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One-Dimensional Hilbert Transform Processing for N-Dimensional Phase Unwrapping KATTOUSH ABBAS HASAN Department of Electrical and Electronics Engineering Al-Isra University P.O. Box 22, 33 AL-ISRA UNIVERSITY P.O. 11622, Amman JORDAN Email:
[email protected] [email protected] Abstract:- True phase including two dimensional (2-D) phase characteristic of different complex functions and transfer functions is calculated applying tangent inverse function which causes different artificial discontinuities or phase jumps to appear in the phase function. Phase unwrapping is the reconstruction of a 2-D phase function given its values modulo 2π . This problem arises as a key step in several imaging technologies, from which we emphasize interferometric synthetic aperture radar and synthetic aperture sonar. In this paper, one-dimensional (1-D) Hilbert Transform (HT) is used to detect phase jumps, and modulo 2π operation is used to eliminate these phase discontinuities. The algorithm uses the property of HT of any function, which gives at any broken point (jump) a spike value which can be easily detected, and the lost 2π phase constant at that point can be estimated as an integer multiple of 2π . Tests of the method presented illustrate the validity and effectiveness of proposed algorithm. Keywords: Phase Unwrapping; SAR interferometry; N-dimensional processing; 1-D Hilbert transform; modulo 2π operation; computational efficiency; filtration.
1. Introduction Over the past several years, there has been a marked increase in the application of coherent signal and image processing. Coherent processing requires an accurate estimate of the phase [1-2]. Examples where coherent processing is required include synthetic aperture radar (SAR) [3-4], synthetic aperture sonar [5], adaptive beamforming, acoustic imaging [6], projection and diffraction tomography [7], adaptive optics, inverse synthetic aperture radar, magnetic resonance imaging [8], field mapping, echo planar imaging [9], velocity measurements [10], ultrasound images [11], optical Doppler tomography systems [12] and others. Unfortunately, one is only able to measure a wrapped version of the phase called measured phase φ ( n ) not the true phase φ T (n) . The measured phases generated from the complex interferogram by use of an arctan2-function, are all mapped into the same interval (e.g. − π , π ), while any absolute phase offset (an integer multiple of 2π ) is lost. The calculation of the unambiguous phase from the measured phase is called phase unwrapping. This
wrapping leads to artificial phase jumps being introduced near boundaries. In many applications, the true phase relates to some physical quantity such as surface topography in interferometry [13-14], and measured nonlinear phase does provide useful information, it must be unwrapped before further use. The phase unwrapping problem is then to obtain an estimate for the true phase from the measured wrapped phase. Thus, wrapped value must be unwrapped through some method to estimate true phase which is the quantity relating to the physical property of interest. Many attempts were done to correct this problem [15-22] however the problem in general is not solved. Basically all the existing phase unwrapping techniques start from the fact that it is possible to estimate the neighboring pixel differences of the unwrapped phase when these differences are less . From these, the unwrapped phase can be than reconstructed up to an additive constant. The methods differ in the way they overcome the difficulty posed by the fact that this hypothesis may be somewhere false, which cause the estimated unwrapped phase differences to be inconsistent, that is their "integral" depends on the integration path.
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sequence of discrete values and the phase does not have mathematical expression to be calculated with. It cannot be sure that phase jumps in the phase characteristic calculated are natural or inserted due to tangent inverse function. Figure 1a shows an example of an unambiguous or true phase φ (n ) , the T corresponding measured phase φ ( n ) calculated by arctan2– function is shown in figure 1b. Jumps of 2π in the measured phase are clearly visible. 10 8
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In all methods, the unwrapping of measured phase starts from finding phase jump or phase discontinuity between phase values of two adjacent entries that exceeds some constant. The most natural way of detecting phase discontinuity is to take the first or second order derivatives and look for maxima or zero crossings in the output [19]. Then unwrapping is achieved by adding an integer k multiple of 2π to each successive element after zero crossing, and updating k at each phase jump. Finding zero crossing problems is similar to edge detection problem, and in this paper we shall apply edge detection methods to phase unwrapping. Traditional edge detection operators like Robert, Sobel and Laplacian [23] detect edges by taking first or second order derivatives and look for maxima. Piggio et al [24] reported that numerical differentiation is an ill-posed problem because its solution depends continuously on the data. Hilbert transform [25-27], provides a means of separating signals based on phase selectivity and uses phase shifts between the pertinent signals to achieve the desired separation. In addition, HT in edge detection has significant advantage in the case of noisy images [28-30]. The main reason for this advantage is that noise smoothing is performed on the same pixels using only 1-D filtering, thus preserving the edge information in the orthogonal direction. The noise suppression and continuity of the edges obtained by the HT are better than that obtained by the 2-D Canny's method and gives also significant computational advantage compared to the 2-D Canny's method, it takes only about 1=10th the time required 2-D Canny's method [30]. In this paper, we extend the same logic of edge detection to propose an algorithm for detecting a phase discontinuity by using 1-D Discrete HT as a phase-jump detection filter. We consider the 2-D phase array as one-dimensional (1-D), and 1-D HT is applied to each row or column of the array to detect crossings. Orthogonal filtering is used after phase reconstruction to eliminate vestiges of discontinuities existing in reconstructed phase.
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Figure 1: Example of true and measured phase, (a) true phase, (b) measured phase.
3. The algorithm The spectrum of N-dimensional (N-D) discrete Hilbert transform was found as [27,-31-32] Z& ( r1 , r 2 ,..., r N ) = H& ( r1 , r 2 ,..., r N ) X& ( r1 , r 2 ,..., r N ) (1) where 1 + N dsign(r ) i ∏ i =1 N 1 − j∏ dsign(ri ) i =1 H& (r1 , r2 ,..., rN ) = N 1 − ∏ dsign( ri ) i =1 N 1 + j ∏ dsign(ri ) i =1
, N = 4K + 1 , N = 2( 2 K + 1 ) (2) , N = 4K + 3 , N = 4(K + 1 )
X& ( r1 , r2 ,..., r N ) - N-D Fourier transform of discrete
signal x(m1 , m2 ,..., mN ) . N - Dimensionality of transform,
2. Problem formulation For many systems and applications phase functions are defined outside the definition interval of the arctan2-function, which means that when calculating the phase characteristic using the arctan2-function many artificial phase jumps being introduced near boundaries in the phase function as shown in figure 1. In most practical cases, the measured signal or transfer function including SAR interferograms are given experimentally as a
− 1 , xi ∈ 1, ( M i 2 ) − 1 dsign ( x i ) = 0 , xi = 0 , ( M i 2 ) 1 , x ∈ ( M 2 ) + 1, M − 1 i i i
(3)
K = ( N − 1) = 0 , 1, 2 , 3, ... M i - quantity of points in the dimension N i Now from equations (1)-(3) we can compute HT for any dimension as the imaginary part of equation (1). For 1-D case the true phase can be
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represented through measured phase and some distortion function d(n):
φT (n) = φ (n) + d ( n) where d(n) = 2πk (n)
(4) (5)
where k is a variable with value depending on number and direction of phase jumps. So we need only to calculate d (n ) and add it to measured phase φ ( n ) to find true phase φT (n ) . Calculating d (n ) starts from finding Hilbert transform of measured phase φˆ(n) then finding maxima or minima of φˆ(n) . After finding them, k (n ) is calculated by the following recurrent equation. k (1) = 0
k ( n − 1) + 1 if there is a maxima k ( n) = k ( n − 1) − 1 if there is a minima (6) k ( n − 1) if there is no maxima or minima
By using equations (1) and (2), the discrete Fourier transform of 1-D Hilbert signal for function φ (n) is given by: Z& ( r1 ) = H& ( r1 )φ& ( r1 ) (7) & H ( r ) = [1 + dsign ( r )) ] (8)
By taking inverse Fourier transform of Z& ( r1 ) we find the needed Hilbert signal in the form: Z& ( n ) = φ ( n ) + j φˆ ( n ) . From the upper discussion the proposed algorithm works as follows: - Find the fast Fourier transform (FFT) of the measured phase to be unwrapped. - Compute H& (r1 ) according to equation (9).
- Compute Z& ( r1 ) according to (8). - Compute inverse fast Fourier transform (IFFT) of Z& ( r1 ) to have the Hilbert signal. - The imaginary part of Z& (n ) is the Hilbert transform of wrapped phase needed for computing k (n) . - k (n) and d (n) are computed by equation (6) and (5) respectively. - Finally we calculate the unwrapped phase φ T (n) according to equation (4).
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4. Examples We shall consider a few examples for 1-D and 2-D cases. As a first example we implement 1-D HT algorithm for 1-D phase provided in figure 1. The results of running algorithm for this example are shown in figure 2. Figure 2a depicts the true phase while figure 2b depicts the measured phase calculated as arctangent of tangent function of true phase. Figure 2c shows the 1-D HT of measured phase and finally figure 2d plots the reconstructed phase applying upper algorithm. 10
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Now using equation (3), we can write equation (8) as r1 ∈ 1, (M 1 2 ) − 1 2, & H ( r1 ) = 1, r1 = 0 , M 1 2 (9) 0, r1 ∈ (M 1 2 ) + 1, M 1 − 1 where M 1 - number of points in the 1-D array H& (r1 )
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This algorithm can work for 1-D, 2-D,…, N-D phase unwrapping. For 1-D array the algorithm is used one time for the array to determine and correct jumps, for 2-D with equal number of points arrays case the algorithm is used N time, for each row or each column, to unwrap a plane. For 3-D with equal number of points arrays case the algorithm is used NxN time to determine and correct jumps in each row or each column in all plane.
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Figure 2: Implementation of 1-D HT to 1-D phase unwrapping. (a) Actual true phase. (b) Measured phase. (c) 1-D HT of measured phase. (d) Reconstructed by proposed 1-D HT technique phase. The first example of implementation proposed 1-D algorithm to 2-D case is a complicated mountainous shape phase characteristic of complex function with transfer function given by: 3 .414 s13 s 23 + 2 .613 s1 s 22 + 5 s14 + 1 Z& 2 − D ( s1 , s 2 ) = 4 s1 + 6 .027 s13 s 23 + 2.613 s1 s 22 + 1
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x 2 y where: 1 Figure 3 shows the steps of running 1-D algorithm to correct the measured phase. Figure 3a depicts the row-by-row 1-D HT of measured phase. Figure 3b plots the path of phase jumps in array (zero crossings). Figure 3c shows measured phase
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characteristic of equation (11) calculated using arctangent-function. It contains a hole like a volcano mouth. The 2-D phase characteristic reconstructed with proposed algorithm is provided in figure 3d. It has no artificial phase jumps. As second example of implementation proposed 1-D algorithm to 2-D case, we consider a phase characteristic of complex function with transfer function given by:
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3.414s13s23 + 2.613s1 s22 + 5s14 + 1 (12) s + 6.027s13s23 + 3.414s12s23 + 12.613s1 s22 + 2 Figure 4 shows the results of implementing 1-D algorithm to correct measured phase of function (12). Figure 4a depicts measured 2-D phase and figure 4b shows reconstructed 2-D phase that has no phase jumps. Z&2−D (s1, s2 ) =
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shown in figure 7. From these plots it can be seen that reconstruction error is very high at low input signal to noise ratios (below 40dB) and this error decreases with increasing input signal to noise ratio. In order to reduce reconstruction error and effects of noise on the algorithm performance, we applied low pass filtration before reconstruction, after reconstruction, and before-after reconstruction. Filtration is done using 1-D second order Butterworth low pass filter to each row when the algorithm used columns for phase reconstruction and to each column when the algorithm used rows for phase reconstruction. So filtration and phase reconstruction was orthogonal operations in this means. Plots for results with filtration are shown in the same figures 6 and 7. Standard diviation of recunstruction error
Figure 5a shows measured phase characteristic of equation (13) calculated using arctangentfunction of tangent of equation (13). The reconstructed phase characteristic of this function using proposed algorithm is provided in figure 5b.
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Figure 5: Phase characteristics of equation (13). (a) Measured phase. (b) Reconstructed phase.
In the deterministic case, proposed algorithm reconstructs phase without any disturbance. Actual and reconstructed phase characteristics are identical and there are no restrictions in its implementation. In practice it is very important to study behavior of proposed algorithm in presence of adaptive noise because it is the case in most practical problems. For this reason we generated random sequences with different power using MATLAB function RAND and added them to known deterministic functions like upper examples. Then we applied proposed algorithm to reconstruct true phase from measured phase corrupted by noise. We calculated standard deviation of reconstruction error and output signal to reconstruction error power ratio as a function of input signal to noise power ratio. Plots for standard deviation are shown in figure 6 and plots for output signal to reconstruction error power ratios are
Filtration after recunstruction Signal to recunstruction error power ratio (dB)
5. Discussion
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Figure 7: Signal to reconstruction error power ratio From plots with filtration it can be seen that filtration reduces reconstruction error levels in different manner especially for low input signal to noise ratio levels. Filtration before reconstruction gives better results than filtration after reconstruction and for low level noise signals (input signal to noise ratio greater than 50dB) filtration is not recommended. In proposed procedure instead of 1-D HT we can use 2-D HT. The procedure and processing now becomes 2-D with all other steps of algorithm not
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changed. We applied such 2-D algorithm for the phase function of equations (11) and (12); the results of running this algorithm were not encouraging. The reconstructed phase was strongly damaged and no type of filtration could retrieve phase to an acceptable form. This is because 2-D HT does not have spikes as 1-D HT, the branch cut path is not similar to that of 1-D and the reconstructed phase is not the actual. In addition to accuracy, computation time is very important for all applications requiring moderate to high throughput. To compare the computational efficiency of proposed 1-D processing algorithm with 2-D processing algorithm we repeated phase characteristic calculations of the upper examples several times varying the number of points in the complex array. Each time we calculated the relative CPU time units used by both algorithms. The relative CPU time units were calculated using MATLAB function CPUTIME. Figure 7 shows the dependence of the relative CPU time units used by both algorithms on the number of points to be calculated. From it we can see that the time needed for 2-D processing algorithm increases very fast with increasing the number of points; the relation very similar to cubic relation. While the relation for 1-D processing algorithm is similar to linear relation.
References: [1] [2]
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In this paper, we extend the same logic of edge detection to propose an algorithm for 2-D phase unwrapping. Detecting phase discontinuity is achieved by using 1-D Discrete HT as a phase-jump detection filter. In this method we treat the N-D array as multi 1-D row arrays or multi 1-D column arrays then 1-D HT is applied to each row or each column arrays to detect crossings. Proposed method is accurate, easy to implement, has no basic limitations and provides the significant computational advantage due to 1-D processing. In presence of noise 1-D filtration reduces reconstruction error levels in different manner especially for low input signal to noise ratio levels. Tests performed demonstrate the validity of this approach.
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6. Conclusions
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