MR IMAGE COMPRESSION BY HAAR WAVELET TRANSFORM

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1 Table of Contents BY HAAR WAVELET TRANSFORM Eva Hošťálková & Aleš Procházka Institute of Chemical Technology in Prague Dept of Computing and Control Engineering Process Control 2007, Štrbské Pleso

2 Table of Contents Table of Contents D Haar Transform 2-D Haar Transform 3 Properties 4 Perfect Perfect Conditions 5

3 Perfect Digital signal processing tasks Noise reduction. Feature extraction classification. Restoration of missing or corrupted components. Image compression and coding. Possible solution - the discrete wavelet transform (DWT). The Haar Transform The Haar function the Haar transform (HT). Simple computation algorithm. Orthonormal transform. Satisfies the perfect reconstruction conditions.

4 Axial spine MR image and its cut (a) SPINAL MR IMAGE (b) IMAGE CUT Perfect Application of the HT to magnetic resonance (MR) images.

5 1-D Haar Transform Perfect One-level decomposition Let us have a signal {x(n)} N 1 n=0. For n =0, 2,..., N 2: ( ) ( ) Xn xn = T X n+1 x n+1 matrix T= 1 ( ) (1) (2) Resulting sequence {X 0, X 2,..., X N 2 } - low-pass decomposition values. {X 1, X 3,..., X N 1 } - high-pass sequence.

6 2-D Haar Transform Perfect One-level decomposition Image [g(n, m)] N,M. For n =0, 2,..., N 2, m =0, 2,..., M 2: Column-wise decomposition: ( ) ( ) G1n,m G1 n,m+1 gn,m g =T n,m+1 G1 n+1,m G n+1,m G1 n+1,m+1 G n+1,m+1 G1 n+1,m g n+1,m g n+1,m+1 Row-wise decomposition: ( ) ( ) Gn,m G n,m+1 G1n,m G1 = n,m+1 T T (4) Resulting matrix The low/low-pass submatrix: G1 n+1,m+1 G 0,0 G 0,2 G 0,M 2 G 2,0 G 2,2 G 2,M 2 G N 2,0 G N 2,2 G N 2,M 2 (3)

7 2-D Haar Transform One-level MR image decomposition (a) ORIGINAL IMAGE (b) 1 LEVEL DECOMPOSITION Low / low Low / high Perfect High/low High/high

8 Perfect Orthonormal matrix A real-valued matrix A of size N N is orthonormal, if I N... identity matrix of size N N. Analysis x... signal x of size N 1. w... coefficients vector of size N 1. A A 1 =A A T =I N (5) w=a x (6) Synthesis x=a 1 w=a T w (7)

9 Perfect Signal energy N 1 ε x = x 2 = x, x = x T x = xn 2 (8) ε x... total energy in a discrete signal x Preserving the energy n=0 ε w = w 2 = w T w = (A x) T A x = = x T A T A x = x T x = x 2 = ε x (9) w... orthonormal transform (matrix A) coefficients vector. ε w... total energy of these coefficients. N 1 N 1 ε x = xn 2 = wk 2 (10) n=0 k=0

10 MR image compression the proportion of energy in each HT coefficients set entropy the extent of compression ( bits per pixel). Perfect Proportions of energy (a) ORIGINAL IMAGE (b) DECOMPOSED IMAGE 94.2% 2% 0.8% 2% 0.4% (c) COMPRESSED IMAGE : 94.2% 0.7% 0.1%

11 Subband coding algorithm Perfect Original signal x(n) X(z) Low pass filter Ld(z) High pass filter Hd(z) Downsampling by 2 yl(n) 2 Yl(z) Downsampling by 2 yh(n) 2 Yh(z) Approximation coefficients w a 1 Detail coefficients w d 1 Upsampling by 2 ŷl(n) 2 Ŷl(z) Upsampling by 2 ŷh(n) 2 Ŷh(z) Low pass filter Lr(z) High pass filter Hr(z) Reconstructed signal ˆx(n) ˆX(z) DECOMPOSITION RECONSTRUCTION Filtering M 1 y l (n) = l d (k)x(n k) Z-tr. Y l (z) = L d (z)x(z) k=0 M 1 y h (n) = k=0 M... filter order. h d (k)x(n k) Z-tr. Y h (z) = H d (z)x(z)

12 Perfect Downsampling by 2 Low-pass sequence - approximations: w1 a = {y l(0), y l (2), y l (4),..., y l (N 2)} High-pass sequence - details: w1 d = {y h(0), y h (2), y h (4),..., y h (N 2)}

13 Perfect Subband coding algorithm Original signal x(n) X(z) Low pass filter Ld(z) High pass filter Hd(z) Upsampling by 2 Downsampling by 2 yl(n) 2 Yl(z) Downsampling by 2 yh(n) 2 Yh(z) Approximation coefficients w a 1 Detail coefficients w d 1 Upsampling by 2 ŷl(n) 2 Ŷl(z) Upsampling by 2 ŷh(n) 2 Ŷh(z) DECOMPOSITION RECONSTRUCTION ŷ l (n) = {y l (0), 0, y l (2), 0,..., y l (N 2), 0} ŷ h (n) = {y h (0), 0, y h (2), 0,..., y h (N 2), 0} Low pass filter Lr(z) High pass filter Hr(z) Reconstructed signal ˆx(n) ˆX(z)

14 Perfect Z-transform Ŷ l (z) = ŷ l (n)z n = 1 2 [Y l(z) + Y l ( z)] (11) n=0 Ŷ h (z) = ŷ h (n)z n = 1 2 [Y h(z) + Y h ( z)] (12) n=0 Filtering X l (z) = 1 2 Lr (z)[y l(z) + Y l ( z)] (13) X h (z) = 1 2 Hr (z)[y h(z) + Y h ( z)] (14) Summation X(z) = X l (z) + X h (z) = = 1 2 Lr (z)[y l(z)+y l ( z)] Hr (z)[y h(z)+y h ( z)] (15)

15 Perfect (PR) Conditions Perfect Substituting for Y l (z) and Y h (z) X(z) = 1 2 X(z)[Lr (z)l d(z)+h r (z)h d (z)] + (16) X( z)[lr (z)l d( z)+h r (z)h d ( z)] Perfect reconstruction requirement X(z) X(z) (17) Perfect reconstruction (PR) conditions L r (z)l d (z) + H r (z)h d (z) 2 (18) L r (z)l d ( z) + H r (z)h d ( z) 0 (19) Eq. (19) - anti-aliasing condition

16 Perfect (PR) Conditions Perfect HT filter bank filters: Matrix T l d = 1/ 2 [1, 1], h d = 1/ 2 [1, 1]. filters: T 1 =T T =T l r = l d and h r = h d, except the inverse time course. from the HT coefficients (a) 1 LEVEL DECOMPOSITION (b) RECONSTRUCTION

17 Perfect Haar transform Simple and useful tool for image compression. Computation algorithm. Description of orthonormal discrete transforms. energy in the coefficients extend of image compression. Perfect reconstruction. PR conditions Derivation of the perfect reconstruction (PR) conditions using the Z-transform.

18 Perfect D. B. Percival and A. T. Walden. Wavelet Methods for Time Series Analysis. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, G. Menegaz and L. Grewe and J. P. Thiran. Multirate Coding of 3D Medical Data. Proceedings of the 2000 International Conference on Image Processing. volume 3, pages , IEEE, J. Wang and H. K. Huang. Medical Image Compression by Using Three-Dimensional Wavelet Transform. IEEE Transactions on Medical Imaging. volume 15(4): , N. Kingsbury. 4F8 Image Coding Course. June 2006.