnitudes: (a) star magnitude = Figure 10. x, S versus well capacity QS and AIT TS for different star magnitudes: (a) star 2; (b) star magnitude = 4; (c) star magnitude = 5; and (d) star magnitude = 6. magnitude = 2; (b) star magnitude = 4; (c) star magnitude = 5; and (d) star magnitude = 6.
In the preceding sections, the centroiding accuracy performance of a single star is analyzed In the preceding sections, the centroiding accuracy performance of a single star is analyzed when when the WCA scheme is adopted. Many stars in the FOV are required to increase the attitude the WCA scheme is adopted. Many stars in the FOV are required to increase the attitude determination accuracy in practice. More dim stars have been generally recorded than bright stars determination accuracy in practice. More dim stars have been generally recorded than bright stars in in the FOV. The overall star centroiding error is defined as the weighted average of centroiding the FOV. The overall star centroiding error is defined as the weighted average of centroiding errors errors for different star magnitudes at this point. The overall centroiding error can directly reflect the for different star magnitudes at this point. The overall centroiding error can directly reflect the attitude determination accuracy of the optical navigation sensor. The star magnitudes range from 0 attitude determination accuracy of the optical navigation sensor. The star magnitudes range from 0 to 7 at 0.5 intervals. The star magnitudes that range from mV − 0.25 to mV + 0.25 are considered to 7 at 0.5 intervals. The star magnitudes that range from mV 0.25 to mV + 0.25 are considered the the magnitude mV to simplify the analysis process. Therefore, the overall star centroiding error is magnitude expressed as: mV to simplify the analysis process. Therefore, the overall star centroiding error is 7 expressed as: ∑ δS,MVi ( NMVi ,FOV − NMVi−1 ,FOV ) i =0 δS,All = (28) NMV7 ,FOV
i 0
S , All where N17, M Vi ,915 FOV Sensors 2017,
S , M Vi
( N MVi , FOV N MVi1 , FOV )
(28)
N MV 7 , FOV
is the average number of stars brighter than magnitude MVi in the FOV, 14and of 23is
expressed as [27]:
where NMVi ,FOV is the average number of stars brighter1than cosmagnitude A / 2 MVi in the FOV, and is (29) N MVi , FOV 6.57 e1.08MVi expressed as [27]: 2 1 − cos( A/2) NMVi ,FOV = 6.57 · e1.08MVi · (29) 2 exponentially with the increase where A is the FOV size. The number of stars in the FOV increases where is the FOV M size. The number of stars in the FOV increases exponentially with the increase in starAmagnitude Vi . There are much more dim stars than the bright ones in the FOV. As a result, inthe star magnitude MVierror . There more dim stars than the overall bright ones in the FOV. AsFigure a result, star centroiding of are dimmuch star contributes more to the centroiding error. 11 the star centroiding error of dim star contributes more to the overall centroiding error. Figure 11 shows the relationship between overall centroiding error and well capacity QS when shows the relationship between overall centroiding error and well capacity QS when TS = 29.6 ms. TS 29.6 ms . The overall centroiding error fluctuation is relatively stable with the increase in QS The overall centroiding error fluctuation is relatively stable with the increase in QS because variation of because variation of star centroiding error of dim star caused by QS is less affected than bright star star centroiding error of dim star caused by QS is less affected than bright star Therefore, the variation Therefore, the variation of overall centroiding error caused by well capacity can be neglected. of overall centroiding error caused by well capacity can be neglected. 0.048
S,All[pixels]
0.046 0.044 0.042 0.04 0.038 0.036 2000
4000
6000
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Figure Figure11. 11.Overall Overallstar starcentroiding centroidingerror errorversus versuswell wellcapacity. capacity.
summary,the the well capacity is mainly responsible for centroiding the centroiding of a single InInsummary, well capacity QS Q isS mainly responsible for the errorerror of a single star. star. However, the attitude accuracy affected well capacity be neglected. However, the attitude accuracy affected by wellbycapacity can be can neglected. 5.5.Exposure ExposureParameter ParameterOptimization Optimization InInthe preceding sections, the celestial body edge star spot model are established. the preceding sections, the celestial bodymodel edge and model and image star spot image model are Both models are piecewise functions, such that the analytical solution of the optimal exposure established. Both models are piecewise functions, such that the analytical solution of the optimal parameter difficult to the optimal exposure parameters obtained are by conducting exposure is parameter is obtain. difficultThus, to obtain. Thus, the optimal exposure are parameters obtained by Monte Carlo simulation. The exposure parameters include the totalinclude integration time integration T, AIT TS , and well T, conducting Monte Carlo simulation. The exposure parameters the total time capacity appropriate of T ensures that sufficient stars are covered in the FOV for star S . An T well capacity Qvalue . An appropriate value of ensures that sufficient stars are covered AIT TS ,Qand S pattern recognition, a prerequisite for aisreliable attitudefor measurement function. The total in the FOV for star which patternisrecognition, which a prerequisite a reliable attitude measurement integration time is generally set to a relatively longer value to ensure that dim stars can be identified as function. The total integration time is generally set to a relatively longer value to ensure that dim reliable, which can causeas the target celestial body to become overexposed. Therefore, the WCA scheme stars can be identified reliable, which can cause the target celestial body to become overexposed. isTherefore, adopted, and the AIT and well capacity are optimized to obtain the best navigation performance. the WCA scheme is adopted, and the AIT and well capacity are optimized to obtain the The optical systemperformance. employed in The this study the following parameters: aperture 40 mm, best navigation opticalhas system employed in this study has D the=following focal length f aperture = 100 mm, and transmission = 80%. sensor utilized isCMV20000, mm ,The D 40 optical mm , focal = 80% . The parameters: length f τ 100 andimage optical transmission the parameters of which are listed in Table 1. image sensor utilized is CMV20000, the parameters of which are listed in Table 1. Table 1. Parameters of the CMV20000 image sensor. Parameter
Value
Parameter
Value
Active pixels Pixel pitch Full well capacity, Q MAX Conversion gain Dark current
5120 × 3840 6.4 µm × 6.4 µm 15, 000e− 0.25DN/e− 125e− /s
PRNU DSNU Read noise Quantization bits Quantum efficiency, ηQE
1% 10e− /s 8e− 12 0.45e− /photon
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5.1. Total Integration Time T The total integration time is determined by the limiting detectable star visual magnitude for the navigation sensor. Navigation sensors must conduct star pattern recognition to obtain attitude information. A sufficient number of navigation stars must be present in the FOV to ensure the effectiveness of the star pattern recognition algorithm. A star can be generally identified as reliable if the SNR of at least five pixels are more than 5. Thus, T must satisfy that the SNR of the darkest pixel of limiting detectable star is more than 5, which can be expressed as: SNR =
KφS ηQE T KφS0 ηQE T · 2.512− MV = q >5 Nnoise n2Shot + n2Dark + n2PRNU + n2DSNU + n2read + n2ADC
(30)
Thus, the following expression can be derived:
T>
5·
q
n2Shot + n2Dark + n2PRNU + n2DSNU + n2read + n2ADC KφS0 ηQE · 2.512− MV
(31)
where nShot , n Dark , n PRNU , n DSNU , nread , and n ADC denote the standard deviations of photon shot noise, dark current noise, photon response non-uniformity noise (PRNU), dark signal non-uniformity noise (DSNU), readout noise and quantization noise, respectively. Photon shot noise follows a Poisson distribution and is dependent on incident flux on the pixel; its variance is equal to the counts of photoelectrons in the imaging process. Dark current noise also follows a Poisson distribution, and its variance is equal to the production of dark√ current and exposure time. The standard deviation of the quantization noise is equal to n ADC = 1/ 12G, where G denote the conversion gain. These noise terms can be derived from the parameters in Table 1. K is the ratio of the energy of the darkest pixel to the total energy of the star signal. K = 0.0287 is set at this point. φS0 = 5.4 × 104 photons/ms is the incident flux of the star, whose magnitude is 0. MV is the limiting detectable star visual magnitude for the navigation sensor. MV = 6 is set at this point. Equation (31) shows that the total integration time must satisfy T > 25.4 ms. T = 30 ms is set to provide the system with a certain degree of redundancy. 5.2. Adjusting Integration Time TS The AIT TS is one of the main factors that influences the edge detection accuracy performance of the celestial body. The edge detection error initially reaches the minimum number and then increases with the increase in TS based on the previous conclusion. However, the variation of star centroiding error caused by TS can be neglected. Therefore, this study focuses on optimizing TS to minimize the edge detection error. Monte Carlo simulation is performed through the following procedure: First, a total of 8000 groups of celestial body images are generated using the proposed model. The well capacity values range from 3000e− to 5000e− at 25e− intervals. The AIT values range from 29.6 ms to 29.8 ms at 0.002 ms intervals. Each group contains 100 celestial body images, whose individual center coordinates are fixed; then, random noise is added. However, the radius of the celestial body obeys uniform distribution over a pixel, which is equivalent to the edge location that changes uniformly over a pixel. Second, edge data of the 100 celestial bodies are extracted using the edge detection algorithm. The absolute fitting radius error is obtained by fitting these edge points utilizing the least square circle fitting algorithm. The absolute radius error is a direct expression of the edge detection error. Then, the standard deviation of these absolute radius errors in one group is considered the average edge detection error at certain well capacity and AIT values. Finally, this procedure is repeated until all groups of celestial body images are processed. The simulation conditions are set as follows: the incident flux of the celestial body on the image plane average in one pixel φP = 7.5 × 104 photons/ms, the total integration time T = 30 ms, the Gaussian radius σPSF = 0.67 pixels, and the Gaussian smoothing kernel radius σ = 0.55 pixels.
error. Then, the standard deviation of these absolute radius errors in one group is considered the average edge detection error at certain well capacity and AIT values. Finally, this procedure is repeated until all groups of celestial body images are processed. The simulation conditions are set as follows: the incident flux of the celestial body on the image plane average in one pixel P = 7.5 10 4 photons/ms , the total integration time T 30 ms , the
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Gaussian radius PSF = 0.67 pixels, and the Gaussian smoothing kernel radius = 0.55 pixels. The
AIT TS that corresponds to the minimum standard deviation of the edge detection error is selected The AIT TS that corresponds to the minimum standard deviation of the edge detection error is selected as the optimal T at a certain well capacity, which is shown as a red scattered point in Figure 12. as the optimal TS Sat a certain well capacity, which is shown as a red scattered point in Figure 12. The simulation results are shown in Figure 12. The blue solid line indicates the linear function The simulation results are shown in Figure 12. The blue solid line indicates the linear function between well capacity Q and optimal T fitted by MATLAB. The optimal T increases with the between well capacity QSS and optimal TSS fitted by MATLAB. The optimal TSS increases with the increaseininwell wellcapacity. capacity.Therefore, Therefore,the theoptimal optimalAIT AITisisderived derivedas: as: increase
Q Q T T Q MAX − QS S TS =S T − MAX QE φPPηQE
(32) (32)
Thesame sameconclusion conclusioncan canalso alsobe beobtained obtainedunder underother othersimulation simulationconditions. conditions.Thus, Thus,the theoptimal optimal The T when utilizing the WCA scheme is given by Equation (32). AIT AIT TS Swhen utilizing the WCA scheme is given by Equation (32). 29.72 Linear fit
T S[ms]
29.7
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29.68
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29.64 3000
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Figure 12. 12. Simulation Simulation results results of of the the optimal optimal TT Figure S .S .
5.3.Well WellCapacity CapacityQQ 5.3. SS The of of well capacity QS isQ determined by two conditions. First, if QS is setifexcessively QS is set Theoptimization optimization well capacity S is determined by two conditions. First, low, then the SNR of a dim star decreases, which can cause the star identification failure of the limiting excessively low, then the SNR of a dim star decreases, which can cause the star identification failure detectable star. Therefore, providing the lower limit of the well capacity is necessary. Second, QS is of the limiting detectable star. Therefore, providing the lower limit of the well capacity is necessary. the main factor thatmain affects the accuracy of the edge detection whendetection the WCAwhen scheme applied, andis QS is the factor that affects accuracy of edge the is WCA scheme Second, the optimal QS must minimize the edge detection error. Therefore, the optimal well capacity QS is applied, and the optimal QS must minimize the edge detection error. Therefore, the optimal well comprehensively determined by the aforementioned conditions. capacity QS is comprehensively determined by the aforementioned conditions. The intensity distribution of the limiting detectable star cannot be degraded to ensure that it is The intensity distribution of the limiting cannotas: be degraded to ensure that it is identified reliably. Thus, the lower limit of welldetectable capacity isstar expressed identified reliably. Thus, the lower limit of well capacity is expressed as: QS ≥ TS K B φS0 ηQE · 2.512− MVtM (33) QS TS K BS 0QE 2.512 Vt (33) By substituting Equation (32) into Equation (33), the following expression is obtained: By substituting Equation (32) into Equation (33), the following expression is obtained: φP TηQE − Q MAX K B φS0 · 2.512− MVt QS ≥ φP − K B φS0 · 2.512− MVt
(34)
where K B is the ratio of the energy of the brightest pixel to the total energy of the star signal, K B = 0.2965 is set at this point. MVt is the star magnitude limit threshold, and the SNR of a star that is dimmer than magnitude MVt does not decrease when the WCA scheme is employed. MVt = 5.5 is set at this point, and QS ≥ 1400e− is obtained. The well capacity QS is another factor that influences the edge detection performance of the celestial body. Monte Carlo simulation is performed to obtain the optimal well capacity value. Given that the relationship between optimal AIT and well capacity is already derived in the previous section, the optimal exposure parameters under current imaging conditions are determined. A total
is set at this point, and QS 1400e is obtained.
The well capacity QS is another factor that influences the edge detection performance of the celestial body. Monte Carlo simulation is performed to obtain the optimal well capacity value. Given that the relationship between optimal AIT and well capacity is already derived in the previous Sensors 2017, 17, 915 17 of 23 section, the optimal exposure parameters under current imaging conditions are determined. A total of 300 groups of celestial body images were generated based on the celestial body edge model. Then, of 300 groups celestial images values were generated celestial edge model. e at 10body e intervals. random noise isof added. Thebody well capacity range frombased 2000 eon tothe 5000 Each − − − Then, random using noise is added. (32) The well capacityQvalues range from 2000e to 5000e at 10e intervals. TS is derived Equation at a certain . Then, the edge data of the 100 celestial bodies are S Each TS isusing derived at a certain QS . Then,fitting the edge dataerror of the celestial extracted theusing edge Equation detection (32) algorithm. The absolute radius is 100 obtained by bodies fitting are extracted using the edge detection algorithm. The absolute fitting radius error is obtained by these edge points utilizing the least square circle fitting algorithm. The standard deviation of these fitting these edge points utilizing the least square circle fitting algorithm. The standard deviation of absolute radius errors in one group is considered the average edge detection error at a certain well these absolute radius errors in group is considered average edge detection errorof at the a certain QSone capacity. The well capacity that corresponds to thethe minimum standard deviation edge well capacity. The well capacity Q that corresponds to the minimum standard deviation of the S QS if it also satisfies the condition expressed in Equation edge (34). detection error is selected as the optimal detection error is selected as the optimal QS if it also satisfies the condition expressed in Equation (34). The simulation results of the optical system in this study are shown in Figure 13. The abscissa The simulation results of the optical system in this study are shown in Figure 13. The abscissa denotes the well capacity QS , whereas the values in the parentheses denote the optimal TS that denotes the well capacity QS , whereas the values in the parentheses denote the optimal TS that Q The edge edge detection detection error error initially initially reaches reaches the the minimum minimum number correspond correspond to to the the current current QSS .. The number and and Q Q . The symbol “*” indicates the optimal . Thus, then increases with the increase in then increases with the increase in QSS . The symbol “*” indicates the optimal QSS . Thus, the the optimal optimal solutions solutions for for the exposure parameters are identified. The The optimal optimal well well capacity capacity of of the navigation navigation − sensor of of the the optical optical system system employed employed in this study is 3750 3750e e , whereas sensor , whereasthe theoptimal optimalAIT AITisis29.67 29.67ms. ms. 0.04 0.035
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0.03 0.025 0.02 0.015 0.01 2000(29.618) 3000(29.647) 4000(29.676) 5000(29.706) QS[e- ](T S[ms]) Figure Figure13. 13.Simulation Simulationresults resultsof ofthe theoptimal optimal Q QSS ..
6. 6. Experimental Experimental Results Results and and Analysis Analysis Laboratorial analysis experiment andand night skysky experiment are Laboratorialsingle-star single-starimaging imagingand andaccuracy accuracy analysis experiment night experiment conducted to validate the correctness of the models, accuracy performance analysis, and are conducted to validate the correctness of proposed the proposed models, accuracy performance analysis, optimal exposure parameters. The image sensor of the navigation sensor adopted in these and optimal exposure parameters. The image sensor of the navigation sensor adopted in these experiments listed in in Table Table 1. 1. experiments is is CMV20000, CMV20000, the the parameters parameters of of which which are are listed 6.1. Laboratorial Single-Star Imaging and Accuracy Analysis Experiment A laboratorial single-star imaging and accuracy analysis experiment is performed to validate the star spot imaging model and centroiding accuracy performance when the WCA scheme is employed. The autocollimator in the laboratory is used to generate infinite distance star signals with different star magnitudes. The navigation sensor is mounted on a turntable, as shown in Figure 14.
6.1. Laboratorial Single-Star Imaging and Accuracy Analysis Experiment 6.1. Laboratorial Single-Star Imaging and Accuracy Analysis Experiment A laboratorial single-star imaging and accuracy analysis experiment is performed to validate the A laboratorial single-star imaging and accuracy analysis experiment is performed to validate the star spot imaging model and centroiding accuracy performance when the WCA scheme is employed. star spot imaging model and centroiding accuracy performance when the WCA scheme is employed. The autocollimator in the laboratory is used to generate infinite distance star signals with different Sensors 2017, 17, 915 18 of 23 The autocollimator in the laboratory is used to generate infinite distance star signals with different star magnitudes. The navigation sensor is mounted on a turntable, as shown in Figure 14. star magnitudes. The navigation sensor is mounted on a turntable, as shown in Figure 14.
Auto-collimator Auto-collimator
Navigation Sensor Navigation Sensor
Turntable Turntable
Figure 14. Setup the laboratorial experiment. Figure Setup for for Figure 14. 14. Setup for the the laboratorial laboratorial experiment. experiment.
The exposure parameters of the navigation sensor are set as follows: total integration time the navigation navigation sensor sensor are are set set as as follows: follows: total total integration integration time The exposure parameters of the T 30 ms , AIT TS 29.67 ms , and well capacity QS 3750e −. Figure 15 shows the star images of 29.67 30 capacity Q QSS = 3750 e .. Figure TT = 30ms ms,, AIT TTSS = 29.67ms ms,, and and well capacity 3750e Figure 15 15 shows shows the the star images of different magnitudes using the normal integration mode and the WCA scheme. The brightness degree different magnitudes using the normal integration mode and the WCA scheme. The brightness degree WCA of the bright star clearly degraded when the WCA scheme is applied. However, the energy distribution of the bright star clearly degraded when the WCA scheme is applied. However, the energy distribution of the dim star is unaffected. Therefore, the star point imaging model is validated by the experiment. of the dim star is unaffected. unaffected. Therefore, Therefore,the thestar starpoint pointimaging imagingmodel modelisisvalidated validatedbybythe theexperiment. experiment.
Figure 15. (a–d) Star images of magnitudes 2, 4, 5 and 6 when using the normal integration mode; and Figure 15. 15. (a–d) images 2, 4, 66 when using the mode; and and Figure (a–d) Star Star images of of magnitudes magnitudes 4, 55 and and when using scheme. the normal normal integration integration mode; (e–h) star images of magnitudes 2, 4, 5 and 2, 6 when using the WCA (e–h) star images of magnitudes 2, 4, 5 and 6 when using the WCA scheme. (e–h) star images of magnitudes 2, 4, 5 and 6 when using the WCA scheme.
The centroiding accuracy performance when utilizing the WCA scheme is validated. The The centroiding accuracy performance when utilizing the WCA scheme is validated. The The centroiding performance when the WCA is validated. 30exposure ms , AIT exposure parametersaccuracy of the navigation sensor areutilizing set as follows: totalscheme integration time T The exposure parameters of the navigation sensor are set as follows: total integration time T 30 ms , AIT integration parameters of the navigation sensor are set as follows: total time T = 30star ms, TS 29.67 ms , and well capacity QS ranges from 2000 e to 8000 e at 500 e intervals. Then, − − − T 29.67 ms Q e e e , and well capacity ranges from 2000 to 8000 at 500 intervals. Then, star S S Q S ranges from 2000e AIT TS = 29.67 ms, and well capacity to 8000e at 500e intervals. Then, centroiding is performed and experimental data are recorded. Although the true position of the star centroiding is performed and experimental data are recorded. Although the truethe position of the star star centroiding is performed and experimental data are recorded. Although true position of is unknown, the average centroid position xcen , ycen of a bright unsaturated star image can be the star is unknown, the average centroid position x , y of a bright unsaturated star image ) unsaturated star image can be is unknown, the average centroid position xcen , ycen( cenof acenbright considered the estimated value of value the true position. Then, the standard the centroiding can be considered the estimated of the true position. Then, thedeviations standard of deviations of the considered the estimated value of the true position. Then, the standard deviations of the centroiding Q are calculated at each certain . The error of eacherror magnitude respectwith to position centroiding of each with magnitude respect to xxposition , y are calculated at each certain ) cen , ycen ( x cen calculated cen at each certain QSS . The error of each magnitude with respect to position cen , ycen are QS . The experimental results areinshown Figure 16. experimental results are shown Figurein16. experimental results are shown in Figure 16.
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Figure Star centroidingerror errorversus versuswell wellcapacity: capacity: (a) (a) star star magnitude = 2; (b) star magnitude = 4; Figure 16.16. Star centroiding magnitude QS[e-] QS[e- ] = 2; (b) star magnitude = 4; (c) star magnitude = 5; and (d) star magnitude = 6. (c) star magnitude = 5; and (d) star magnitude = 6. Figure 16. Star centroiding error versus well capacity: (a) star magnitude = 2; (b) star magnitude = 4;
In(c) Figure 16, the experimental are =denoted with red solid lines, whereas the simulation star magnitude = 5; and (d) starresults magnitude 6. In Figure 16, the with experimental arecentroiding denoted with red solid lines, whereas the simulation results are denoted blue solid results lines. The accuracy performance from the experiment results are denoted with blue solid lines. The centroiding accuracy performance from the experiment is consistent with simulation results. the laboratorial single-star imaging accuracy is In Figure 16, the the experimental results Thus, are denoted with red solid lines, whereas the and simulation consistent the simulation results. Thus, the laboratorial single-star imaging andthe accuracy analysis resultswith are denoted with blue solidconclusions lines. The centroiding accuracy performance from experiment analysis experiment validates the of this study. experiment validates of thisThus, study.the laboratorial single-star imaging and accuracy is consistent with the the conclusions simulation results. analysis experiment validates the conclusions this study. 6.2. Night Sky Observation and Accuracy AnalysisofExperiment 6.2. Night Sky Observation and Accuracy Analysis Experiment nightSky skyObservation observation and accuracy analysis experiment is performed to validate the correctness 6.2.ANight and Accuracy Analysis Experiment night sky observation and accuracy analysis experiment is performed validate theiscorrectness of A the celestial body edge model and optimal exposure parameters when the to WCA scheme applied. A night sky observation and accuracy analysis experiment performed to the correctness of the celestial body edge model and optimal exposure parameters when thevalidate WCA scheme is applied. Moreover, whether the optical navigation measurements of theisstars and target celestial body from the of the whether celestial body edge model and optimal exposure parameters when the WCAcelestial scheme is applied. Moreover, the optical navigation measurements of the stars and target body from image by a single exposure with the optimal exposure parameters satisfy the requirements ofthe Moreover, whether the optical navigation measurements of the stars and target celestial body from the image by a single exposure with the optimal exposure requirements of navigation navigation accuracy is validated. In Figure 17, the parameters navigation satisfy sensor the is installed on a tripod. The image by a single exposure with the optimal exposure parameters satisfy the requirements of hardware configurations of the navigation sensor are the sameisasinstalled those previously mentioned. accuracy is validated. In Figure 17, the navigation sensor on a tripod. The hardware navigation accuracy is validated. In Figure 17, the navigation sensor is installed on a tripod. The configurations of the navigation are sensor the same as those previously mentioned. hardware configurations of the sensor navigation are the same as those previously mentioned.
Moon Moon
Navigation Sensor Navigation Sensor
Tripod Tripod
Figure 17. Setup of the night sky experiment. Figure 17. Setup of the night sky experiment.
Figure 17. Setup of the night sky experiment.
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6.2.1. Observations of the Moon and Accuracy Analysis Experiment Sensors 2017, 17, 915
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The images of the Moon are obtained utilizing the normal integration and WCA schemes, as shown 6.2.1.The Observations the Moon and Accuracy Experiment of of the Moon are obtained normal integration WCA schemes, in Figure 18. images Figure 18b shows that the image utilizing is Analysis largelythe overexposed, and theand apparent diameter as of the shown in Figure 18. Figure 18b shows that the image is largely overexposed, and the apparent Moon is The extended significantly. 18c shows the Moon imaged with optimal exposure parameters images of the MoonFigure are obtained utilizing the normal integration and WCA schemes, as diameter the Moon is extended significantly. 18c shows in theFigure Moon imaged with optimal shownthe inof Figure 18. Figure 18b shows that the image isshown largely overexposed, the apparent utilizing WCA scheme. Compared with theFigure image 18b,and although the total exposure parameters utilizing the WCA scheme.Figure Compared with the image shown inwith Figure 18b, diametertime of the Moon is extended significantly. 18cexposed, shows the Moon imagedexhibited optimal integration is the same, the image of the Moon is well and the image suitable although the total integration time is the same, the image of the Moon is well exposed, and the image exposure parameters utilizing the WCA scheme. Compared with the image shown in Figure 18b, 18d performance of navigation measurement extraction when the WCA scheme is applied. Figure exhibited suitable performance of isnavigation measurement extraction when the WCA scheme is although the total integration time the same, the image of the Moon is well exposed, and the image − shows the image of the Moon when the well capacity is set to 6093e . The energy ring around the 6093e applied. 18dperformance shows the image of the Moon when the wellextraction capacity iswhen set tothe . The energy exhibitedFigure suitable of navigation measurement WCA scheme Moon isaround clearly visible. The observation results validate the validate correctness of the celestial body isedge the celestial ring the Moon is clearly visible. The observation results the correctness of applied. Figure 18d shows the image of the Moon when the well capacity is set to 6093e . The energy model when WCA scheme is adopted. body edgethe model when the WCA scheme is adopted. ring around the Moon is clearly visible. The observation results validate the correctness of the celestial body edge model when the WCA scheme is adopted.
Figure 18. Lunar images with different exposure parameters: (a) T 0.4 ms utilizing the normal
Figure 18. Lunar images with different exposure parameters: (a) T = 0.4 ms utilizing the normal 29.67 ms, utilizing exposure the normal integration (a) mode; integration mode; (b) T 30 ms different T 30utilizing ms， TS the Figure 18. Lunar T (c) 0.4 integration mode; (b)images T = 30with ms utilizing the normalparameters: integration mode; (c)ms T = 30 ms, TS =normal 29.67 ms, T 29.67 ms, Q 3750 e Q 6093 e utilizing the scheme; and (d) the T 30 ms， S mode; (c) T S 30 ms， T utilizing S 29.67 ms, utilizing the(d) normal integration mode; (b)the T WCA WCA 30 msscheme; S − QS = 3750e− utilizing and T = integration 30 ms, TS = 29.67 ms, QS = 6093e utilizing the WCA scheme. 3750e utilizing the WCA scheme; and (d) T 30 ms， TS 29.67 ms, QS 6093e utilizing the WCAQSscheme. WCA scheme.
The theoretical value of the apparent diameter of the Moon is estimated to be 71.02 pixels on the The theoretical value ofthe theSTK apparent diameter of the Moon is estimated to be 71.02location pixels on image plane by applying software to simulate theMoon distance from theto observation to the The theoretical value of the apparent diameter of the is estimated be 71.02 pixels on the image plane by applying the STK software to simulate thetodistance fromedge the observation to the the Moon. edge detection algorithm the Moon.location The least image planeThe by applying the STK softwareistoemployed simulate the extract distancethe from theof observation location to Moon. The edge detection algorithm is employed to extract the edge of the Moon. The least squares squares circle fitting is utilized to obtain the apparent radius and centroid of the Moon image. The the Moon. The edge detection algorithm is employed to extract the edge of the Moon. The least results are in to Figure 19.the circle fitting isshown utilized apparent and centroid of the Moon results squares circle fitting is obtain utilized to obtain the radius apparent radius and centroid of theimage. Moon The image. The are
shown in Figure 19. in Figure 19. results are shown
Figure 19. Edge detection and circle fitting results of the Moon image. Figure 19. Edge detection and circle fitting results of the Moon image.
Figure 19. Edge detection and circle fitting results of the Moon image.
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In Figure 19, the red scatter points are the extracted edge points, the yellow circle is the fitting circle,Inand “+” symbol is the centroid position. The minimum relative deviation of the apparent Figure 19, the red scatter points are the extracted edge points, the yellow circle is the radius fitting with tosymbol the theoretical value isposition. obtainedThe when the optimal exposure parameters are adopted. circle,respect and “+” is the centroid minimum relative deviation of the apparent radius Table 2 shows thetheoretical average extraction results when of thethe apparent under different with respect to the value is obtained optimal radius exposure parameters areexposure adopted. conditions. However, theextraction apparentresults radius errors are larger simulation results. This Table 2 shows the average of the apparent radius than underthe different exposure conditions. phenomenon be attributed to atmospheric turbulence and lens calibration error, whichmay are However, the may apparent radius errors are larger than the simulation results. This phenomenon beyond the scope of this study.turbulence Thus, these factors not considered in the are model. In summary, be attributed to atmospheric and lens are calibration error, which beyond the scopethe of Moon observations validate theare reliability of errorin analysis and parameter optimization. this study. Thus, these factors not considered the model. In summary, the Moon observations validate the reliability of error analysis and parameter optimization. Table 2. Apparent radius of the Moon under different exposure conditions. Table 2. Apparent radius of the Moon under different exposure conditions.
Exposure Conditions T 0.4 ms Normal integration, Exposure Conditions
T ms 30 ms Normal integration, Normal integration, T = 0.4 Normal integration, T = ms, 30 ms T 30 ms, TS =29.65 QS 3046e WCA, WCA, T = 30 ms, TS = 29.65 ms, QS = 3046e− 30 Tms,=TS 29.67 = 29.67 3750 WCA, − e WCA, T =T 30ms, ms, ms, QS Q =S3750e S − WCA, T =T 30ms, ms, ms, QS Q = 4453e S = 30 Tms, TS 29.69 = 29.69 e WCA, S 4453 WCA, T = 30 ms, TS = 29.74 ms, QS = 6093e− WCA, T 30 ms, TS = 29.74 ms, QS 6093e
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6.2.2. Observations of the Moon and Stars in the Same FOV 6.2.2. Observations of the Moon and Stars in the Same FOV Images of stars and the Moon in the same FOV are taken utilizing the WCA scheme and optimal Images of stars and MoonininFigure the same utilizing the WCAalgorithm scheme and optimal exposure parameters, asthe shown 20. FOV The are startaken pattern identification is applied, exposure parameters, as shown in Figure 20. The star pattern identification algorithm is applied, and and the identified stars are denoted with the yellow “+” symbol. A total of 12 stars are identified. the identified stars are denoted with the yellow “+” symbol. A total of 12 stars are identified. The The brightest star magnitude is 3.0, whereas the dimmest star magnitude is 6.2. The centroid positions brightest star magnitude is 3.0, whereas theimage dimmest star magnitude and magnitude of the identified stars in the are listed in Table 3. is 6.2. The centroid positions and magnitude of the identified stars in the image are listed in Table 3.
MV = 6.2
MV = 3.0
Figure 20. 20. Observations Observations of of the the Moon Moon and and stars stars in in the the same same FOV. Figure Table 3. Centroid positions and magnitude of the identified stars. Table
Number Number 1 1 22 33 4 55 6 6
Centroid/Pixels Centroid/Pixels (3439.886, 3084.843) (3439.886, 3084.843) (2985.837, 2934.429) (2985.837, 2934.429) (1707.333, 2326.095) (1707.333, 2326.095) (2575.164, 2232.000) (2575.164, 2232.000) (1012.179, 256.769) (1012.179, 256.769) (3025.250, 1638.643) (3025.250, 1638.643)
Magnitude Magnitude 3.8 3.8 3.0 3.0 4.4 4.4 5.2 5.2 5.5 5.5 5.4 5.4
Number Centroid/Pixels Number Centroid/Pixels 7 (57.533, 1493.233) 7 (57.533, 1493.233) 88 (3045.821, (3045.821,1069.692) 1069.692) 99 (3131.244, (3131.244,1017.171) 1017.171) (3624.311,684.475) 684.475) 1010 (3624.311, 11 (977.373, 675.311) 11 (977.373, 675.311) 12 (4056.727, 59.644) 12 (4056.727, 59.644)
Magnitude Magnitude 4.9 4.9 6.2 6.2 5.3 5.3 4.8 4.8 4.3 4.3 3.1 3.1
Under the the optimal optimal exposure exposure parameters, parameters, the the navigation navigation sensor sensor can can simultaneously simultaneously identify identify the the Under stars that that are are dimmer dimmer than than the the limiting limiting detectable detectable star star magnitude magnitude and and extract extract the the high-accuracy high-accuracy edge edge stars
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location of the celestial body (results listed in Table 2). In summary, by utilizing the WCA scheme, the navigation sensor can image the stars and target celestial body well-exposed simultaneously within a single exposure and can reliably extract high-accuracy optical navigation measurements that satisfy the navigation demand. The night sky observation and accuracy analysis experiment validates our study conclusions. 7. Conclusions In this paper, we first analyze the irradiance characteristics of a celestial body. This study aims at solving the problem that an optical navigation sensor is unable to image and expose the target celestial body and stars well-exposed simultaneously. Given that their irradiance difference is generally large, a solution that utilizes the WCA scheme is proposed. Then, celestial body edge model and star spot imaging model are established when the WCA scheme is adopted. The effect of exposure parameters on the accuracy of the star centroid estimation and edge extraction is analyzed based on the models. The AIT TS and well capacity QS are the main factors that influence the edge detection accuracy performance of the celestial body. The edge detection error initially reaches the minimum number and then increases with the increase in TS . An interval exists that indicates that the edge detection error is significantly small. The edge detection error initially reaches the minimum number and then increases with the increase in QS when we set TS to an appropriate value. The well capacity is the main factor that influences the centroiding accuracy performance of a single star. The star centroiding error of a bright star decreases with the increase in QS . However, the centroiding error of a dim star is mainly caused by random noise, and more dim stars are recorded than bright stars in the FOV. Therefore, the overall centroiding error variation caused by the exposure parameters can be neglected. The exposure parameters are optimized to ensure that the optical navigation measurements satisfy the requirement of navigation accuracy. The optimal QS and analytical solution of the optimal TS are obtained by conducting Monte Carlo simulation. The laboratorial and night sky experiments validate the correctness of the models, the proposed optimal exposure parameters, and other study conclusions. This study validates the feasibility of extracting attitude information and LOS vector from the sensor to the centroid of the target celestial body simultaneously by utilizing a miniaturized single FOV optical navigation sensor. Acknowledgments: This work was supported by National Natural Science Fund of China (NSFC) (No. 61222304) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20121102110032). We gratefully acknowledge the supports. Author Contributions: Hao Wang is responsible for the overall work, performance of the simulations, analysis and experiments, and writing of this paper. Jie Jiang provided partial research ideas and modified the paper. Guangjun Zhang is the research group leader who provided general guidance during the research and approved this paper. Conflicts of Interest: The authors declare no conflict of interest.
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El Gamal, A.; Eltoukhy, H. CMOS image sensors. IEEE Circuits Devices Mag. 2005, 21, 6–20. [CrossRef] Kühl, C.T. Combined Earth-/Star Sensor for Attitude and Orbit Determination of Geostationary Satellites. Ph.D. Dissertation, Universität Stuttgart, Stuttgart, Baden-Württemberg, Germany, March 2005. Jinno, T.; Okuda, M. Multiple exposure fusion for high dynamic range image acquisition. IEEE Trans. Image Process. 2012, 21, 358–365. [CrossRef] [PubMed] Zhang, W.; Cham, W.-K. Reference-guided exposure fusion in dynamic scenes. J. Vis. Commun. Image Represent. 2012, 23, 467–475. [CrossRef] Goshtasby, A.A. Fusion of multi-exposure images. Image Vis. Comput. 2005, 23, 611–618. [CrossRef] Mertens, T.; Kautz, J.; Van Reeth, F. Exposure fusion: A simple and practical alternative to high dynamic range photography. In Computer Graphics Forum; Blackwell Publishing Ltd.: Oxford, UK, 2009; Volume 28, pp. 161–171.
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