Modeling Visual Perception for Image Processing. - CiteSeerX

Modeling Visual Perception for Image Processing. Jeanny Hérault, Barthélémy Durette GIPSA-Lab, INPG, 46 Ave. Félix Viallet, 38000 Grenoble, France {Jeanny.Herault, Barthelemy.Durette}@lis.inpg.fr.

Abstract. This paper presents a model of the retina with its properties with respect to sampling, spatiotemporal filtering, color-coding and non-linearity, and their consequences on the processing of visual information. It's formalism points out the architectural and algorithmic principles of neuromorphic circuits which are known to improve compactness, consumption, robustness and efficiency, leading to direct applications in engineering science. It's biological aspect, strongly based neural and cellular descriptions makes it suitable as an investigation tool for neurobiologists, allowing the simulation of experiences difficult to set up and answering fundamental theoretical questions. Keywords: Model of the retina, spatiotemporal filtering, color processing, irregular sampling, non-linear processing, adaptation.

1 The retina, Architecture and Functionalities In this section, we review some basic data about the biology of the primate's retina. The various topics that will be approached here are selected with respect to their interest from the engineer's point of view: signal processing, efficiency, and robustness. We will insist on a principle that seems general in living systems: the seek of economy in order to spare matter and processing time for a given task. 1.1 The Photoreceptor Layer There are two categories of photoreceptor: cones and rods (fig. 1). The first ones are dedicated to photopic vision and colors. The second ones, which we will not study here, are active in scotopic vision. Cones are of 3 types according to their spectral sensitivity curve [23, 34, 35]: L (red-orange), M (green-yellow) and S (blue). Notice that there is a significant overlap between the spectra of L and M cones, whereas the S cones spectrum is shifted towards shorter wavelengths. Cones are disposed more or less regularly on the surface of the retina [28, 33], according to a triangular mesh (about 15% increase of spatial frequency coverage relatively to a square mesh), according to a spatial multiplexing of colors [31]. This sampling scheme, with a limited number of photoreceptor types, results from a trade-

off between spatial and chromatic resolutions [4]. The number of 3 for the cone color types is consistent with the results of Principal Component Analysis carried on the color spectra of natural scenes [30]. Moreover, photoreceptors are electrically coupled by so-called gap junctions [5]. This implies a reduction of the structural noise.

Fig. 1. Representation of the principal retinal cells and of their interconnections in the outer(OPL) and inner- (IPL) plexiform layers.

1.2 Outer Plexiform Layer In the Outer Plexiform Layer (OPL), we find a special type of connections: the synaptic triad. Cones deliver their output signal, both to Bipolar and Horizontal cells. Here again, gap junctions occur between Horizontal cells, resulting in a low-pass spatiotemporal filter, with non-separable time and space variables, for the cone signal [6]. This kind of connection provides an Infinite Impulse Response (IIR) spatiotemporal filter much more efficient and economic than direct connections between cones and Horizontal cells would do. Horizontal cells are known to provide an inhibition, either directly on bipolar cells [42, 20], or recursively on cones [27], or both. In this latter case, the feedback may involve the Calcium current of the photoreceptor, leading to a gain modulation [41] capable of some implication in the color-constancy phenomenon [26]. Midget Bipolar (MB) cells (mostly present in fovea) are connected to only one cone, but in periphery, Diffuse Bipolar cells (DB) are connected to an increasing number of cones as eccentricity increases. They are of ON or OFF type according to their response to the onset or to the offset of light. 1.3 Inner Plexiform Layer and retino-geniculate pathway A second level of processing occurs in the Inner Plexiform layer (IPL), where bipolar cells deliver their signal to Ganglion cells whose axons form the optic nerve, under the mediation of Amacrine cells. There are two classes of ganglion cells according to their target in the Lateral Geniculate Nucleus (the relay center to cortical area V1): 1. In the Parvocellular pathway, we find the Midget GC, directly linked to the seemingly named bipolar cells and, like them, can be divided into ON and OFF cells. They are said to be of X type, they are spatially High-Pass and temporally

Low-Pass and they carry color as spectral oppositions (Red-Green) or (BlueYellow). However, a single Parvo cell transmits as well Luminance information as Chromatic information. Remark that, in the retina, only ganglion cells and some amacrine cells carry impulses. For all other cells the signal is of analog type. 2. In the Magnocellular pathway, small and large Parasol GC collect the signals of several Diffuse Bipolar cells, under the mediation of amacrine cells that mainly introduce a temporal high-pass behavior. This results in a selection of the low spatial frequencies of the OPL signal, with a transient response. Small and large Parasol GC are partly of X type (linear) but some are of Y type: the response of Y cells is non-linear (ON/OFF), they respond as well when light is switched on as when it is switched off. The magnocellular GC are known to be spatially low-pass and temporally high-pass for luminance, they do not carry chrominance.

2 Model of Signal Processing in the retina 2.1 Spatiotemporal Transfer Function of Cells Linked by Gap Junctions The membrane potential c(k, t ) of a cone at location k responds to the flux of incident photons like an electric circuit with resistor rC and capacitor CC, excited by a voltage generator of internal resistor r . The gap-junctions between cones are modeled by simple resistors RC between location k and neighboring locations k ! 1 and k + 1 . Such a circuit is described by equation (1): C

d s(k,t )!c(k,t ) c(k !1,t )!c(k,t ) c(k +1,t )!c(k,t ) . c(k, t ) = + + c dt rc Rc Rc

(1)

The solution of this system equation is obtained by successively taking the Fourier transforms of equation (1) with respect to the discrete variable k and the continuous variable t, leading to the transfer function of the cone circuit: 1 C( f , f ) = S( f , f ) , s t s t 1+2! c [1"cos(2# a fs )]+ j 2# $ c ft

(2)

where fs and ft are the spatial and temporal frequencies, ! c = rc Rc is the space constant linked to the “gap-junction” and ! c = rcCc is the time constant of the cellular membrane. The transfer function of the photoreceptor's layer is a low-pass spatiotemporal filter with non-separable variables. The horizontal cells respond to the same scheme of interconnections and present the same type of response, with different coefficients ! h = rh Rh and ! h = rh Ch .

2.2 Spatiotemporal Transfer Function of the OPL As horizontal cells (of low-pass type) inhibit the bipolar cells, they are responsible for a lateral inhibition effect, then the bipolar cell (or OPL) signal is spatiotemporally high-pass filtered, as shown in figure 2.

Fig. 2. (a) Spatiotemporal Transfer Function of the OPL for 1D spatial signals, (b): Spatial Transfer Function for 2D space.

Several remarks can be done at this stage: 1. The principle of economy applies here by the filtering induced by the gap junctions: as already said, it is an IIR filter, much more efficient than direct synapses would do. 2. The low-pass filtering of cones allows significant reduction of the structural noise due to the natural dispersion of the cone gains and thresholds. This circuit is very robust, it has been shown [39] that a ±20% random mismatch of resistor values results in only a ±2% signal dispersion at the bipolar outputs. 3. This low-pass filtering is wide-band and concerns only the high frequencies, that is, the ones where information is less important. However, the high-pass filtering at the OPL output concerns precisely the region of useful frequencies for natural scenes. As the frequency spectrum of natural scenes is in 1/f, the retina performs a spectral whitening, particularly useful for further cortical processing.

Fig. 3. Time-course of the spatial Transfer Function. (a) OPL. (b): parasol ganglion cells.

4. We have said that the filtering is of non-separable variable type: this means that the spatial transfer function depends on the temporal content of the input signal. This results in a special “coarse-to-fine” analysis. Let us suppose that the stimulation is an image presented on the retina during a period of time T and let us study the temporal evolution of the spatial transfer function (figure 3a): at the beginning, the spatial transfer function is low-pass, meaning that the image is first globally analyzed. Then, this spatial transfer function evolves towards a high-pass mode, allowing more details to be processed.

2.3 Neuromorphic circuits Conventional vision systems are based on a video camera associated to several digital processors. Because of their size, complexity and cost, they are limited to specific industrial or military applications, but they do not allow large-scale consumer or onboard applications where small size and low energy consumption are mandatory. Vision chips and neuromorphic circuits [9] are good candidates for such applications. Firstly proposed by [32], they are based on retina models and they integrate a parallel processing at the very level of the sensor [11]. They are part of “smart sensors”, that is sensors where the function of processing is linked to the function of transduction. This concept is very interesting for it suppresses the analog to digital conversion and the temporal sampling. Many extensions have been derived from this technology [9, 27], as in particular motion estimation circuits with dense estimation of the optic flow [39]. These techniques are particularly suitable for autonomous mobile robots.

3 Color Coding The three types of cones are identified by their spectral sensitivity curves: L (orangered), M (yellow-green) and S (blue). It should be remarked that there is a significant overlap in the spectral sensitivities of L and M cones whereas the S cones sensitivity is more shifted towards short wave lengths. ! L M

êl (k) êm(k)

S ês (k) k2 s(k1,k2) k1

Fig. 4. Spatiochromatic sampling of the cone mosaic. Three kinds of photoreceptors sample the wavelength spectrum and are spatially multiplexed: there is only one cell per retinal location.

Cones are disposed on a centered hexagonal grid, the mesh of which is more or less regular, on the retinal surface [28, 33]. This is true in the fovea but some perturbation of this scheme appears with increasing eccentricity because of the presence of rods. Because we have only one photoreceptor per spatial position, the three cone types are spatially multiplexed [31]. This kind of spatiochromatic sampling with a limited number of receptor types results from a trade-off between the needs of an accurate spatial sampling and an accurate spectral representation of objects. In fact, anatomical data show that the proportions of L, M and S cones in the retina vary from 1 to 10% for S cones and that the ratio of L to M cones varies from 1/3 to 3

in human primate (see fig. 4). We are now facing the problem that even when considering a regular sampling grid, relative densities of the three different receptor types are not uniform: the color sampling process is fundamentally stochastic and should thus be addressed with the appropriate mathematical tools. The first study of this model is due to [31], a model in the regular case has been proposed by [21] and its extension to the random sampling [3, 22] is summarized here after. 3.1 Model of the Retinal Sampling of Color Let us consider figure 4 and let us suppose that the three kinds of receptors L, M, and S are randomly distributed on the retinal surface according to a regular grid. There is only one receptor type at each location and their respective probabilities to be present at any location are: pl , pm and ps , with pl + pm + ps =1 . These probabilities are different between species and between individuals within a same species. We will choose experimental values as proposed in [17], that is pl = 10 / 16 , pm = 5 / 16 , and ps = 1 / 16 , knowing that according to authors, the ratio between L and M cones may vary from 1/3 to 3 and that ps may be as low as 1%.

Sampling function Model Pi 0

Pi 1

êi(x)

1-Pi 0

!l

Pi "!l(x)

x

1-Pi x

+

~ ei(x)

0 -Pi

x

Fig. 5. Random sampling function eˆ ( x) and its model seen as the sum of a regular sampling (x) h amplitude Pi and a zero-mean random function ˜e( x) .

In order to simplify the expressions, we will study a one-dimensional model. The two dimensional case would give the same properties. Let C i (x) , i=l, m, or s, three fictive color components continuous signals in space, sampled by corresponding random functions: eˆ i (x) : C i (x) => Cˆ i (x) = C i (x)! eˆ i (x) . Functions eˆ i (x) are Dirac impulses (distributions) at positions k!l and the probability to find an impulse at position k!l is pi , the probability to find no impulse at this position is 1- pi . The random sampling function eˆ i (x) may be represented as the sum of a regular sampling like a "Dirac Comb" ! "l (x) with the weight pi , and a zero-mean random function e˜ i (x) with positive values (1- pi ) and negative values (- pi ), as in fig. 5:

eˆ i (x) = pi ! "l (x)+ e˜ i (x) ,

(3)

Each color receptor signal comprises two components: the first one is regularly sampled with a spatial mesh !l , the second one modulates a zero-mean random function. The global discrete signal s(k) is the sum of the three channels C i (k) , sampled by the random functions eˆ i (k) . By replacing the functions eˆ i (k) by their decomposition into a regular sampling plus a zero-mean random function, we get:

$ s(k) = % &

# i"l,m,s pi !C i (k)'() + $%&# i"l,m,sC i (k) !e˜i (k)'() ,

(4)

The first term represents the weighted sum of the three color components. Introducing the photoreceptors' wavelength sensitivity C i (! ,k) , it represents the wellknown human luminance visibility function V! , which we will name A(!,k ) as an achromatic component. The second term represents the chromatic modulations. We name it chrominance Chr(!, k) . Here, an important fact appears: chromatic oppositions. Multiplying this term by each of the three sampling functions eˆ i (k) gives, in front of each photoreceptor:

L ! M ! S !

( pm + ps ) Cl (k) " pm Cm (k) " ps Cs (k) ( pl + ps ) Cm (k) " pl Cl (k) " ps Cs (k) , ( pm + pl ) Cs (k) " pm Cm (k) " pl Cl (k)

(5)

that is, roughly (red-green), (green-red) and (blue-yellow). These signals represent what is named "chromatic oppositions" in neuro- and in psychophysiology. They are the components of color perception. Particularly, a grey stimulus (achromatic), for which C l = C m = C s will give a value of 0 for this second term. These signals strikingly compare to the Blue/Yellow and Red/Green opposition components found by [24] in psychophysics. Remark that for the S photoreceptor the chromatic opposition signal is not directly weighted by ps , which provides it with the same relative importance as the others, even if the S cones are in small number! So, the luminance and chrominance signals are conveyed in a multiplexed form through the optic nerve to the brain. This multiplexing is invariant with respect to the variable cone type distribution and relative proportion. Color opposition coding is a consequence of this multiplexing. 3.2 Frequency Spectrum If the random sampling is a Markov process [22], the power frequency spectrum of the multiplexed signal exhibits interesting properties (see fig. 6). The luminance component is a broadband low-pass signal, whereas the chrominance component is a narrow-band signal modulating a random carrier of mean frequency 1/2. Considering the spatiotemporal transfer function of the retina (fig. 2), we remark that, at the output of the retina, the luminance signal will be high-pass filtered as already seen, whereas the chrominance signal will not be affected. This has been

experimentally demonstrated [16] by recording cells in the LGN: they presented a high-pass response to luminance and a low-pass response to chrominance. Now, what has the visual cortex to do in order to separate the signals? The answer is simple: luminance is extracted by a simple spatial low-pass filtering, and chrominance is extracted by a demodulation followed by a low-pass filtering [8]. This scheme of color decoding in the brain is very economic: decoding color in the retina would have led to convey three signals in the optic nerve, instead of one. In fact, there is special coding for the Blue channel: because the S cones are very few, their random carrier would interfere with the frequency band of luminance, producing a strong aliasing. It is probably the reason why Nature has provided a special circuit for the Blue/Yellow signal, as it has been recently discovered [12, 15].

Fig. 6. Power spectrum of the signal sampled according to a Markov process: chrominance appears as a modulation of a random carrier of mean frequency 1/2.

3.3 Application: Color Demozaicing In consumer applications, low-cost video cameras are built from only one CCD sensor where the pixels are "painted" according to a Bayer Color Filter Array (CFA), i.e. with one color per pixel just like in the retina. But if the designers of this principle had known the biology of the retina, they would have avoided many drawbacks.

Fig. 7. Extraction of the luminance signal in mono CCD cameras: (left) the Bayer CFA, (center) the spatial frequency spectrum with filter limits, (right) the luminance signal.

The algorithms to decode color (demozaicing) generally try to interpolate each color plan and then combine the result to produce three colors per pixel. This operation produces a dramatic aliasing effect (remember the color artifact that occurs

when somebody wears a striped black and white suit on TV). In fact, engineers did not exploit the natural redundancy of color signal (what Nature did). Taking the model of the retino-cortical coding of color, we derive a very simple and efficient algorithm [2]: 1. First apply a low-pass spatial frequency filter, regardless of the colors of pixels, to extract luminance with maximum accuracy, 2. Second, demodulate then low-pass filter the result to get color-opponent signals, 3. Finally, combine the results to provide three colors per pixel.

4 Non-Linear Processing Two of the most important aspects of signal processing on the retina rely on nonlinearity: the spatial sampling processes and the coding of signal amplitudes. 4.1 Random Sampling We have said that photoreceptors were approximately distributed according to a triangular grid. In fact, looking at the recent images of the retina's photoreceptors obtained by [33], we notice a second order variability: a random positioning around the vertexes of the regular grid. This variability, which has been some times seen as a drawback, is in fact particularly interesting. It has been shown that it drastically reduces the moiré effects (aliasing) seen in regular sampling [43]. Moreover, it should be reminded that the eye is continuously moving (micro saccades). In this case of movements, [29] has shown a considerable improvement of aliasing effects. 4.2 Space Variant Sampling The mesh of the photoreceptor sampling grid is not constant. The density of the photoreceptors, very high at the retinal fovea (fig. 8a) decreases progressively with eccentricity [14, 19].

a)

b)

Fig. 8. Retinal sampling. a) density of cones versus retinal eccentricity in mm (from [14]). b) number of cones per midget ganglion cell versus retinal eccentricity in degrees (from [19]).

Similarly, ganglion cells recruit more and more bipolar cells signals as the eccentricity increases (fig. 8b). As a first result, our retina analyses the fine details in

the fovea, keeping only the gross structures in the periphery. This is a strategy of focus/context analysis. Now let us consider the projection x of the world coordinates X and Y of an object M onto the retinal surface. The object is seen under an angle α (fig. 9). By extracting the linear density d(x) of retinal samples (midget ganglion cells) from data of figure 9, the distance between output samples is 1/d(x). Then, by a simple calculus of geometry, it is possible to derive that the number n of the sample at retinal position x is roughly proportional to the logarithm of the tangent of α.

Fig. 9. Geometry of the retinal projection of external world. An object M at coordinates X and Z is seen under an angle α, and projects in x on the retinal surface.

! X$ n ! n0 ln # & . "Y %

(6)

The coefficient n0 is almost constant over a wide range of eccentricity, from a few degrees to 70°. This fact has two important consequences: 1. When reading a newspaper, a word or a letter of size ΔX at distance X from the fixation point is analyzed by a constant number of samples !n = n0 !X X , whatever the reading distance. This property implies a remarkable processing efficiency: in peripheral vision, an object is described by the same number of pixels, whatever the viewing distance. 2. When approaching at a constant velocity V Z toward an object, the velocity of the retinal image in samples per second is dn = n0 dZ / dt = n0 VZ = n0 , that is,

dt

Z

Z

Tc

proportional to the inverse of the time to contact, an highly important ecological information obtained without the need of velocity and distance. It should be remarked that this information gives the relative distances of objects viewed at various eccentricities (an elegant solution for the depth from motion problem), except at the fovea. This fact is well known in computer vision and useful in robotic applications, see [38, 10], and more recently [13]. 4.3 Amplitude Compression

3.3.1 At the Photoreceptor Level The range of light intensities coded by a photoreceptor is incredibly wide: from 1 to 106, which do not comply with neurons, where a maximal range of 1.5-2 decades is

allowable. Fortunately, in current life we never meet sudden variations of a 106 range of intensities. This makes possible the use of an adaptive process to the mean ambient light. The adaptation law resides at the level of photoreceptors, it is typically a Michaelis-Menten law, for which the response x to a stimulus intensity X is:

x=

Xn . X n + X0n

(7)

The exponent n usually takes values between 1 and 2, we will consider n=1 in the sequel. The X0 parameter (the half-response stimulus) is capable of adaptation [7], mainly under the molecular dynamics of light transduction [40]. Its adaptation is twofold: 1. Temporal. The value of X0 evolves with the history of the cell activity: it can be modeled in first approximation by a temporal low-pass filter of the input signal X. 2. Spatial. Due to the Horizontal cells feedback and Calcium ions dynamics in photoreceptors [41], it also adapts to the neighborhood activity. The model is that of a spatial low-pass filtering. As a temporal consequence: when the ambient light is high, X0 increases and the overall gain dx / dX decreases; when it is low, X0 decreases and the overall gain increases. As a spatial consequence: at locations where shadows are present in the image the input signal is low, then X0 takes a low value and the local gain is increased (see fig. 10, center). An important aspect of this gain adaptation to local intensity is that the local histograms of the image signal are equalized: hence according to information theory, more information can be extracted from the signal [44]. 3.3.2 At the Ganglion Cell Level In the Inner Plexiform layer, ganglion cells also adapt to their input signal [37], in a similar manner to that of photoreceptors (temporal and spatial adaptation).

Fig. 10. Intensity and contrast gain adaptation in the retina. Left: original image; Center: with adaptation to local intensity by photoreceptors; Right: with adaptation to local contrast by ganglion cells.

As a consequence, this phenomenon compensates for a drawback of photoreceptor adaptation: in strongly illuminated regions, the overall gain is reduced and details of texture may be less visible (i.e. on the front wall in the center of fig. 10). The local contrast adaptation helps to recover details in the front wall region (fig.10, right), this phenomenon is known as "sharpness constancy" [18].

3.3.3 Consequence for Color Notice that the three photoreceptor types are individually submitted to the same L M S adaptation law: l = , m= and s = . Though not evident at L + L0 M + M0 S + S0 first glance, the luminance signal a = Pl l + Pm m + Ps s behaves approximately in the same manner: replacing L, M and S in the preceding formulas respectively by cL, cM c and cS leads to a ! . By simulating this property, it is possible to account for c + c0 the inter-individual variations of color thresholds in the MacAdam ellipses experiment [1]. An other important consequence is the implication of this phenomenon in the color constancy mechanism [36].

5 Summary The retinal characteristics and the resulting properties for signal and image processing are summarized in the following table Topic Spatiotemporal filtering: High-Pass Non separable variables

Property Compensates for the 1/f spectrum of images Coarse-to-fine processing

Color multiplexing

-

Retinal filtering applies only to luminance Color decoding occurs later in the cortex

Irregular sampling

Broadens the Nyquist limit (anti-aliasing)

Space-variant sampling

-

Focus / context representation Independence with respect to zoom Direct estimation of time to contact

Photoreceptors' compression

-

Local histogram equalization Color constancy

Ganglion cells compression

Local contrast equalization

For more illustrations (movies, color examples), the reader can look at the following links. Retina model: http://www.lis.inpg.fr/pages_perso/durette/resultats_demos.html.en Motion tracking: http://www.lis.inpg.fr/pages_perso/benoit/index.html.en Color: http://david.alleysson.free.fr/ Vision models: http://www.lis.inpg.fr/pages_perso/herault/enseignement.html

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