IN many image processing applications involving wavelets

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1 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 7, JULY Phase-Shifting for Nonseparable 2-D Haar Wavelets Mais Alnasser and Hassan Foroosh, Senior Member, IEEE Abstract In this paper, we present a novel and efficient solution to phase-shifting 2-D nonseparable Haar wavelet coefficients. While other methods either modify existing wavelets or introduce new ones to handle the lack of shift-invariance, we derive the explicit relationships between the coefficients of the shifted signal and those of the unshifted one. We then establish their computational complexity, and compare and demonstrate the superior performance of the proposed approach against classical interpolation tools in terms of accumulation of errors under successive shifting. Index Terms Nonseparable, phase-shifting, 2-D Haar transform, wavelets. I. INTRODUCTION AND RELATED WORK IN many image processing applications involving wavelets [1] [12], data may need to be phase shifted. Due to lack of shift-invariance of wavelets, simple operations such as translation and scaling require back and forth transformations [13]. One can achieve shift-invariance by relaxing the orthogonality condition as in [14]. However, nonorthogonality makes the interpretation of the correlation properties among the transform coefficients more difficult. Another approach is to introduce redundancy by avoiding decimation [15] [19], which unfortunately undermines wavelets use in compression and coding. A more recent approach achieves shift-invariance through complex wavelets [20] [23]. Complex wavelets prove to be useful in solving the shift-invariance problem without compromising many other properties. However, they often suffer from lack of speed and poor inversion properties. A more successful attempt in this category is perhaps the dual-tree complex wavelet transform (DT-CWT) [24], [25]. Although DT-CWT provides a good tradeoff between fully decimated wavelets and the redundant complex wavelet transform, it does so by trading off the compression capabilities. Rather than modifying classical wavelets or designing new ones, our goal is to establish a relationship between the coefficients of a transformed signal and those of its shifted version. Of course, such a relation would be wavelet-dependent and may not be straightforward. The key idea is that as long as the relation is known, one can tackle shift-variance, since all coefficients of Manuscript received July 26, 2007; revised December 19, This work was supported in part by a grant from Electronic Arts Tiburon. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Pier Luigi Dragotti. The authors are with the Computational Imaging Lab (CIL), Department of Computer Science, University of Central Florida, Orlando, FL USA ( nasserm@cs.ucf.edu; foroosh@cs.ucf.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TIP Fig. 1. Haar transform of the 2-D signal x(m; n) is composed of the dc value A and the detail coefficients a, b and c, where l =0;...;N 0 1, i =0;...; 2 0 1, and j =0;...; The blur coefficients A at each level are used to help derive the equation for phase-shifting the signal x(m; n), but are not used in the final form of the equation. Only the first three levels of the tree are shown with one l to avoid cluttering the figure. a shifted signal can be mapped to those of the original 0-shift signal. On the other hand, shift-variance is tackled without compromising speed and compression properties. Furthermore, establishing explicit relations between the coefficients of a signal and those of its shifted version, would allow us to perform compressed domain processing without requiring a chain of forward and backward transforms. This is particularly of interest in data compression and progressive transmission. Our focus in this paper is on the nonseparable 2-D Haar wavelet transform due to its additional desirable properties, such as symmetry and conservation of image aspect ratio through different compression levels, which is important to applications that require progressive reconstruction. We derive explicit expressions for shifting data directly in the Haar domain using only the available coefficients of the original 0-shift signal. We also show how our solution can be expanded for noninteger phase shifts, and evaluate our approach against popular interpolation methods in terms of accumulation of errors through successive shifts. II. NOTATIONS AND SETUP Let be a 2-D signal of size, where is a positive integer. The Haar transform of can be expressed using a tree as in Fig. 1. The tree is constructed of levels with residing at the leaves, i.e., the th level. The th node at level in the tree is made to hold the th blur coefficient and the detail coefficients,, and, where /$ IEEE

2 1062 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 7, JULY 2008 Fig. 2. Blur coefficients at level l =3. In (a) (c), respectively, the rectangular windows show the coefficients included in evaluating a, b, and c under a horizontal odd shift of one. These are evaluated by summing and subtracting the highlighted quadrants in the windows as shown in the diagrams.,, and. We denote as and let with,, and. For brevity, we will sometimes refer to the horizontal, vertical and diagonal detail coefficients by,, and, respectively. Let The following formula shows the relation between the blur coefficient and its parent at level : is even, is even, is odd, is even is odd is even is odd, is odd. (2) We let be the difference between and, then (1) By substituting (3) into (2), can be computed recursively solely in terms of the detail coefficients as follows: is even, is even is even, is odd is odd, is even is odd, is odd. (4) A 2-D signal can be shifted horizontally or vertically. Both types of shifts affect the, and at all levels. At level, there are nonredundant coefficient sets each of size [26], where. A horizontal shift or a vertical shift can be one of the following possibilities: a shift that is divisible by ; an odd shift; an even shift that is not divisible by. We derive the formulae for evaluating the detail coefficients under a horizontal shift, for each of the three possibilities, followed by similar derivations for and detail coefficients. Formulae for vertical shifts can be derived in a similar manner. III. HORIZONTAL COEFFICIENTS FOR HORIZONTAL SHIFT A. Shifting by a Multiple of This is the simplest case. A shift in the discrete time domain that is equal to is a horizontal circular shift of the 0-shift detail coefficients at level by, that is (3) (5) (6) where (7)

3 ALNASSER AND FOROOSH: PHASE-SHIFTING FOR NONSEPARABLE 2-D HAAR WAVELETS 1063 where and% is the mod operation. Notice that for levels a horizontal shift of in the time domain is a horizontal circular shift of the coefficients at those levels by, respectively. In other words, a horizontal shift of in the time domain shifts the coefficients at level horizontally by, while shifting the coefficients at level horizontally by twice as much, and the coefficients at level by four times as much, and so on. B. Shifting by an Odd Amount By examining the tree in Fig. 1, we notice that [see (6) and (7), shown at the bottom of the previous page]. Equations (6) and (7) evaluate by summing the first half and subtracting the second half of each row of blur coefficients at the leaves level, which fall inside the window determined by the two corners and. Fig. 2(a) illustrates the evaluation of using the blur coefficients at level under a horizontal odd shift of one. Substituting (3), (4), and then (1) into (7), we get the coefficients of the shifted signal as follows [see (8), shown at the bottom of the page]. Note that at, is a noninteger value. When that is the case we set to 0. C. Shifting by an Even Amount That is Not Divisible by In this case, is divisible by, where and is the highest power of 2 by which is divisible. This allows us to let, where is an odd integer in the range. This means that the coefficients at levels are a subset of the first case. In other words, the 0-shift coefficients at levels are circularly shifted in the horizontal direction by, respectively. On the other hand, is verified to be an odd shift of the blur details at level. Therefore, at level, can be evaluated using the following modification of (7) [see (9), shown at the bottom of the page]. Following the same steps as above, we get (10), shown at the bottom of the page. Note that the second case is the same as the third case where. This gives rise to the following final formula for evaluating the detail coefficients under a horizontal shift [see (11), shown at the bottom of the next page]. IV. VERTICAL COEFFICIENTS FOR HORIZONTAL SHIFT For this section and the next one, again we let be a horizontal shift, where,, is the highest power of 2 by which is divisible, and is an odd positive integer. Then for all (12) where (8) where (9) where (10)

4 1064 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 7, JULY 2008 To compute the coefficient for after a horizontal shift, we write in terms of the coefficients at level [see (13), shown at the bottom of the page]. Equation (13) defines a window that encloses the blur coefficients at level that fall between the two corners and, which are the same bounds used to compute in (11). is found by summing each row of blur coefficients in the upper half of the window and subtracting the sum of each row in the lower half. Fig. 2(b) illustrates the computation of using the blur coefficients at level under a horizontal odd shift of one. By substituting (1), (3), and (4) in (13), and combining the result with (12) we get the coefficients of the shifted signal [see (14), shown at the bottom of the page]. where (11) where (13) where (14)

5 ALNASSER AND FOROOSH: PHASE-SHIFTING FOR NONSEPARABLE 2-D HAAR WAVELETS 1065 TABLE I COMPARISON WITH OTHER METHODS: ACCUMULATED RESIDUAL ERRORS IN ROOT MEAN SQUARE (RMS) AFTER SEVERAL SUCCESSIVE SHIFTS (SEE TEXT FOR MORE DETAILS) V. DIAGONAL COEFFICIENTS FOR HORIZONTAL SHIFT Similar to 5 and 12, we can apply the following equation when the horizontal shift is divisible by : (15) To compute the coefficient for after a horizontal shift, we write in terms of coefficients at level [see (16), shown at the bottom of the page]. Equation (16) defines a window which has the same bounds used to compute and in (11) and (14), respectively. By examining the example in Fig. 2(c), one can see that is computed by summing the first half and subtracting the second half of each row in the upper half of the window, while subtracting the first half and summing the second half of each row in the lower half of the window, hence, (16). By substituting (1), (3), and (4) in (16), and combining the result with (15), we get (17), shown under (16). where (16) where (17)

6 1066 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 7, JULY 2008 where (18) where (19)

7 ALNASSER AND FOROOSH: PHASE-SHIFTING FOR NONSEPARABLE 2-D HAAR WAVELETS 1067 The relations (11), (14), and (17) can now be used to compute the new detail coefficients of the Haar transform at all different levels after any horizontal shift using only the 0-shift coefficients. The equations for the vertical shift can be derived following the same steps that we used for horizontal shifting. They turn out to look the same as the horizontal case after interchanging the s with s, s with s and s with s. VI. SUBPIXEL SHIFTING Let the size of the signal be, and, where is the number of added levels. Equations (11), (14), and (17) can now be modified to allow for noninteger shifting by a precision of by substituting for each in the equations. On the other hand, we can verify that, where. This combined with (4) allows us to modify (18) and (19) (shown on the previous page), as well as (20) (shown at the bottom of the page), in such a way that avoids having to actually up-sample the signal for noninteger shifts, thus saving memory space in actual implementation, especially that the size increases exponentially. However, we have to split the equation into two cases. The first is when, which is when the coefficients at the added levels are being used to compute, and. The second is when is large enough for the coefficients at the original levels of the tree to be used. This leads to the phase shifting relation for noninteger values as in (18) (20). VII. EXPERIMENTAL RESULTS AND DISCUSSION For integer shifting, our solution for Haar-domain phase shifting, given by (11), (14), and (17), is exact and does not incur any errors. For the subpixel case, a shift is approximated by modeling the process as upsampling by a factor of TABLE II COMPLEXITY AND PROBABILITY AT EACH REDUCTION LEVEL k FOR EVALUATING THE 2-D DETAIL COEFFICIENTS a followed by an integer shift. However, our solution does not require to actually upsample the image i.e., the solution derived performs subpixel shifts directly from the original coefficients. For a factor of, we can obtain a precision of, i.e., subpixel shifts are approximated by the closest value in multiples of. In order to evaluate our method, we compared our results with bilinear, bicubic, and cubic spline interpolations. For this purpose, using each method, we performed successive noninteger shifts until the test image was shifted by some prescribed large integer value. We then used the residual error accumulated over these successive shifts as a measure of performance. We performed this experiment on numerous test images, some of which are shown in Table I. As shown intable I, the accumulated residual errors in root mean square (rms) are on average an order of magnitude smaller in our method than the other three methods, i.e., our method accumulates far less error over successive shifts. where (20)

8 1068 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 17, NO. 7, JULY 2008 On the other hand, the low complexity of the derived solutions allow for fast processing directly in the transform domain. By examining (11), one can find that the complexity of evaluating can be expressed by the difference of the bounds of the two inner sums in the equation multiplied by the difference of the bounds of the outer sum, that is. Substituting the values for and, the complexity is shown to be when. When the complexity becomes. Therefore, one can determine that the worst case is when, that is when the shift is odd. In that case the complexity of computing becomes. Let be the size of the 2-D signal, then the number of the detail coefficients is. The detail coefficients are one third of the total number of coefficients, i.e.,. At reduction level, i.e., the root, the complexity of computing is with a probability of. At the next reduction level, the complexity is with a probability of. Table II shows the complexity and its probability at each reduction level. By multiplying the complexities and the probabilities in Table II and summing them up, the average performance of the worst case for evaluating is found to be. Following the same analysis, one can find that the worst case complexities for evaluating and are found to be, as well. REFERENCES [1] T. Chang and C. C. J. Kuo, Texture analysis and classification with tree-structured wavelet transform, IEEE Trans. Image Process., vol. 2, no. 4, pp , Oct [2] N. Saito and R. R. Coifman, Local discriminant bases, in Proc. Wavelet Applications in Signal and Image Processing II, A. F. Laine and M. A. Unser, Eds., 1994, pp [3] N. Saito and R. Coifman, On local orthonormal bases for classification and regression, in Proc. ICASSP, 1995, vol. 3, pp [4] M. Do and M. Vetterli, Rotation invariant texture characterization and retrieval using steerable wavelet-domain hidden Markov models, IEEE Trans. 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Kingsbury, Fast segmentation using level set curves of complex wavelet surfaces, presented at the Int. Conf. Image Processing, [26] H. Sari-Sarraf and D. Brzakovic, A shift-invariant discrete wavelet transform, IEEE Trans. Signal Process., vol. 45, no. 10, pp , Oct Mais Alnasser, received the Ph.D. degree in computer science from the University of Central Florida, Orlando, in She is currently a Senior Research Engineer in Advanced Micro Devices (AMD). Her research interests include global illumination, 3-D modeling, and the common grounds of computer vision and computer graphics. Hassan Foroosh (M 02 SM 03) recieived the M.S. and Ph.D. degrees in computer science, specializing in computer vision and image processing, from INRIA-Sophia Antipolis, France, in 1993 and 1996, respectively. He is an Assistant Professor of electrical engineering and computer science and the Director of the Computational Imaging Laboratory (CIL), University of Central Florida (UCF), Orlando. He has authored and co-authored over 70 peer-reviewed journals and conference papers. Dr. Foroosh is an Associate Editor of the IEEE TRANACTIONS ON IMAGE PROCESSING and has been on the organizing and the technical committees of numerous international colloquia. In 2004, he was a recipient of an academic excellence award from Sun Microsystems and the Piero Zamperoni award from the International Association for Pattern Recognition (IAPR).