sparse image representation via combined transforms - Convex ...
sparse image representation via combined transforms - Convex ... sparse image representation via combined transforms - Convex ...
52 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES 3.1.5 2-D DCT The 2-D DCT is an extension of the 1-D DCT. On a matrix, we simply apply a 1-D DCT to each row, and then apply a 1-D DCT to each column. The basis function of the 2-D DCT is the tensor product of (two) basis functions for the 1-D DCT. The optimality of the 2-D DCT is nearly a direct extension of the optimality of the 1-D DCT. As long as the homogeneous components of images are defined consistently with the definition corresponding to 1-D homogeneous components, we can show that the 2-D DCT will diagonalize the covariance matrices of the 2-D signal (images) too. Based on this, we can generate a slogan: “The 2-D DCT is good for images with homogeneous components.” Another way to show the optimality of the DCT is to look at its basis functions. The properties of the basis functions represent the properties of a transform. A coefficient after a transform is equal to the inner product of an input and a basis function. Intuitively, if basis functions are similar to the input, then we only need a small number of basis functions to represent the original input. Note here that the input is a signal. When the basis functions are close to the features we want to capture in the images, we say that the corresponding transform is good for this type of images. To show that the 2-D DCT is good for homogeneous images, let’s look at its basis functions. Figure 3.2 shows all 2-D DCT basis functions on an 8 × 8 block. From these figures, we should say that the basis functions are pretty homogeneous, because it is impossible to say that the statistical properties in one area are different from the statistical properties in another. 3.2 Wavelets and Point Singularities It is interesting to review the history of the development of wavelets. The following description is basically from Deslauriers and Dubuc [60, Preface]. In 1985 in France, the first orthogonal wavelet basis was discovered by Meyer. Shortly thereafter Mallat introduced multiresolution analysis, which explained some of the mysteries of Meyer’s wavelet basis. In February, 1987, in Montreal, Daubechies found an orthonormal wavelet basis that has compact support. Next came the biorthogonal wavelets related to splines. In Paris, in May 1985, Deslauriers and Dubuc used dyadic interpolation to explain multiresolution, a term that by then was familiar to many. Donoho and Johnstone developed the wavelet shrinkage
3.2. WAVELETS AND POINT SINGULARITIES 53 Figure 3.2: Two-dimensional discrete cosine transform basis functions on an 8 × 8block. The upper-left corner corresponds to the low-low frequency, and the lower-right corner corresponds to the high-high frequency.
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3.2. WAVELETS AND POINT SINGULARITIES 53<br />
Figure 3.2: Two-dimensional discrete cosine transform basis functions on an 8 × 8block.<br />
The upper-left corner corresponds to the low-low frequency, and the lower-right corner<br />
corresponds to the high-high frequency.
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