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An introduction to discrete wavelet transforms
- 1. Advisor : Jian-Jiun Ding, Ph. D. Presenter : Ke-Jie Liao NTU,GICE,DISP Lab,MD531 1
- 2. Introduction Continuous Wavelet Transforms Multiresolution Analysis Backgrounds Image Pyramids Subband Coding MRA Discrete Wavelet Transforms The Fast Wavelet Transform Applications Image Compression Edge Detection Digital Watermarking Conclusions 2
- 3. Why WTs? F.T. totally lose time-information. Comparison between F.T., S.T.F.T., and W.T. f f f t t t F.T. S.T.F.T. W.T. 3
- 4. Difficulties when CWT DWT? Continuous WTs Discrete WTs need infinitely scaled wavelets to represent a given function Not possible in real world Another function called scaling functions are used to span the low frequency parts (approximation parts)of the given signal. Sampling F.T. , 1 ( ) ( )s x x ss 0 0 , 00 1 ( ) ( ) j s jj x k s x ss Sampling 0, 0 0( ) exp ]( [ 2 ( )) j s j x A j ss fx k 4[5]
- 5. MRA To mimic human being’s perception characteristic 5 [1]
- 6. Definitions Forward where • Inverse exists only if admissibility criterion is satisfied. ,( , ) ( ) ( )sW s f x x dx , 1 ( ) ( )s x x ss 2 0 1 , x f x W s d ds sC s s 2 | ( ) | | | f C df f C 6
- 7. An example -Using Mexican hat wavelet 7[1]
- 8. Image Pyramids Approximation pyramids Predictive residual pyramids 8N*N N/2*N/2 N/4*N/4 N/8*N/8
- 9. Image Pyramids Implementation 9 [1]
- 10. Subband coding Decomposing into a set of bandlimited components Designing the filter coefficients s.t. perfectly reconstruction 10[1]
- 11. Subband coding Cross-modulated condition Biorthogonality condition 0 1 1 1 0 ( ) ( 1) ( ) ( ) ( 1) ( ) n n g n h n g n h n 1 0 1 1 0 ( ) ( 1) ( ) ( ) ( 1) ( ) n n g n h n g n h n (2 ), ( ) ( )i jh n k g k i j 11 or [1]
- 12. Subband coding Orthonormality for perfect reconstruction filter Orthonormal filters ( ), ( 2 ) ( ) ( )i jg n g n m i j m 1 0( ) ( 1) ( 1 )n eveng n g K n ( ) ( 1 )i i evenh n g K n 12
- 13. The Haar Transform 1 11 1 12 2H 0 1 ( ) 2 0 2 H k 1 1 ( ) 0 2 2 H k DFT 1 1 ( ) 1 1 2 h n 0 1 ( ) 1 1 2 h n 13 [1]
- 14. Any square-integrable function can be represented by Scaling functions – approximation part Wavelet functions - detail part(predictive residual) Scaling function Prototype Expansion functions /2 , ( ) 2 (2 )j j j k x x k 2 ( ) ( )x L R ,{ ( )}j j kV span x 14
- 15. MRA Requirement [1] The scaling function is orthogonal to its integer translates. [2] The subspaces spanned by the scaling function at low scales are nested within those spanned at higher scales. 1 0 1 2V V V V V V 15 [1]
- 16. MRA Requirement [3] The only function that is common to all is . [4] Any function can be represented with arbitrary precision. jV ( ) 0f x {0}V 2 { ( )}V L R 16
- 17. Refinement equation the expansion function of any subspace can be built from double-resolution copies of themselves. 1j jV V ( 1)/2 1 , ( ) ( )2 (2 )j j j k n x h n x n , 1,( ) ( ) ( )j k j n n x h n x 1/2 ( ) ( )2 (2 ) n x h n x n Scaling vector/Scaling function coefficients 17 /2 , ( ) 2 (2 )j j j k x x k
- 18. Wavelet function Fill up the gap of any two adjacent scaling subspaces Prototype Expansion functions ( )x /2 , ( ) 2 (2 )j j j k x x k ,{ ( )}j j kW span x 1j j jV V W 0 0 0 2 1( ) j j jL V W W R 18 [1]
- 19. Wavelet function Scaling and wavelet vectors are related by 1j jW V , 1,( ) ( ) ( )j k j n n x h n x ( 1)/2 1 , ( ) ( )2 (2 )j j j k n x h n x n 1/2 ( ) ( )2 (2 ) n x h n x n Wavelet vector/wavelet function coefficients ( ) ( 1) (1 )n h n h n 19
- 20. Wavelet series expansion 0 0 0 , , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a d j j k j j k k j j k f x f x f x f x c k x d k x 0 0 0 2 1( ) j j jL V W W R ( )f x ( )af x ( )df x 0jW 0jV 0 1jV 0 ( ) 0jd k 0j j 20
- 21. Discrete wavelet transforms(1D) Forward Inverse 00 , 1 ( , ) ( ) ( )j k n W j k f n n M , 0 1 ( , ) ( ) ( ) ,j k n W j k f n n for j j M 0 0 0 , , 1 1 ( ) ( , ) ( ) ( , ) ( )j k j k k j j k f n W j k n W j k n M M 21
- 22. Fast Wavelet Transforms Exploits a surprising but fortune relationship between the coefficients of the DWT at adjacent scales. Derivations for ( ) ( ) 2 (2 ) n p h n p n ( , )W j k (2 ) ( ) 2 2(2 )j j n p k h n p k n 1 ( 2 ) 2 2j m h m k p m 2m k n 22
- 23. Fast Wavelet Transforms Derivations for ( , )W j k /2 /2 1 ( 1)/2 1 1 ( , ) ( )2 (2 ) 1 ( )2 ( 2 ) 2 (2 ) 1 ( 2 ) ( )2 (2 ) ( 2 ) ( 1, ) j j n j j n m j j m n m W j k f n n k M f n h m k n m M h m k f n n m M h m k W j k , 1 ( , ) ( ) ( )j k n W j k f n n M 1 (2 ) ( 2 ) 2 2j j m n k h m k n m 2 , 0( , ) ( ) ( 1, ) |n k kW j k h n W j n 23
- 24. Fast Wavelet Transforms With a similar derivation for An FWT analysis filter bank ( , )W j k 2 , 0( , ) ( ) ( 1, ) |n k kW j k h n W j n 24[1]
- 25. FWT 25[1]
- 26. Inverse of FWT Applying subband coding theory to implement. acts like a low pass filter. acts like a high pass filter. ex. Haar wavelet and scaling vector ( )h n ( )h n DFT 1 ( ) 1 1 2 h n 1 ( ) 1 1 2 h n 1 ( ) 2 0 2 H k 1 ( ) 0 2 2 H k 26 [1]
- 27. 2D discrete wavelet transforms One separable scaling function Three separable directionally sensitive wavelets ( , ) ( ) ( )x y x y ( , ) ( ) ( )H x y x y ( , ) ( ) ( )V x y y x ( , ) ( ) ( )D x y x y x y 27
- 28. 2D fast wavelet transforms Due to the separable properties, we can apply 1D FWT to do 2D DWTs. 28 [1]
- 29. 2D FWTs An example LL LH HL HH 29 [1]
- 30. 2D FWTs Splitting frequency characteristic 30 [1]
- 31. Image Compression have many near-zero coefficients JPEG : DCT-based JPEG2000 : FWT-based , ,H V D W W W DCT-based FWT-based 31 [3]
- 33. Digital watermarking Robustness Nonperceptible(Transparency) Nonremovable Digital watermarking Watermark extracting Channel/ Signal processin g Watermark Original and/or Watermarked data Secret/Public key Secret/Public key H o s t d a t a Watermark or Confidence measure 33
- 34. Digital watermarking An embedding process 34
- 35. Wavelet transforms has been successfully applied to many applications. Traditional 2D DWTs are only capable of detecting horizontal, vertical, or diagonal details. Bandlet?, curvelet?, contourlet? 35
- 36. [1] R. C. Gonzalez, R. E. Woods, "Digital Image Processing third edition", Prentice Hall, 2008. [2] J. J. Ding and N. C. Shen, “Sectioned Convolution for Discrete Wavelet Transform,” June, 2008. [3] J. J. Ding and J. D. Huang, “The Discrete Wavelet Transform for Image Compression,”,2007. [4] J. J. Ding and Y. S. Zhang, “Multiresolution Analysis for Image by Generalized 2-D Wavelets,” June, 2008. [5] C. Valens, “A Really Friendly Guide to Wavelets,” available in http://pagesperso- orange.fr/polyvalens/clemens/wavelets/wavelets.html 36