Wavelet transform in two dimensions
- 1. WAVELET TRANSFORM IN TWO DIMENSIONS Presented By: Ayushi Gagneja
- 2. Wavelet Transform The wavelet transform is a tool that cuts up data, functions or operators into different frequency components, and then studies each component with a resolution matched to its scale Uses a variable length window, e.g.: Narrower windows are more appropriate at high frequencies Wider windows are more appropriate at low frequencies
- 3. Wavelet Functions ■ Different purposes, different wavelets
- 4. Two-Dimensional Wavelets • For image processing applications we need wavelets that are two-dimensional. • This problem reduces down to designing 2D filters. • We will focus on a particular class of 2D filters: separable filters (can be directly designed from their 1D counterparts)
- 5. 2-D Separable WT •For images we use separable Wavelet Transform First we apply a 1-D filter bank to the rows of the image Then we apply same transform to the columns of each channel of the result Therefore, we obtain 3 high-pass channels corresponding to vertical, horizontal, and diagonal, and one approximation image
- 6. ■ In two dimensions, a two-dimensional scaling function, ϕ(x, y) , and three two-dimensional wavelet , and are required. ■ Excluding products that produce one dimensional results, like ϕ(x) 𝛹(x) , the four remaining products produce the separable scaling function ϕ(x, y) = ϕ(x)ϕ(y) And separable, “directionally sensitive” wavelets 𝛹 𝐻 𝑥, 𝑦 𝛹 𝑉 𝑥, 𝑦 𝛹 𝐷 𝑥, 𝑦 𝛹 𝐻 𝑥, 𝑦 = 𝛹(x)ϕ(y) 𝛹 𝑉 𝑥, 𝑦 = ϕ(x)𝛹(y) 𝛹 𝐷 𝑥, 𝑦 = 𝛹(x)𝛹(y) corresponds to variations along diagonals measures variations along columns (like, horizontal edges) responds to variations along rows (like vertical edges)
- 7. The scaled and translated basis functions: j / 2 j j j,m,n (x, y) 2 (2 x m,2 y n) i H,V, D i j,m,n (x, y) 2 j / 2 (2 j x m,2 j y n), where i is the subscript that assumes the values H, V and D
- 8. The Discrete Wavelet Transform of function f(x,y) of size M*N is then 1 M 1N 1 x0 y 0MN Wj0 ,m,n f (x, y) j0 ,m,n (x, y) 1 M 1N 1 x0 y 0 i i H,V, D MN W j,m,n f (x, y) i j ,m,n (x, y) Here, j0 is an arbitrary starting scale Wj0,m,n coefficients define an approximation of of f(x,y) at scale j0 coefficients add horizontal, vertical and diagonal details for the scales j>=j0 i W j, m,n
- 9. , f(x,y) is obtained via the inverse discrete wavelet transform i Given the W and W i m n MN iH ,V ,D j j m n i j,m,n 0 (x, y) MN 1 1 W ( j, m,n) f (x, y) W ( j0, m, n)j0 ,m,n (x, y) Inverse Wavelet Transforms in Two Dimensions
- 10. 2-D Analysis Filter Bank 10 1h 0h 1h 1h 0h 0h x diagonal vertical horizontal approximation Like 1-D DWT, the 2-D DWT can be implemented using digital filters and downsamplers.
- 11. ■ The 2-D FWT filters the scale j+1 approximation coefficients to construct the scale j approximation and detail coefficients. ■ The single scale filter can be “iterated” to produce P scale transforms in which scale j is equal to J-1, J-2,……., J-P. ■ Image f(x,y) is used as the W j0, m, n input. ■ High pass or detail component characterizes the image’s high frequency information with vertical orientation. ■ Lowpass approximation component contains its low frequency vertical information. ■ Both images are then filtered columnwise and downsampled to yield four quarter size output subimages 𝑊𝜙,𝑊 𝛹 𝐻 , 𝑊 𝛹 𝑉 𝑎𝑛𝑑 𝑊 𝛹 𝐷 .
- 13. 2D Discrete Wavelet Transformation d2 h2 v2 a2 a1 h1d1 v1 Original image NxN a3 d3 h3 v3 Level/Band/Scale 1 Level/Band/Scale 2 Level/Band/Scale 3 d = diagonal detail (LOW/LOW) h = horizontal detail (HIGH/LOW) v = vertical detail (LOW/HIGH) a = approximation (HIGH/HIGH)
- 14. 2D Discrete Wavelet Transformation (cont.) d2 h2 v2 h1 d1v1 a3 d3 h3 v3 Original image NxN Wavelet coefficients NxN
- 15. 2-D Synthesis Filter Bank ˆxdiagonal vertical horizontal approximation 1g 1g 1g 0g 0g 0g
- 16. ■ Synthesis Filter bank reverses the process. ■ At each iteration, four scale j approximation and detail subimages are upsampled and convolved with the 2-D filters, one operating on subimages columns and other on its rows. ■ Addition of the results yields the scale j+1 approximation, and process is repeated until the original image is reconstructed.
- 17. computer-generated image consisting of 2-D sine-like pulses on a black background Each pass through the filter bank produced four quarter-size output images that were substituted for the input from which they were derived. Example:
- 18. Next example • The decomposition filters used in the preceding example are part of a well known family of wavelets called symlets, short for "symmetrical wavelets.“ • Figure 7.26(a) and (d), shows the corresponding decomposition and reconstruction filters. • The coefficients of lowpass reconstruction filter go(n) = hφ(n) for 0 < n < 7 . • Figure 7.26(g), a low-resolution graphic depiction of wavelet 𝛹 𝑣 𝑥, 𝑦 , is provided as an illustration of how a one- dimensional scaling and wavelet function can combine to form a separable, two-dimensional wavelet .
- 21. As in the Fourier domain, the basic approach is to – Step 1. Compute a 2-D wavelet transform of an image. – Step 2. Alter the transform. – Step 3. Compute the inverse transform. How?? ■ In Fig 7.27(a) the lowest scale approximation component of the discrete wavelet transform shown in Fig 7.25(c) has been eliminated by setting its values to zero. ■ Fig 7.27(b) shows, the net effect of computing the inverse wavelet transform using these modified coefficients is edge enhancement.
- 22. Note how well the transitions between signal and background are delineated, despite the fact that they are relatively soft, sinusoidal transitions. By zeroing the horizontal details as well—see Figs. 7.27(c) and (d)—we can isolate the vertical edges .
- 23. Next Example Thresholding hard thresholding, means setting to zero the elements whose absolute values are lower than the threshold soft thresholding, involves first setting to zero the elements whose absolute values are lower than the threshold and then scaling the nonzero coefficients toward zero General wavelet-based procedure for denoising the image : Step 1. Choose a wavelet (Haar, symlet) and number of levels (scales), P, for the decomposition. Then compute the FWT of the noisy image. Step 2. Threshold the detail coefficients. That is, select and apply a threshold to the detail coefficients. Soft thresholding eliminates the discontinuity(at the threshold) that is inherent in hard thresholding. Step 3. Compute the inverse wavelet transform (i.e., perform a wavelet reconstruction) using the original approximation coefficients at level J - P and the modified detail coefficients for levels J — 1 to J — P.
- 24. shows the information that is lost. note the increase in edge information in(f) and the corresponding decrease in edge detail in (e). generated by simply zeroing the highest-resolution detail coefficients and reconstructing shows the information that is lost in the process which was generated by computing the inverse FWT of the two-scale transform with all but the highest-resolution detail coefficients zeroed Reconstruction of the DWT in which the details at both levels of the two-scale transform have been zeroed;
- 25. Wavelets Applications Feature Extraction Hardware Implementation Document Analysis Classification Image matching and retrieval Recognition Image compression •Noise filtering
- 26. THANK YOU