IN many image processing applications involving wavelets

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1 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. Y, MM Phase-Shifting for Non-separable 2D Haar Wavelets (This correspondence extends our earlier paper also under review with IEEE TIP to 2D non-separable Haar. See the attched supplementary document.) Mais Alnasser, and Hassan Foroosh, Senior Member, IEEE Abstract In this correspondence, we present a novel and efficient solution to phase-shifting two-dimensional non-separable Haar wavelet coefficients. While other methods either modify existing wavelets or introduce new ones to handle the lack of shiftinvariance, 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. l2, kn-2 A 2 0,0 A 2 1,0 l1, kn-1 A 1 0,0 a 1 0,0 b 1 0,0 c 1 0,0 A 2 0,1 a 2 0,1 b 2 0,1 c 2 0,1 A 2 1,1 l0, kn A 0 0,0 a 0 0,0 b 0 0,0 c 0 0,0 A 2 0,2 A 2 0,3 A 1 0,1 a 1 0, ,2 b 1 0,1 c 1 0,1 1,2 A 2 2 1,2 1,2 A 2 1,3 Index Terms Phase-shifting, 2D Haar transform, non-separable, wavelets. A 2 2,0 A 2 2,1 A 2 2,2 A 2 2,3 A 1 1,0 a 1 1,0 A 1 1,1 a 1 1,1 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, non-orthogonality 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 third more recent approach achieves shiftinvariance through complex wavelets as in [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 trade-off 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. We call the original unshifted signal the 0-shift signal. Of course such a relation would be wavelet-dependent and may not be a straightforward one. The key idea is that as long as the relation is known, one can tackle shift variance, since all coefficients of a shifted signal can be mapped to those of the 0-shift signal. On the other hand, shift-variance is tackled without compromising speed and compression properties. Furthermore, establishing the explicit and direct relations between the coefficients of a signal and its shifted version, would allow us to perform compressed domain processing of signals or images without requiring a chain of forward and backward transforms. This is particularly of interest in applications such A 2 3,0 b 1 1,0 c 1 1,0 A 2 3,1 A 2 3,2 b 1 1,1 c 1 1,1 A 2 3,3 Fig. 1. The Haar transform of the two-dimensional signal x(m, n) is composed of the dc value A 0 0,0 and the detail coefficients al i, j, bl i, j and c l i, j, where l 0,...,2 N1, i 0,...,2 l 1 and j 0,...,2 l 1. The blur coefficients A l i, j 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. as data compression and progressive transmission. Our focus in this paper is on the non-separable two-dimensional 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 the explicit expressions for shifting the Haar compressed data in the transform domain solely based on using the available coefficients of the original 0-shift signal. We also show how our solution can be expanded for non-integer phase shifts, and evaluate our approach against popular interpolation methods in terms of accumulation of errors through sucessive shifts. II. NOTATIONS AND SETUP Let x(n,m) be a two-dimensional signal of size 2 N 2 N, where N is a positive integer. The Haar transform of x(n,m), can be expressed using a tree as in Fig. 1. The tree is constructed of N levels with x(n,m) residing at the leaves, i.e. the N th level. The ij th node at level l in the tree is made to hold the ij th blur coefficient A l i, j and the detail coefficients a N i, j, bn i, j and c N i, j, where l 0,...,N 1, i 0,...,2l 1 and j 0,...,2 l 1. We denote x(i, j) as A N i, j and let l 0,...,N with a N i, j 0, b N i, j 0 and cn i, j 0. For brevity, we will sometimes refer to the horizontal, vertical and diagonal detail coefficients by a, b and c, respectively.

2 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. Y, MM Let Xi, l j a l1 i/2, j/2 + bl1 i/2, j/2 + cl1 i/2, j/2 Yi, l j a l1 i/2, j/2 + bl1 i/2, j/2 cl1 i/2, j/2 Zi, l j a l1 i/2, j/2 bl1 i/2, j/2 cl1 i/2, j/2 Wi, l j a l1 i/2, j/2 bl1 i/2, j/2 + cl1 i/2, j/2 (1) The following formula shows the relation between the blur coefficient A l i, j and its parent at level l 1: A l1 i/2, j/2 + X i/2, l j/2, i is even, j is even A l A l1 i/2, j/2 i, j +Y i/2, l j/2, i is even, j is odd A l1 i/2, j/2 + Zl i/2, j/2, (2) i is odd, j is even A l1 i/2, j/2 +W i/2, l j/2, i is odd, j is odd We let D l i, j be the difference between A 0 0,0 and Al i, j, then D l i, j A l i, j A 0 0,0 + Dl i, j (3) By substituting (3) into (2), D l i, j can be computed recursively solely in terms of the detail coefficients as follows: D l1 i/2, j/2 + X i/2, l j/2, i is even, j is even D l1 i/2, j/2 +Y l D l1 i/2, j/2, i is even, j is odd i is odd, j is even i/2, j/2 + Zl i/2, j/2, D l1 i/2, j/2 +W i/2, l j/2, 0, i j l 0 i is odd, j is odd A two-dimensional signal can be shifted horizontally or vertically. Both types of shifts affect the a l i, j, bl i, j and c l i, j at all levels. At level N k, there are 2 k 2 k non-redundant coefficient sets each of size 2 Nk 2 Nk [26], where k 1,...,N. A horizontal shift s h 0,...,2 N 1 or a vertical shift s v 0,...,2 N 1 can be one of the following possibilities: A shift that is divisible by 2 k. An odd shift. An even shift that is not divisible by 2 k. We derive the formulae for evaluating the a detail coefficients under a horizontal shift s h, for each of the three possibilities, followed by similar derivations for b and c 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 2 k This is the simplest case. A shift s h in the discrete time domain that is equal to 2 k u is a horizontal circular shift of the 0-shift detail coefficients at level N k by u, that is, (4) a Nk a Nk i,( j+u)%2 Nk, k 1,...,N (5) where 0 u 2 Nk 1 and % is the mod operation. Notice that for levels N (k 1),N (k 2),...,N 1 a horizontal shift of 2 k u in the time domain is a horizontal circular shift of the coefficients at those levels by 2u,2 2 u,...,2 k1 u respectively. In other words, a horizontal shift of 2 k u in the time domain shifts the coefficients at level N k horizontally by u, while shifting the coefficients at level N (k 1) horizontally by twice as much, and the coefficients at level N (k 2) by four times as much and so on. B. Shifting by an Odd Amount By examining the tree in Fig. 1, we notice that: 2 k (i+1) a Nk ((A N m, j 1 %2 N A N m,( )%2 N ) m2 k i j 1 2 k j + s h (A N m, j 2 %2 N A N m,( j 3 1)%2 N )))/4 k j 2 2 k1 (2 j + 1) + s h j 3 2 k ( j + 1) + s h (6) The above equation evaluates a Nk by summing the first half and subtracting the second half of each row of blur coefficients at the leaves level N, which fall inside the window determined by the two corners (2 k i,2 k j+s h ) and (2 k (i+1),2 k ( j+1)+s h ). Figure 2-(a) illustrates the evaluation of a 2 0,0 using the blur coefficients at level l 3 under a horizontal odd shift of one. Substituting (3), (4) and then (1) into (6), we get the coefficients of the shifted signal as follows: 2 k1 (i+1)1 a Nk 2 m2 k1 i j 3 1 (D N1 m, j 1 %2 N1 + 2 D N1 m,n%2 N1 D N1 m, j 3 %2 N1 D N1 m,n%2 N1 a N1 m, j 1 %2 N1 +2a N1 m, j 2 %2 N1 a N1 m, j 3 %2 N1 )/2 4 k1 where j 1 2 k1 j + s h /2 j 2 2 k2 (2 j + 1) + s h /2 j 3 2 k1 ( j + 1) + s h /2 (7) Note that at k 1, j 2 is a non-integer value. When that is the case we set a N1 i, j 2 %2 N1 to 0. C. Shifting by an Even Amount that is Not Divisible by 2 k In this case, s h is divisible by 2 t, where 1 t k 1 and 2 t is the highest power of 2 by which s h is divisible. This allows us to let s h 2 t u, where 0 u 2 Nt 1, which means that the coefficients at levels N 1,...,N t follow the first case. In other words, the 0-shift coefficients at levels N 1,N 2,...,N t are circularly shifted in the horizontal direction by 2 t1 u,2 t2 u,...,u, respectively. On the other hand, a Nk is verified to be an odd shift of the blur details at level N t. In other words, at level N k, a Nk can be evaluated using the following modification of equation (6): 2 kt (i+1) a Nk ((A Nt m, j 1 %2 Nt A Nt m,( )%2 Nt ) m2 kt i (A Nt m, j 2 %2 Nt A Nt m,( j 3 1)%2 Nt )))/4 kt j 1 2 kt j + s h /2 t j 2 2 kt1 (2 j + 1) + s h /2 t j 3 2 kt ( j + 1) + s h /2 t (8)

3 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. Y, MM A 3 0,0 A 3 0,1 A 3 0,2 A 3 0,3 A 3 0,4 A 3 0,5 A 3 0,6 A 3 0,7 A 3 0,0 A 3 0,1 A 3 0,2 A 3 0,3 A 3 0,4 A 3 0,5 A 3 0,6 A 3 0,7 + - A 3 0,0 A 3 0,1 A 3 0,2 A 3 0,3 A 3 0,4 A 3 0,5 A 3 0,6 A 3 0,7 A 3 1,0 A 3 1,1 A 3 1,2 A 3 1,3 A 3 1,4 A 3 1,5 A 3 1,6 A 3 1,7 A 3 2,0 A 3 2,1 A 3 2,2 A 3 2,3 A 3 2,4 A 3 2,5 A 3 2,6 A 3 2,7 A 3 3,0 A 3 3,1 A 3 3,2 A 3 3,3 A 3 3,4 A 3 3,5 A 3 3,6 A 3 3,7 - A 3 1,0 A 3 1,1 A 3 1,2 A 3 1,3 A 3 1,4 A 3 1,5 A 3 1,6 A 3 1,7 A 3 2,0 A 3 2,1 A 3 2,2 A 3 2,3 A 3 2,4 A 3 2,5 A 3 2,6 A 3 2,7 A 3 3,0 A 3 3,1 A 3 3,2 A 3 3,3 A 3 3,4 A 3 3,5 A 3 3,6 A 3 3,7 A 3 1,0 A 3 1,1 A 3 1,2 A 3 1,3 A 3 1,4 A 3 1,5 A 3 1,6 A 3 1,7 A 3 2,0 A 3 2,1 A 3 2,2 A 3 2,3 A 3 2,4 A 3 2,5 A 3 2,6 A 3 2,7 A 3 3,0 A 3 3,1 A 3 3,2 A 3 3,3 A 3 3,4 A 3 3,5 A 3 3,6 A 3 3,7 A 3 4,0 A 3 4,1 A 3 4,2 A 3 4,3 A 3 4,4 A 3 4,5 A 3 4,6 A 3 4,7 A 3 4,0 A 3 4,1 A 3 4,2 A 3 4,3 A 3 4,4 A 3 4,5 A 3 4,6 A 3 4,7 A 3 4,0 A 3 4,1 A 3 4,2 A 3 4,3 A 3 4,4 A 3 4,5 A 3 4,6 A 3 4,7 A 3 5,0 A 3 5,1 A 3 5,2 A 3 5,3 A 3 5,4 A 3 5,5 A 3 5,6 A 3 5,7 A 3 5,0 A 3 5,1 A 3 5,2 A 3 5,3 A 3 5,4 A 3 5,5 A 3 5,6 A 3 5,7 A 3 5,0 A 3 5,1 A 3 5,2 A 3 5,3 A 3 5,4 A 3 5,5 A 3 5,6 A 3 5,7 A 3 6,0 A 3 6,1 A 3 6,2 A 3 6,3 A 3 6,4 A 3 6,5 A 3 6,6 A 3 6,7 A 3 6,0 A 3 6,1 A 3 6,2 A 3 6,3 A 3 6,4 A 3 6,5 A 3 6,6 A 3 6,7 A 3 6,0 A 3 6,1 A 3 6,2 A 3 6,3 A 3 6,4 A 3 6,5 A 3 6,6 A 3 6,7 A 3 7,0 A 3 7,1 A 3 7,2 A 3 7,3 A 3 7,4 A 3 7,5 A 3 7,6 A 3 7,7 A 3 7,0 A 3 7,1 A 3 7,2 A 3 7,3 A 3 7,4 A 3 7,5 A 3 7,6 A 3 7,7 A 3 7,0 A 3 7,1 A 3 7,2 A 3 7,3 A 3 7,4 A 3 7,5 A 3 7,6 A 3 7,7 (a) (b) (c) Fig. 2. The above figures show the blur coefficients at level l 3. In (a), (b), and (c), respectively, the rectangular windows show the coefficients included in evaluating a 2 0,0 new, b 2 0,0 new, and c 2 0,0 new 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. Following the same steps as above, we get: 2 kt1 (i+1)1 a Nk 2 m2 kt1 i j 3 1 (D Nt1 m, j 1 %2 Nt1 + 2 D Nt1 m,n%2 Nt1 D Nt1 m, j 3 %2 Nt1 D Nt1 m,n%2 Nt1 a Nt1 m, j 1 %2 Nt1 +2a Nt1 m, j 2 %2 Nt1 a Nt1 m, j 3 %2 Nt1 )/2 4 kt1 j 1 2 kt1 j + s h /2 t+1 j 2 2 kt2 (2 j + 1) + s h /2 t+1 j 3 2 kt1 ( j + 1) + s h /2 t+1 (9) Note that the second case is the same as the third case where t 0. This gives rise to the following final formula for evaluating the a detail coefficients under a horizontal shift: k > t : 2 kt1 (i+1)1 a Nk (D Nt1 + 2 m, j 1 %2 Nt1 2 m2 kt1 i j 3 1 D Nt1 m,n%2 Nt1 D Nt1 m, j 3 %2 Nt1 D Nt1 m,n%2 Nt1 a Nt1 +2a Nt1 a Nt1 )/2 4 kt1 m, j 1 %2 Nt1 m, j 2 %2 Nt1 m, j 3 %2 Nt1 k t : a Nk a Nk i,( j+s h /2 k )%2 Nk j 1 2 kt1 j + s h /2 t+1 j 2 2 kt2 (2 j + 1) + s h /2 t+1 j 3 2 kt1 ( j + 1) + s h /2 t+1 (10) IV. VERTICAL COEFFICIENTS FOR HORIZONTAL SHIFT For this section and the next one, again we let s h 2 t u be a horizontal shift, where 0 s h 2 N 1, 0 t N, 2 t is the highest power of 2 by which s h is divisible, and 0 u 2 Nt 1 is an odd positive integer. Then for all k t, b Nk b Nk i,( j+s h /2 k )%2 Nk (11) To compute the b Nk coefficient for k > t after a horizontal shift, we write b Nk in terms of the coefficients at level N t: 2 kt1 (2i+1)1 b Nk m2 kt i 2 kt (i+1)1 (A Nt m2 kt1 (2i+1) j 1 2 kt j + s h /2 t (A Nt m, j 1 %2 Nt A Nt m,( )%2 Nt ) m, j 1 %2 Nt A Nt m,( )%2 Nt ))/4 kt j 2 2 kt ( j + 1) + s h /2 t (12) The above equation defines a window that encloses the blur coefficients at level N t that fall between the two corners (2 kt i,2 kt j + s h /2 t ) and (2 kt (i + 1),2 kt ( j + 1) + s h /2 t ), which are the same bounds used to compute a Nk in (10). bi, Nk j new 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. Figure 2-(b) illustrates the computation of b 2 0,0 using the blur coefficients at level l 3 under a horizontal odd shift of one. By substituting (3), (4) and (1) in the above, and combining the result with (11) we get the coefficients of the shifted signal: k > t + 1 : 2 kt2 (2i+1)1 b Nk m2 kt1 i (D Nt1 m, j 1 %2 Nt1 + 2 D Nt1 m,n%2 Nt1 + D Nt1 m, j 2 %2 Nt1 a Nt1 m, j 1 %2 Nt1 + a Nt1 m, j 2 %2 Nt1 ) 2 kt1 (i+1)1 (D Nt1 m2 kt2 (2i+1) + 2 m, j 1 %2 Nt1 D Nt1 m,n%2 Nt1 + D Nt1 m, j 2 %2 Nt1 a Nt1 m, j 1 %2 Nt1 + a Nt1 m, j 2 %2 Nt1 ) )/2 4 kt1 k t + 1 : b Nk b Nk c Nk i,( j+ s h /2 k )%2 Nk i,( j+ s h /2 k )%2 Nk + b Nk i,( j+ s h /2 k +1)%2 Nk + c Nk i,( j+ s h /2 k +1)%2 Nk )/2

4 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. Y, MM k t : b Nk b Nk i,( j+s h /2 k )%2 Nk j 1 2 kt1 j + s/2 t+1 h j 2 2 kt1 ( j + 1) + s h /2 t+1 (13) V. DIAGONAL COEFFICIENTS FOR HORIZONTAL SHIFT Similar to 5 and 11, we can apply the following equation when the horizontal shift s h is divisible by 2 k : c Nk c Nk i,( j+s h /2 k )%2 Nk (14) To compute the c Nk coefficient for k > t after a horizontal shift, we write c Nk in terms of coefficients at level N t: 2 kt1 (i+1)1 c Nk ((A Nt m, j 1 %2 Nt A Nt m,( )%2 Nt ) m2 kt i (A Nt m, j 2 %2 Nt A Nt m,( j 3 1)%2 Nt )) 2 kt (i+1)1 ((A Nt m2 kt1 (i+1) m, j 1 %2 Nt A Nt m,( )%2 Nt ) (A Nt m, j 2 %2 Nt A Nt m,( j 3 1)%2 Nt )))/4 kt j 1 2 kt j + s h /2 t j 2 2 kt1 (2 j + 1) + s h /2 t j 3 2 kt ( j + 1) + s h /2 t (15) The above equation defines a window which has the same bounds used to compute a Nk and b Nk in (10) and (13), respectively. By examining the example in Fig. 2-(c), one can see that c Nk 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, the upper equation. By substituting (3), (4) and (1) in the above, and combining the result with (14), we get: k > t + 1 : 2 kt2 (2i+1)1 c Nk 2 m2 kt1 i j m, j 1 %2 Nt1 (D Nt1 D Nt1 m,n%2 Nt1 D Nt1 m, j 3 %2 Nt1 D Nt1 m,n%2 Nt1 a Nt1 j 1 %2 Nt1 + 2a Nt1 j 2 %2 Nt1 a Nt1 j 3 %2 Nt1 ) 2 kt1 (i+1)1 (D Nt1 m2 kt2 (2i+1) + 2 m, j 1 %2 Nt1 D Nt1 m,n%2 Nt1 j D Nt1 D Nt1 m,n%2 Nt1 m, j 3 %2 Nt1 a Nt1 + 2a Nt1 a Nt1 ) j 1 %2 Nt1 j 2 %2 Nt1 j 3 %2 Nt1 )/2 4 kt1 k t + 1 : c Nk b Nk c Nk i,( j+ s h /2 k )%2 Nk i,( j+ s h /2 k )%2 Nk k t : c Nk c Nk i,( j+s h /2 k )%2 Nk b Nk i,( j+ s h /2 k +1)%2 Nk c Nk i,( j+ s h /2 k +1)%2 Nk )/2 j 1 2 kt1 j + s h /2 t+1 j 2 2 kt2 (2 j + 1) + s h /2 t+1 j 3 2 kt1 ( j + 1) + s h /2 t+1 (16) The relations (10), (13), and (16) can now be used to compute the new detail coefficients of the Haar transform at all different levels after any horizontal shift s h 0,...,2 N 1 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 a s with b s, i s with j s and m s with n s. VI. SUBPIXEL SHIFTING Let the size of the signal be 2 N 2 N, N N + h and k 1 + h,...,n + h, where h is the number of added levels. Equations (10), (13) and (13) can now be modified to allow for noninteger shifting by a precision of 1 2 h by substituting N for each N in the equations. On the other hand, we can verify that D N+h 0 i, j D N, where 0 h i/2 h 0, j/2 h 0 0 h. This combined with (4) allows us to modify (17), (18) and (19) in such a way that avoids having to actually up-sample the signal for non-integer shifts, saving thus 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 h t + 1, which is when the coefficients at the added levels are being used to compute a N k, b N k and c N k. The second is when t is large enough for the coefficients at the original levels of the tree to be used. This leads to the phase shifting relation for non-integer values as follows: h t + 1 : 2 kt1 (i+1)1 a N k +2 m2 kt1 i N t1, j 1 %2N t1 N t1 D N m N t1, n%2n t1 N t1 j D N m N t1, n%2n t1 N DN t1 m N t1, j 3 %2N t1 N t1 h < t + 1, k > t : 2 kt1 (i+1)1 a N k 2 m2 kt1 i j 3 1 j m, j 1 %2 N t1 (D N t1 D N t1 m,n%2 N t1 DN t1 m, j 3 %2 N t1 )/2 4 kt1 D N t1 m,n%2 N t1

5 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. Y, MM a N t1 m, j 1 %2 N t1 +2aN t1 m, j 2 %2 N t1 an t1 )/2 m, j 3 %2 N 4kt1 t1 h < t + 1,k t : a N k a N k i,( j+s h /2 k )%2 N k j 1 2 kt1 j + s h /2 t+1 j 2 2 kt2 (2 j + 1) + s h /2 t+1 j 3 2 kt1 ( j + 1) + s h /2 t+1 (17) h t + 1, k > t + 1 : 2 kt2 (2i+1)1 b N k +2 2 kt1 (i+1)1 m2 kt2 (2i+1) +D N m m2 kt1 i N t1, j 1 %2N t1 N t1 D N + m N t1, n%2n t1 N DN t1 m N t1, j 2 %2N t1 N t1 N t1, j 2 %2N t1 N t1 h < t + 1, k > t + 1 : 2 kt2 (2i+1)1 b N k N t1, j 1 %2N t1 N t1 m2 kt1 i ))/2 4 kt1 (D N t1 + 2 j m, j 1 %2 N t1 ) D N m N t1, n%2n t1 N t1 D N t1 m,n%2 N t1 +D N t1 m, j 2 %2 N t1 an t1 m, j 1 %2 N t1 + an t1 m, j 2 %2 N t1 ) 2 kt1 (i+1)1 (D N t1 m2 kt2 (2i+1) j m, j 1 %2 N t1 D N t1 m,n%2 N t1 +D N t1 m, j 2 %2 N t1 an t1 + m, j 1 %2 N t1 an t1 ))/2 m, j 2 %2 N 4kt1 t1 h < t + 1, k t + 1 : b N k (b N k i,( j+ s h /2 k )%2 N k cn k i,( j+ s h /2 k )%2 N k +b N k + i,( j+ s h /2 k +1)%2 N k cn k )/2 i,( j+ s h /2 k +1)%2 N k h < t + 1, k t : b N k b N k i,( j+s h /2 k )%2 N k j 1 2 kt1 j + s/2 t+1 h j 2 2 kt1 ( j + 1) + s h /2 t+1 (18) h t + 1, k > t + 1 : 2 kt2 (2i+1)1 c N k +2 D N m2 kt1 i m N t1, n%2n t1 N t1 D N m N t1, j 3 %2N t1 N t1 +2 D N m N t1, n%2n t1 N t1 N t1, j 1 %2N t1 N t1 j kt1 (i+1)1 ) m2 kt2 (2i+1) j31 2 D N m N t1, n%2n t1 N t1 N t1, j 1 %2N t1 N t1 D N m N t1, n%2n t1 N t1 D N m N t1, j 3 %2N t1 N t1 h < t + 1, k t + 1 : ))/2 4 kt1 c N k (b N k i,( j+ s h /2 k )%2 N k cn k i,( j+ s h /2 k )%2 N k b N k i,( j+ s h /2 k +1)%2 N k cn k )/2 i,( j+ s h /2 k +1)%2 N k h < t + 1, k t : c N k c N k i,( j+s h /2 k )%2 N k j 1 2 kt1 j + s h /2 t+1 j 2 2 kt2 (2 j + 1) + s h /2 t+1 j 3 2 kt1 ( j + 1) + s h /2 t+1 (19) VII. EXPERIMENTAL RESULTS AND DISCUSSION For integer shifting, our solution for Haar-domain phase shifting, given by (10), (13), and (16), 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 h 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 h, we can obtain a precision of 1 2 h, i.e. subpixel shifts are approximated by the closest value in multiples of 1 2 h. In order to evaluate the accuracy of our Haar-domain phaseshifting method, we shifted the test images first by a large amount using our method. We then performed the same shift but using a succesive set of smaller shifts, and used the accumulated error as a measure for performance. We performed this test on numerous images, some of which are shown in table I. The performance was compared with different interpolation methods such as bilinear, bicubic, and spline. As shown in table I, accumulated errors are on average an order of magnitude smaller in our method. On the other hand, the low complexity of the derived solutions allow for fast processing directly in the transform domain. By examining (10) one can find that the complexity of evaluating a Nk 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 O(( j 3 j 1 ) (2 kt1 )). Substituting the values for j 1 and j 3, the complexity is shown to be O(2 kt1 2 kt1 ) when k > t. When k t the complexity becomes O(1). Therefore, one can determine that the worst case is when t 0, that is when the shift is odd. In that case the complexity of computing a Nk i new becomes O(2 k1 2 k1 ). Let L L 2 N 2 N be the size of the two-dimensional signal, then the number of the detail coefficients is L N 1. The a detail coefficients are one third of the total number of coefficients, i.e. 4N 1 3. At reduction level k N, i.e. the root, the complexity of computing a 0 0,0 new is O(2 N1 2 N1 ) O(( L 2 )2 ) with a 3 probability of. At the next reduction level k N 1, L 2 1 the complexity is O(2 N2 2 N2 ) O(2 2 ( L ) 2 ) with a 2 2

6 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. X, NO. Y, MM Bilinear Bicubic Spline Our Method TABLE I QUANTIFICATION AND COMPARISON OF THE ACCUMULATED RESIDUAL ERROR ON SEVERAL TEST IMAGES. Reduction Level Complexity Prob. Number of Coefficients at k Number of Coefficients k N O(( L 2 )2 ) 3 L 2 1 k N 1 O(( L 2 2 ) 2 ) (2 1 ) 2 3 L 2 1 k N 2 O(( L 2 3 ) 2 ) (2 2 ) 2 3 L 2 1 k N 3 O(( L 2 4 ) 2 ) (2 3 ) 2 3 L 2 1 : : : k 1 O(( L 2 N ) 2 ) (2 N1 ) 2 3 L 2 1 TABLE II TABLE OF THE COMPLEXITY AND PROBABILITY AT EACH REDUCTION LEVEL k FOR EVALUATING THE 2D DETAIL COEFFICIENTS a Nk. probability of Table (II) shows the complexity and L 2 1 its probability at each reduction level k. By multiplying the complexities and the probabilities in table (II) and summing them up, the average performance of the worst case for evaluating a Nk is found to be O(lg(L)). Following the same analysis, one can find that the worst case complexities for evaluating b Nk and ci, Nk j new are found to be O(lg(L)) as well. REFERENCES [1] T. Chang and C. C. J. Kuo, Texture analysis and classification with treestructured wavelet transform. IEEE Transactions on Image Processing, vol. 2, no. 4, pp , [2] N. Saito and R. R. Coifman, Local discriminant bases, in Wavelet Applications in Signal and Image Processing II, Proc. SPIE 2303, A. F. Laine and M. A. Unser, Eds., 1994, pp [3] N. Saito and R. Coifman, On local orthonormal bases for classification and regression, icassp, vol. 3, pp , [4] M. Do and M. Vetterli, Rotation invariant texture characterization and retrieval using steerable wavelet-domain hidden markov models. [5] M. Do, S. Ayer, and M. Vetterli, Invariant image retrieval using wavelet maxima moment, in Visual Information and Information Systems, 1999, pp [6] M. Do and M. Vetterli, Texture similarity measurement using kullbackleibler distance on wavelet subbands, [7] R. R. Coifman and M. V. Wickerhauser, Adapted waveform analysis as a tool for modeling, feature extraction, and denoising, Optical Engineering, vol. 33, no. 7, pp , [9] D. L. Donoho, De-noising by soft-thresholding, IEEE Transactions on Information Theory, vol. 41, no. 3, pp , [10] M. Figueiredo, J. Bioucas-Dias, and R. Nowak, Majorizationminimization algorithms for wavelet-based image restoration, Preprint, [11] D. D. H. Krim, S. Mallat and A. Willsky, Best basis algorithm for signal enhancement, icassp, vol. 3, pp , [12] M. Vetterli and J. Kovačevic, Wavelets and subband coding. Upper Saddle River, NJ, USA: Prentice-Hall, Inc., [13] H.-W. Park and H.-S. Kim, Motion estimation using low-band-shift method for wavelet-based moving-picture coding. IEEE Transactions on Image Processing, vol. 9, no. 4, pp , [14] E. P. Simoncelli, W. T. Freeman, E. H. Adelson, and D. J. Heeger, Shiftable multi-scale transforms, IEEE transactions on informations theory, vol. 38, no. 2, [15] J. M. M. Holschneider, R. Kronland-Martinet and P. Tchamitchian, A Real-Time Algorithm for Signal Analysis with the Help of the Wavelet Transform, in Wavelets. Time-Frequency Methods and Phase Space, J.- M. Combes, A. Grossmann, and P. Tchamitchian, Eds., 1989, pp [16] P. Dutilleux, An Implementation of the algorithme à trous to Compute the Wavelet Transform, in Wavelets. Time-Frequency Methods and Phase Space, J.-M. Combes, A. Grossmann, and P. Tchamitchian, Eds., 1989, pp [17] S. Mallat, A wavelet tour of signal processing. Academic Press, [18] C. S. Burrus, R. A. Gopinath, and H. Guo, Introduction to Wavelets and Wavelets Transforms. Prentice Hall, August [19] M. Lang, H. Guo, J. E. Odegard, C. S. Burrus, and R. O. Wells, Noise reduction using an undecimated discrete wavelet transform, IEEE Signal Processing Letters, vol. 3, no. 1, [20] B. S. Manjunath and W. Y. Ma, Texture features for browsing and retrieval of image data, IEEE Trans. Pattern Anal. Mach. Intell., vol. 18, no. 8, pp , [21] D. E. Newland, Harmonic wavelet analysis, Proc. R. Soc. Lond. A, vol. 443, pp , [22], Harmonic wavelets in vibrations and acoustics, Phil. Trans. R. Soc. Lond. A, vol. 357(1760), pp , [23] J. Magarey and N. Kingsbury, Motion estimation using complex wavelets, in ICASSP 96: Proceedings of the Acoustics, Speech, and Signal Processing, on Conference Proceedings., 1996 IEEE International Conference. Washington, DC, USA: IEEE Computer Society, 1996, pp [24] P. de Rivaz and N. Kingsbury, Complex wavelet features for fast texture image retrieval, in Proc. ICIP, 1999, pp. I: [25] P. D. Rivaz and N. G. Kingsbury, Fast segmentation using level set curves of complex wavelet surfaces. in Proc. ICIP, [26] H. Sari-Sarraf and D. Brzakovic, A shift-invariant discrete wavelet transform, IEEE Transactions on Signal Processing, vol. 45, no. 10, pp , [8] R. R. Coifman and D. L. Donoho, Translation-invariant de-noising, Department of Statistics, Tech. Rep., 1995.