Digital Image Processing. Chapter 7: Wavelets and Multiresolution Processing ( )

Transcription

1 Digital Image Processing Chapter 7: Wavelets and Multiresolution Processing ( )

2 7.4 Fast Wavelet Transform Fast wavelet transform (FWT) = Mallat s herringbone algorithm Mallat, S. [1989a]. "A Theory for Multiresolution Signal Deco mposition: The Wavelet Representation," IEEE Trans. Pattern A nal. Mach. Intel!., vol. PAMI-ll, pp Mallat, S. [1989b]. "Multiresolution Approximation and Wavel et Orthonormal Bases of L2," Trans. American Mathematical S ociety, vol. 315, pp Stephane Mallat unified wavelets, filter banks (subband codding), multiresolution in 1989.

3 7.4 Fast Wavelet Transform Dilation equation for scale function Wavelet equation for wavelet function

4 7.4 Fast Wavelet Transform dx Change notation for FWT Recursive relation for FWT

5 7.4 Fast Wavelet Transform Order-reversed

6 7.4 Fast Wavelet Transform Two-scale FWT

7 Example 7.10 (7.4-13) (7.4-14) These are the functions used to build the FWT filter banks; they provide the filter coefficients. Note that because Haar scaling and wavelet functions are orthonormal, Eq.(7.1-14) can be used to generate the FWT filter coefficients from a single prototype filter. We can use Eq.(7.4-12) to compute W ψ 1, k = h ψ n W ψ 2, n n=2k,k 0 = h ψ ( n) f(n) n=2k,k 0 = = l 1 h ψ l 2k x(l) k=0,1 2 x 2k 1 2 x(2k + 1) k=0,1

8 Example 7.10 f( n) {1,4, 3,0}

9 7.4 Fast Wavelet Transform Inverse fast wavelet (FWT -1 )

10 Example 7.11 Computation of the inverse fast wavelet transform mirrors its forward counterpart. Figure7.22 illustrates the process for the sequence considered in Example 7.10.

11 7.4 Fast Wavelet Transform Meaning of WT : (Uncertainty principle) No Freq. Info No Time. Info Compromise in time & freq info FT for stationary signal WT for non-stationary signal whose frequency vary in time

12 7.4 Fast Wavelet Transform Remark I Fourier basis functions guarantee the existence of the FFT The existence of the FWT depends upon - the availability of a scaling functions being used - Orthogonality or biorthogonality of the scaling function and corresponding wavelets FWT is inapplicable to the Mexican hat wavelet (no corre sponding scaling function)

13 WT vs. FT At what time the frequency change? ml

14 Ch7.5 Wavelet Transforms in Two Dimensions Three Two- Dimensional wavelets : Different directions 을따라 Image 의 intensity 를변화

15 Ch7.5 Wavelet Transforms in Two Dimensions An arbitrary starting scale f(x, y) : Descrete function x, y : Descrete variables Add vertical, horizontal and diagonal detail

16 Ch7.5 Wavelet Transforms in Two Dimensions

17 Ch7.5 Wavelet Transforms in Two Dimensions Down sampling Horizontal resolutions are Reduced by a factor 2 Vertical Horizontal Diagonal Down sampling Vertical resolutions are Reduced by a factor 2 FIGURE 7.24 The 2-D fast wavelet transform: (a) the analysis filter bank; (b) the resulting decomposition;

18 Ch7.5 Wavelet Transforms in Two Dimensions Up sampling Vertical resolutions are Reduced by a factor 2 Reconstruction of image FIGURE 7.24 (c) the synthesis filter bank. Up sampling Horizontal resolutions are Reduced by a factor 2

19 Example /4 Horizontal Vertical Diagonal 1/4 1/4 FIGURE 7.25 Computing a 2-D three-scale FWT: (a) the original image; (b) a one-scale FWT; (c) a two-scale FWT; (d) a three-scale FWT.

20 Example 7.11 Figure 7.25(a) is a computer-generated image consisting of 2-D sine-like pulses on a black background. Figure 7.25(b) through (d) show the FWTs of the image in Fig.7.25(a). The 2-D filter bank of Fig.7.24(a) and the decomposition filters shown in Figs.7.26(a) and (b) were used to generate all three results. Figure 7.25(b) shows the one-scale FWT of the image in Fig.7.25(a). To compute this transform, the original image was used as the input to the filter bank of Fig.7.24(a). The four resulting quarter-size decomposition outputs(the approximation and horizontal, vertical, and diagonal details) were then arranged in accordance with Fig.7.24(b) to produce the image in Fig.7.25(b). A similar process was used to generate the two-scale FWT in Fig.7.25(c). Finally, Fig.7.25(d) is the three-scale FWT that resulted when the subimage from the upper-left-hand corner of Fig.7.25(c) was used as the filter bank input.

21 Ch7.5 Wavelet Transforms in Two Dimensions FIGURE 7.26 Fourth-order symlets: (a)-(b) decomposition filters; (c)-(d) reconstruction filters; (e) the one-dimensional wavelet; (f) the one-dimensional scaling function;

22 Ch7.5 Wavelet Transforms in Two Dimensions FIGURE 7.26 (g) one of three twodimensional wavelets, ψ V (x, y). See Table 7.3 for the values of h φ (n) for 0 n 7. TABLE 7.3 Orthonormal fourth-order symlet filter coefficients for h φ n. The coefficients of the remaining orthonormal filters are obtained using Eq.(7.1-14).

23 Ch7.5 Wavelet Transforms in Two Dimensions We conclude this section with two examples that demonstrate the usefulness of wavelets in image processing. 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. Because the DWT`s scaling and wavelet vectors are used as lowpass and highpass filters, most Fourier-based filtering techniques have an equivalent wavelet domain counterpart.

24 Example 7.13 Edge enhancement Isolate the vertical edges FIGURE 7.27 Modifying a DWT for edge detection: (a) and (c) twoscale decompositions with selected coefficients deleted; (b) and (d) the corresponding reconstructions.

25 Example 7.13 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. As Fig.7.27(b) shows, the net effect of computing the inverse wavelet transform using these modified coefficients is edge enhancement, reminiscent of the Fourier-based image sharpening results discussed in Section 4.9. 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 Fig.7.27(c) and (d) we can isolate the vertical edges.

26 Example 7.14 FIGURE 7.28 Modifying a DWT for noise removal: (a) a noisy CT of a human head: (b), (c) and (e) various reconstructions after thresholding the detail coefficients; (d) and (f) the information removed during the reconstruction of (c) and (e).

27 Example 7.14 As a second example, consider the CT image of a human head shown in Fig.7.28(a). As can be seen in the background, the image has been uniformly corrupted with additive white noise. A general waveletbased procedure for denoising the image is as follow: Step 1. Choose a wavelet 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 from scales J 1 to J P. This can be accomplished by hard thresholding, which means setting to zero the elements whose absolute values are lower than the threshold, or by soft thresholding, which involves first setting to zero the elements whose absolute values are lower than the threshold and then scaling the nonzero coefficients toward zero. Soft thresholding eliminates the discontinuity that is inherent in hard thresholding.

28 Example 7.14 Step 3. Compute the inverse wavelet transform using the original approximation coefficients at level J P and the modified detail coefficients for levels J 1 to J P. Fig.7.28(b) shows the result of performing these operations with fourthorder symlets, two scale(p=2), and a global threshold that was determined interactively. Note the reduction in noise and blurring of image edges. This loss of edge detail is reduced significantly in Fig.7.28(c), which was generated by simply zeroing the highest-resolution detail coefficients and reconstructing the image. The difference image in Fig.7.28(d) show the information that is lost in the process. This result was generated by computing the inverse FWT of the two-scale transform with all but the highest-resolution detail coefficients zeroed. As can be seen, the resulting image contains most of the noise in the original image and some of the edge information. Figures 7.28(e) and (f) are included to show the negative effect of deleting all the detail coefficient.

29 7.6 Wavelet packets Low frequency content scaling wavelet functions with narrow bandwidth High frequency content functions with higher bandwidth Time-frequency scale increase in height as you move up the frequency axis FWT must generalized to yield more flexible decomposition wavelet packet P={1,2,3} : 3 개의분해능

30 7.6 Wavelet packets W j 1,AD J-1 parent, A approximation, D detail filtering 26 개분해능 : D(p+1)=D(p) 2 +1 V J = V J 3 W J 3 W J 2,A W J 2,D W J 1,AA W J 1,AD W J 1,DA W J 1,DD

31 7.6 Wavelet packets 2 dimensional 4 bank filter Row/column filtering D(p+1)=D(p) 4 +1

32 Example *480 Three-scales wavelet packet trees Decomposition : E f = m,n f(m, n) Maximize the number of non0zero values Minimize cost of the leaf nodes in decomposition tree Beginning with the root and proceeding level by level leaves Step 1. calculate E P E A E H E V E D Step 2. include if E P < E A + E H + E V + E D

33 The subimages that are not split (further decomposed) in Fig.7.37 are relatively smooth and composed of pixels that are middle gray in value. Because all but the approximation subimage of this figure have been scaled so that gray level 128 indicates a zero-valued coefficient, these subimages contain little energy. FIGURE 7.37 An optimal wavelet packet decomposition for the fingerprint of Fig.7.36(a). FIGURE 7.38 The optimal wavelet packet analysis tree for the decomposition in Fig.7.37.

34 7.6 Wavelet packets Cohn-Daubechies-Feauveau Biorthogonal wavelet h 0 (n) : low-pass, h 1 (n) : high pass g 0 (n)=(-1) n+1 h 1 (n) g 1 (n)=(-1) n h 0 (n) h φ (-n)=h 0 (n) H ψ (-n)=h 1 (n)