PRIMERA GUERRA MUNDIAL (1914-1918)

While judging the quality of the processed images due to compressed domain processing, we observe degrees of compression for both the input and output of a processing. The performance of a scheme may vary with the level of com- pression of images and videos. The level of compression is usually measured by the compression ratio, which is expressed as the ratio between the size of raw input data and the size of compressed data. The compression ratio may also be expressed in percentage (%). Further, assuming 8 bits per pixel for a gray-level image and 24 bits per pixel for color images, compression lev- els are expressed by the average bits per pixel (bpp) required for representing the compressed data. For video, the preferred unit in this case is the bits per second (bps), by taking care of the frame rate of a video.

1.8

Summary

In the compressed domain, images and videos are represented in alternative forms that are quite different from their representations in the spatial or spa- tiotemporal domain. In today’s world, due to the digital revolution in our daily life, in particular due to the stupendous advancement in the multimedia and communication technology, these images and videos are increasingly available in the compressed format. Compression standards such as JPEG, JPEG2000, MPEG-2, MPEG-4, H.264, etc., are not known only to specialists but they have gained popularity among modern consumers and users as well. Hence, in applications involving image and video processing, computations directly in the domain of alternative representation are often felt necessary as it re- duces the time and storage required for the processing. In this chapter, with an introduction to different popular compression technology, motivation and performance issues related to the compressed domain processing have been discussed.

Chapter 2

Image Transforms

2.1 Orthogonal Expansion of a Function . . . 50 2.1.1 Trivial Expansion with Dirac Delta Functions . . . 53 2.1.2 Fourier Series Expansion . . . 53 2.1.3 Fourier Transform . . . 53 2.1.3.1 Properties of Fourier Transform . . . 54 2.1.4 Shannon’s Orthonormal Bases for Band-limited Functions . . . 57 2.1.5 Wavelet Bases . . . 57 2.1.5.1 Multiresolution Approximations . . . 58 2.1.5.2 Wavelet Bases for Multiresolution Approximations . . . 60 2.2 Transforms of Discrete Functions . . . 60 2.2.1 Discrete Fourier Transform (DFT) . . . 61 2.2.1.1 The Transform Matrix . . . 62 2.2.1.2 Discrete Fourier Transform as Fourier Series of a

Periodic Function . . . 62 2.2.1.3 Circular Convolution . . . 63 2.2.1.4 Energy Preservation . . . 64 2.2.1.5 Other Properties . . . 64 2.2.2 Generalized Discrete Fourier Transform (GDFT) . . . 65 2.2.2.1 Transform Matrices . . . 66 2.2.2.2 Convolution–Multiplication Properties . . . 66 2.2.3 Discrete Trigonometric Transforms . . . 67 2.2.3.1 Symmetric Extensions of Finite Sequences . . . 68 2.2.3.2 Symmetric Periodic Extension . . . 68 2.2.3.3 Different Types of Discrete Trigonometric Transforms 74 2.2.3.4 Convolution Multiplication Properties . . . 77 2.2.4 Type-II Even DCT . . . 79 2.2.4.1 Matrix Representation . . . 79 2.2.4.2 Downsampling and Upsampling Properties of the

DCTs . . . 79 2.2.4.3 Subband Relationship of the type-II DCT . . . 80 2.2.4.4 Approximate DCT Computation . . . 81 2.2.4.5 Composition and Decomposition of the DCT Blocks . 81 2.2.4.6 Properties of Block Composition Matrices . . . 82 2.2.4.7 Matrix Factorization . . . 86 2.2.4.8 8-Point Type-II DCT Matrix (C8) . . . 86

2.2.4.9 Integer Cosine Transforms . . . 87 2.2.5 Hadamard Transform . . . 89 2.2.6 Discrete Wavelet Transform (DWT) . . . 89 2.2.6.1 Orthonormal Basis with a Single Mother Wavelet . . . . 89 2.2.6.2 Orthonormal Basis with Two Mother Wavelets . . . 90 2.2.6.3 Haar Wavelets . . . 90 2.2.6.4 Other Wavelets . . . 91 2.2.6.5 DWT through Filter Banks . . . 92

2.2.6.6 Lifting-based DWT . . . 95 2.3 Transforms in 2-D Space . . . 97 2.3.1 2-D Discrete Cosine Transform . . . 99 2.3.1.1 Matrix Representation . . . 99 2.3.1.2 Subband Approximation of the Type-II DCT . . . 99 2.3.1.3 Composition and Decomposition of the DCT Blocks

in 2-D . . . 100 2.3.1.4 Symmetric Convolution and

Convolution–Multiplication Properties for 2-D DCT . 100 2.3.1.5 Fast DCT Algorithms . . . 100 2.3.2 2-D Discrete Wavelet Transform . . . 102 2.3.2.1 Computational Complexity . . . 103 2.4 Summary . . . 104

Image transforms [52] are used for decomposing images in different structures or components so that a linear combination of these components provide the image itself. We may consider an image as a 2-D real function f (x, y) ∈ L2_(R2_),

(x, y) ∈ R2_{, where R is the set of real numbers and L}2_(R2_{) is the space of}

all square integrable functions. In a transform, the function is represented as a linear combination of a family (or a set) of functions, known as basis functions. The number of basis functions in the set may be finite or infinite. Let B = {bi(x, y)| − ∞ < i < ∞} be a set of basis functions, where bi(x, y)

may be in real (R) or complex space (C). In that case, a transform of the function f (x, y) with respect to the basis set B is defined by the following expression: f (x, y) = ∞ X i=−∞ aibi(x, y), (2.1)

where ai’s are in R or C depending upon the space of the basis functions.

They are known as transform coefficients. This set of coefficients provides an alternative description of the function f (x, y).

In this chapter we discuss a number of different image transforms and their properties. As all our transforms are extended from their counterparts in 1-D, for the sake of simplicity, we initially restrict our discussion to 1-D. Their extensions in 2-D are discussed subsequently.

2.1

Orthogonal Expansion of a Function

Let f (x) ∈ L2_{(R) be a function over the support [a, b] ⊆ R, and B = {b} i(x)| −

∞ < i < ∞, x ∈ [a, b]} be a set of basis functions (either in R or C). In our discussion we consider every function to be integrable in [a, b]. An expansion

of f (x) with respect to B is defined as follows: f (x) = ∞ X i=−∞ λibi(x), (2.2)

where λi’s are in R or C depending upon the space of the basis functions.

Definition 2.1 Inner Product: The inner product of two functions h(x) and g(x) in [a, b] ⊆ R is defined as

< h, g >= Z b

a

h(x)g∗(x)dx, (2.3) where g∗_{(x) is the complex conjugate of g(x). }

Definition 2.2 Orthogonal Expansion: An expansion of a function f (x) in the form of Eq. (2.2) is orthogonal when the inner products of pairs of basis functions in B have the following properties:

< bi, bj > = 0, for i 6= j,

= ci, Otherwise (for i = j), where ci> 0. (2.4)

 Theorem 2.1 Given an orthogonal expansion of f (x) with a base B, trans- form coefficients are computed as follows:

λi= 1 ci < f, bi> . (2.5) Proof: < f, bi> = Rb af (x)b∗i(x)dx (from Def. 2.1) = R_ab   X∞ j=−∞ λjbj(x)   b∗ i(x)dx (from Eq. (2.2)) = λi Rb abi(x)b∗i(x)dx + 0 = λici (2.6)  Corollary 2.1 For an orthogonal base, if ci= 1 for all the i-th basis function,

we can simply obtain the transform coefficients as

λi=< f, bi> . (2.7)

Such a family of basis functions is known as orthonormal, and corresponding basis functions are called orthonormal basis functions. Any subset of orthog- onal base B also satisfies the property of orthogonality as expressed in Eq. (2.4). However, it may not be sufficient or complete for full reconstruction of the function f (x). The set of orthogonal (orthonormal) functions that is complete for every f (x) is known as complete orthogonal (orthonormal) base. The set of transform coefficients Λ = {λi| − ∞ < i < ∞} provides the

alternative description of f (x) and is known as its transform, while the space of these coefficients is termed as the transform space. Eq. (2.5) or (2.7) denotes the forward transform, while the expansion with transform coefficients (Eq. (2.2)) is termed as the inverse transform. In continuous space for i, Eq. (2.2) is expressed as

f (x) = Z ∞

i=−∞

λibi(x)di. (2.8)

One of the important facts about the orthonormal expansion of the func- tion is that it preserves the energy of the function as expressed in the following theorem.

Theorem 2.2 For an orthonormal expansion of the function as given in Eq. (2.2), the sum of squares of the transform coefficients remains the same as the sum of squares of functional values within its support (or energy of the function) as expressed in the following relation

∞ X i=−∞ λ2i = Z b a f2(x)dx. (2.9) Proof: Rb a f 2_{(x)dx =} Rb a f (x)f∗(x)dx, = R_abf (x) ∞ X i=−∞ λ∗ib∗i(x) ! dx, = ∞ X i=−∞ λ∗i Z b a f (x)b∗i(x)dx, = ∞ X i=−∞ λ∗ iλi, = ∞ X i=−∞ λ2 i. (2.10)  In the following text we examine a few examples of orthogonal transforms.