Dual Tree Complex Wavelet Transform for Adaptive Interferogram Residual Reduction

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1 Middle-East Journal of Scientific Research 0 (9): , 04 ISSN IDOSI Publications, 04 DOI: 0.589/idosi.mejsr Dual Tree Complex Wavelet Transform for Adaptive Interferogram Residual Reduction V. Khanaa, K.P. Thooyamani and R. Udayakumar Information Technology Bharath University Chennai , India School of Computing, Bharath University, Chennai, India Abstract: A novel scheme for interferogram filtering is presented in this paper. The main concern of the proposed scheme is to lower the residues count mean while preserving the location and jump height of the lines of phase discontinuity. The real interferogram is converted to complex form by using wrapped phase values as an argument of complex exponentially, placing the data on the unit circle on the complex plane and thus removing the jump discontinuity. Dual Tree Complex Wavelet Transform (DT-CWT) is then used to perform -D multiresolution discrete wavelet transform for complex interferogram. For each sub-band, a simple bivariate shrinkage rule is applied to the coefficients and its parents, taking into account the statistical dependencies among wavelet coefficients. After thresholding, the coefficients are retransformed using synthesis filter bank. The results are compared to both of Daubechies real orthogonal max flat filter and pivot median filter. Results show that the proposed scheme surplus other techniques in terms of reducing residuals count and unwrapping accuracy. Key words: Dual Tree Complex Wavelet Transform Interferogram Filter INSAR INTRODUCTION noise in () due to, temporal de-correlation introduces local inconsistencies of the data referred to as model Synthetic aperture radar (SAR) enables imaging of (DEM) which represents the topography of the terrain. the ground by processing microwave backscattering data The SAR interferogram presents two main features collected along the flight path of an airborne or space preventing its direct use. The first one is that the phase is borne platform. This results in high-resolution images of wrapped within the interval (-, ), due to the periodic the local complex reflectivity of the ground. For the SAR nature of the phase signal. In order to obtain the absolute images obtained from slightly different flight paths, the phase and therefore, to obtain the height information, it is complex-valued pixels of one image are multiplied with the necessary to unwrap the phase signal []. From the other co-registered complex conjugate pixels of the other, the side, the interferometric phase signal is corrupted by phase of the resulting product image constitutes a SAR noise. This noise, on a first stage, will affect the interferogram. The significance of this phase image is that unwrapping process as it induces phase residues, which it contains information on terrain height. With proper make the unwrapping process difficult. On a second processing, it yields a so-called digital elevation. stage, the own noise degrades the derived height (n, n ) = Arg {e} () information. In order to overcome these problems it is necessary to introduce a phase filtering process before phase unwrapping. This filtering process can be also where is the noisy wrapped interferogram, are thought as an estimation process. samples of a smooth phase surface and Arg(. ) is The key computational problem in obtaining the DEM determination of an estimate from. A major challenge lies in the fact that the measured phase differences are of -D phase unwrapping comes from the fact that the undetermined within multiples of. They are given as a Corresponding Author: R. Udayakumar, Information Technology Bharath University Chennai , India. 059

2 Middle-East J. Sci. Res., 0 (9): , 04 wrapped phase field of principal values with range from - = x + v (4) to. It is necessary to perform a -D phase unwrapping operation, which removes phase jumps between where is the measured interferometric phase, xis the neighboring pixels larger than by adding or subtracting real interferometric phase and v represents the noise with multiples of so that the resulting distribution can be the distribution listed in eq. (3). The measured phase considered as samples of an underlying smooth function. can be encoded in the complex plane as a point in the unit This is a noise-sensitive problem and subject of recent circle: research []. The -D phase unwrapping problem can be stated as follows: j e = cos( ) + j sin( ) i[ (n, n )+ noise] residues. Any phase unwrapping algorithm is sensitive to the presence of residues that make contour integrations path-dependent. Many interferogram filtering are introduced recently. Lee et al. demonstrated that the interferometric phase noise can be modeled as an additive noise [3], other algorithm is based on a new interferometric phase noise model in the complex plane []. Interferometric Residual Model: The interferometric phase is due to the interaction between two SAR images. The behaviour of statistical the interferometric phase depends on this interaction. The interferometric coherence is the amplitude of the correlation coefficient between the two complex SAR images = EI { * I} EI EI { } { } e j where I and I represent the two SAR images, is the coherence and is the corresponding interferometric phase. The interferometric phase has been completely characteried in the real domain. For Gaussian scattering model and distributed scatterers, the interferometric phase follows the following distribution []: P (n +)( ) ( ) n ( ) = + F (n, ; /; (n)( ) n + ½ where the location of the peak of the distribution, F represents the Gauss hyper-geometric functions and n is the number of looks. The pdf (3) is symmetric (mod ) about its mode, which occurs at. Based on the distribution of the interferometric phase eq. (), Lee et al. demonstrated that the interferometric phase noise can be modeled as an additive noise [4]: Nc F = (/,/ );; 4 ( ) 0.68 vc = vs = Using the distribution (3) together with the properties of the trigonometric functions, the real and imaginary parts of (5) can be modeled as [5]: cos( ) = N cos( ) + v (6) c x c sin( ) = N sin( ) + v (7) c x s Based on (), it can be demonstrated that the value of N, for n =, is: c As equations (6), (7) and (8) show, N c only depends on. The terms v c and v s can be considered as noise terms, as their means are ero. These noise terms depend on the interferometric phase x. This dependence does not affect the mean, which is ero, but only the standard deviation [5]. The dependence with the interferometric phase x can be neglected and then, the variance of the noise terms can be approximated by the function: Eqs. (6) and (7) can be seen respectively as a noise model for the real and imaginary parts of the interferometric phase in the complex plane. In each case, the signals to recover are cos( x) and sin( x). These signals are multiplied by N c. As shown before, this parameter behaves in the same way as the coherence does, so instead assuming it as a noise parameter, it can be considered as a useful parameter to recover. It was demonstrated that N c can be employed to estimate the coherence [6]. The main feature of this way to estimate the coherence is that the coherence information can be estimated with high spatial resolution. 060

3 3 3 P w(w) = exp( w + w ) Middle-East J. Sci. Res., 0 (9): , 04 Bayesian Residual Reduction with Bivariate Shrinkage The Proposed Scheme: Recogniing the residuals Model: In the wavelet domain, if we use an orthogonal sensitivity of -D phase unwrapping algorithms, it would wavelet transform, the residual reduction problem can be be useful to reduce the residuals from the measured formulated as: interferograms before processing them further. The main concern in the proposed scheme is not to affect the y = w + n (0) fringes (line of discontinuity) with respect to their location, sharpness and jumps of height. Other where y is the wavelet coefficient corresponding to requirements are low computational complexity, noisy interferogram, w is the original coefficient and robustness and ease of use. We propose a scheme for n represents residuals, which is independent interferogram filtering using Dual Tree Complex Wavelet Gaussian function [6]. This is a classical problem in Transform (DT CWT). It is a form of discrete wavelet estimation theory. The aim is to estimate w from transform which generates complex coefficients by the noisy observation y. The maximum a posteriori using a dual tree of wavelet filters to obtain their real (MAP) considered in this work to take into account and imaginary parts. This introduces limited redundancy the statistical dependency between adjacent (m: for m-dimensional signals) and allows the transform wavelet coefficients [7, 8]. Let w represents the to provide approximate shift invariance and directionally parent of w (w is the wavelet coefficient at the selective filters (properties lacking in the traditional same position as w, but at the next coarser scale.) wavelet transform) while preserving the usual properties Then of perfect reconstruction and computational efficiency with good well-balanced frequency responses. y = w + n () The summary of the proposed scheme is as follows: y = w + n The measured phase is encoded in the complex where y and y are noisy observations of w and w ; and plane as a point in unit circle according to, n and n are noise samples. We can write j e = cos( ) + j sin( ) y = w + n () Thus, the complex interferogram are situated on the where w = (w, w ), y = (y, y ) and n = (n, n ). unit circle. Consequently, there are no fringes in the The standard MAP estimator for w given the complex interferogram [6]. corrupted observation y is Apply DT-CWT to complex interferogram. w(y) = arg max [pn(y-w).pw(w)] (3) Estimate the noise variance wavelet coefficients of interferogram, using mean absolute deviation defined a non-gaussian bivariate pdf for the coefficient and its as: parent is defined as[7] N n = i= Xi X N For each wavelet coefficients (k = number of The marginal variance is also dependent on the wavelet coefficients): coefficient index. Using (3) with (), the MAP estimator of w is derived to be [8, 9]. Marginal variance ä for the K coefficient is This can be interpreted as bivariate shrinkage calculated using neighbour coefficients in the region N function. This estimator requires a prior knowledge of (k). Where N(k) is defined as all coefficients within a the noise variance th n and marginal variance for each square-shaped window that is centered at the K wavelet. coefficient [8]. 06

4 Middle-East J. Sci. Res., 0 (9): , 04 Fig. : Demonstrate the Input Phase Images a) Original Unwrapped Phase. b) Reference Wrapped Phase. c) Residues of Original Phase = y yi M yien ( K ) where M is the sie of the neighbourhood N (k). New wavelet coefficient is calculated using equation 4. Inverse DT-CWT is calculated. Filtered interferogram is obtained by Appling natural logarithm operation. Experimental Results The Dual Tree Complex Wavelet Transform [6] is applied to simulated interferometric SAR (InSAR) example. The data set was generated on the basis of a real digital elevation model of mountainous terrain around Long and isolation Peak Colorado, united state, using a high-fidelity InSAR simulator that models the SAR point spread function, InSAR geometry, speckle noise, layover and shadow phenomena []. The sie of the image in pixels is 458 (aimuth) 5 (range). The estimated surfaces are compared with the reference digital elevation model []. Figure (a) shows the phase image of original interferogram. Two other filtering algorithm are used for evaluation, they are ) pivoting median filter [5] with 3X3 window sie, ) max-flat filter defined in [3]. In the available literature on interferogram filtering, the method of choice is often judged by visually comparing original and processed interferograms. In addition to visual judgment a quantitative evaluation of a filter to the interferogram is used during this work includes: Signal to noise ratio between filtered and original phase data. Reduction of the residues count. Preservation of the edges and fine details. Fig. : Residues Distribution after Interferogram Filtering a) DT-CWT b) Max-Flat c) pivoting Median Table : Performance Evaluation Measures Original Pivoting Interferogram DTCWT Max-flat Medium PSNR (db) Number of Residues Table : DEM without DEMafter DEM after DEM after filtering DT-CWT Max-flat Pivoting Median Average error Time (sec) Quantitative effect on phase unwrapping. In order to evaluate the effect of complex interferogram filtering on -D phase unwrapping quantitatively, we consider determination of the phase field estimation by means of Goldstein algorithm []. The unwrapped phase value and compared with the reference data shown in Figure (b). Figure (c) shows the residues of the phase data of Figure (a). Most of the 5578 residues are concentrated within the de-correlated regions. The performance measures used to quantify the performance of various filter types used in this study are listed in Table (). In general, it can be stated that the signal to noise ratio for proposed scheme is the highest one. It can be seen from Table () that the three filters have the ability to reduce the residues count with different percentages (47% with proposed scheme, 58% with Max-flat and pivoting median reduce the count by 84.8%). The distributions of residues after filtering process are shown in Figure (). Another evaluation parameter applied to the complex interferogram filtering is subsequent phase unwrapping including height inversion (DEM). This is done by taking the noisy phase data with and without preprocessing as input to Goldstein algorithm [, 9]. Table lists the 06

5 Middle-East J. Sci. Res., 0 (9): , 04 Fig. 3: Output Unwrapped Phase Data after Different Filtering Process a) DTCWT b) Max-Flat c) Pivoting Median high unwrapping accuracy. Since the DT-CWT, Max-Flat and Pivoting Median have the ability to reduce the residues count by 47%, 58% and by 84.8% respectively. While median filter achieves higher average error followed by Max-flat and the DT-CWT comes with the lowest average error the results of the unwrapped phase image show that the proposed scheme has the ability to reduce the residues count while preserving the phase discontinuity. It also indicates that the filtering process is necessary since it improves the accuracy of height inversion process and accelerates phase unwrapping process. REFERENCES. Ghiglia, D.C. and M.D. Pritt, 998. Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software, Wiley, New York, USA.. L ope Mart ýne, C., X. F`abregas C`anovas and M. Chandra, 00. High resolution coherence Fig. 4: Perspective Phase Unwrapped Images Using estimation, Open Symp. on Prop. Rem. Sens. URSI Goldstein Algorithm a) Reference b) DT-CWT c) Commission F,,Garmisch-Partenkirchen, Germany. Max-Flat d) Median 3. Lee, J.S., K.P. Papathanassiou, T.L. Ainsworth, M.R Grunes and A. Reigber, 998. A new technique average errors between wrapped phase with and without for noise filtering of SAR interferometric phase preprocessing and the original height values. It is verified images. IEEE Transactions on Geoscience and that the proposed scheme introduces the best average Remote Sensing, 36(5): error, affecting directly the accuracy of terrain height 4. Selesnick, W., 004. The double-density dual- [0-4]. tree DWT. IEEE Trans. on Signal Processing, On the other hand, it needs more time to unwrap the 5(5): phase data. This is due to fact that the Goldstein 5. Kingsbury, N.G., 00. Complex wavelets for shift algorithm needs to unwrap the data around the existence invariant analysis and filtering of signals, Applied residuals. and Computational Harmonic Analysis, pp: Qualitative measures of the output phase data, as 6. Kingsbury, N.G., 999. Image processing with shown in Figure (3), indicate the superiority of the complex wavelets, Phil. Trans. Royal Society DT-CWT. It demonstrates good adaptively of proposed London A. scheme as it reduces the residues count while preserving 7. Sendur, L. and I.W. Selesnick, 00. Bivariate edges and fine details. The pivoting median filter gives shrinkage functions for wavelet-based denoising good results of residues reduction but introduces exploiting interscale dependency, IEEE Transactions smoothness at level of phase details. Max-flat filter is on Signal Processing, 50(): better than pivoting median filter. The output wrapped 8. Sendur, L. and I.W. Selesnick, 00. Bivariate phase data and the derived 3d images are shown in shrinkage with local variance estimation, IEEE Signal Figures (3), 4). Processing Letters, 9(): Mallat, S., 998. A Wavelet Tour of Signal CONCLUSION Processing (Second Ed.), Academic Press, San Diego, USA. To obtain a more accurate unwrapped phase, a new 0. Prajapati Hetal Ritesh, Brahmkshatriya Pathik scheme has been proposed based on Dual Tree Complex Subhashchandra, Vaidya Hitesh Bharatbhai and V. Wavelet Transform using a bivariate shrinkage rule for Thakkar Dinesh, 008. Avian Influena (Bird Flu) in reduction residual counts in interferogram. It has been Humans: Recent Scenario, Global Journal of proved that lowering residues count does not guarantee Pharmacology, ():

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