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The forward unitary transform of the cer-image array results in a transformed image array as defined by Digital Image Processing: PIKS Inside, Third Edition... The basis functions of the

8.1 GENERAL UNITARY TRANSFORMS

A unitary transform is a specific type of linear transformation in which the basic

lin-ear operation of Eq 5.4-1 is exactly invertible and the operator kernel satisfies tain orthogonality conditions (1,2) The forward unitary transform of the

cer-image array results in a transformed image array as defined by

Digital Image Processing: PIKS Inside, Third Edition William K Pratt

Copyright © 2001 John Wiley & Sons, Inc.ISBNs: 0-471-37407-5 (Hardback); 0-471-22132-5 (Electronic)

where represents the forward transform kernel A reverse orinverse transformation provides a mapping from the transform domain to the imagespace as given by

where the kernel subscripts indicate row and column one-dimensional transformoperations A separable two-dimensional unitary transform can be computed in twosteps First, a one-dimensional transform is taken along each column of the image,yielding

GENERAL UNITARY TRANSFORMS 187

Unitary transforms can conveniently be expressed in vector-space form (3) Let F and f denote the matrix and vector representations of an image array, and let and

be the matrix and vector forms of the transformed image Then, the sional unitary transform written in vector form is given by

(8.1-10)

and A is said to be a unitary matrix A real unitary matrix is called an orthogonal

matrix For such a matrix,

where and denote rows m1 and m2 of the unitary matrices BC and

BR, respectively The vector outer products of Eq 8.1-14 form a series of matrices,

called basis matrices, that provide matrix decompositions of the image matrix F or

its unitary transformation F

There are several ways in which a unitary transformation may be viewed Animage transformation can be interpreted as a decomposition of the image data into ageneralized two-dimensional spectrum (4) Each spectral component in the trans-form domain corresponds to the amount of energy of the spectral function within theoriginal image In this context, the concept of frequency may now be generalized toinclude transformations by functions other than sine and cosine waveforms Thistype of generalized spectral analysis is useful in the investigation of specific decom-positions that are best suited for particular classes of images Another way to visual-ize an image transformation is to consider the transformation as a multidimensionalrotation of coordinates One of the major properties of a unitary transformation isthat measure is preserved For example, the mean-square difference between twoimages is equal to the mean-square difference between the unitary transforms of theimages A third approach to the visualization of image transformation is to consider

Eq 8.1-2 as a means of synthesizing an image with a set of two-dimensional matical functions for a fixed transform domain coordinate

mathe- In this interpretation, the kernel is called a sional basis function and the transform coefficient is the amplitude of thebasis function required in the synthesis of the image

two-dimen-In the remainder of this chapter, to simplify the analysis of two-dimensional

uni-tary transforms, all image arrays are considered square of dimension N

Further-more, when expressing transformation operations in series form, as in Eqs 8.1-1and 8.1-2, the indices are renumbered and renamed Thus the input image array is

denoted by F(j, k) for j, k = 0, 1, 2, , N - 1, and the transformed image array is

rep-resented by F(u, v) for u, v = 0, 1, 2, , N - 1 With these definitions, the forward

uni-tary transform becomes

The indices (u, v) are called the spatial frequencies of the transformation in analogy

with the continuous Fourier transform It should be noted that Eq 8.2-1 is not versally accepted by all authors; some prefer to place all scaling constants in theinverse transform equation, while still others employ a reversal in the sign of thekernels

uni-Because the transform kernels are separable and symmetric, the two dimensionaltransforms can be computed as sequential row and column one-dimensional trans-forms The basis functions of the transform are complex exponentials that may bedecomposed into sine and cosine components The resulting Fourier transform pairsthen become

(8.2-2a)

(8.2-2b)

Figure 8.2-1 shows plots of the sine and cosine components of the one-dimensional

Fourier basis functions for N = 16 It should be observed that the basis functions are

a rough approximation to continuous sinusoids only for low frequencies; in fact, the

highest-frequency basis function is a square wave Also, there are obvious dancies between the sine and cosine components.

redun-The Fourier transform plane possesses many interesting structural properties.The spectral component at the origin of the Fourier domain

(8.2-3)

is equal to N times the spatial average of the image plane Making the substitutions

, in Eq 8.2-1, where m and n are constants, results in

FIGURE 8.2-1 Fourier transform basis functions, N = 16.

FOURIER TRANSFORM 191

(8.2-4)

For all integer values of m and n, the second exponential term of Eq 8.2-5 assumes

a value of unity, and the transform domain is found to be periodic Thus, as shown in

valid, the field must be periodic Thus, as shown in Figure 8.2-2b, the original image

must be considered to be periodic horizontally and vertically The right side of theimage therefore abuts the left side, and the top and bottom of the image are adjacent.Spatial frequencies along the coordinate axes of the transform plane arise from thesetransitions

If the image array represents a luminance field, will be a real positivefunction However, its Fourier transform will, in general, be complex Because thetransform domain contains components, the real and imaginary, or phase andmagnitude components, of each coefficient, it might be thought that the Fouriertransformation causes an increase in dimensionality This, however, is not the casebecause exhibits a property of conjugate symmetry From Eq 8.2-4, with m and n set to integer values, conjugation yields

FIGURE 8.2-2 Periodic image and Fourier transform arrays.

one-Figure 8.2-4 shows a monochrome test image and various versions of its Fourier

transform, as computed by Eq 8.2-1a, where the test image has been scaled over

unit range Because the dynamic range of transform components ismuch larger than the exposure range of photographic film, it is necessary to com-press the coefficient values to produce a useful display Amplitude compression to aunit range display array can be obtained by clipping large-magnitude valuesaccording to the relation

FIGURE 8.2-3 Fourier transform frequency domain.

FIGURE 8.2-4. Fourier transform of the smpte_girl_luma image

(a) Original (b) Clipped magnitude, nonordered

(c) Log magnitude, nonordered (d) Log magnitude, ordered

=

where a and b are scaling constants Figure 8.2-4b is a clipped magnitude display of the magnitude of the Fourier transform coefficients Figure 8.2-4c is a logarithmic display for a = 1.0 and b = 100.0.

In mathematical operations with continuous signals, the origin of the transformdomain is usually at its geometric center Similarly, the Fraunhofer diffraction pat-tern of a photographic transparency of transmittance produced by a coher-ent optical system has its zero-frequency term at the center of its display Acomputer-generated two-dimensional discrete Fourier transform with its origin at itscenter can be produced by a simple reordering of its transform coefficients Alterna-

tively, the quadrants of the Fourier transform, as computed by Eq 8.2-la, can be

reordered automatically by multiplying the image function by the factor

prior to the Fourier transformation The proof of this assertion follows from Eq.8.2-4 with the substitution Then, by the identity

(8.2-10)

Eq 8.2-5 can be expressed as

(8.2-11)

Figure 8.2-4d contains a log magnitude display of the reordered Fourier

compo-nents The conjugate symmetry in the Fourier domain is readily apparent from thephotograph

The Fourier transform written in series form in Eq 8.2-1 may be redefined invector-space form as

(8.2-12a)

(8.2-12b)

where f and are vectors obtained by column scanning the matrices F and F,

respectively The transformation matrix A can be written in direct product form as

(8.2-13)

F x y( , )

1( )j k+

COSINE, SINE, AND HARTLEY TRANSFORMS 195

where

(8.2-14)

with As a result of the direct product decomposition of A, the

image matrix and transformed image matrix are related by

sequence of N values requires on the order of complex multiply and add tions A fast Fourier transform (FFT) requires on the order of operations.For large images the computational savings are substantial The original FFT algo-rithms were limited to images whose dimensions are a power of 2 (e.g.,

opera-) Modern algorithms exist for less restrictive image dimensions.Although the Fourier transform possesses many desirable analytic properties, ithas a major drawback: Complex, rather than real number computations arenecessary Also, for image coding it does not provide as efficient image energycompaction as other transforms

8.3 COSINE, SINE, AND HARTLEY TRANSFORMS

The cosine, sine, and Hartley transforms are unitary transforms that utilizesinusoidal basis functions, as does the Fourier transform The cosine and sinetransforms are not simply the cosine and sine parts of the Fourier transform In fact,the cosine and sine parts of the Fourier transform, individually, are not orthogonalfunctions The Hartley transform jointly utilizes sine and cosine basis functions, butits coefficients are real numbers, as contrasted with the Fourier transform whosecoefficients are, in general, complex numbers

8.3.1 Cosine Transform

The cosine transform, discovered by Ahmed et al (12), has found wide application

in transform image coding In fact, it is the foundation of the JPEG standard (13) forstill image coding and the MPEG standard for the coding of moving images (14).The forward cosine transform is defined as (12)

(8.3-1a)

(8.3-1b)

where and for w = 1, 2, , N – 1 It has been observed

that the basis functions of the cosine transform are actually a class of discrete byshev polynomials (12)

Che-Figure 8.3-1 is a plot of the cosine transform basis functions for N = 16 A graph of the cosine transform of the test image of Figure 8.2-4a is shown in Figure 8.3-2a The origin is placed in the upper left corner of the picture, consistent with

photo-matrix notation It should be observed that as with the Fourier transform, the imageenergy tends to concentrate toward the lower spatial frequencies

The cosine transform of a image can be computed by reflecting the imageabout its edges to obtain a array, taking the FFT of the array and thenextracting the real parts of the Fourier transform (15) Algorithms also exist for thedirect computation of each row or column of Eq 8.3-1 with on the order of

real arithmetic operations (12,16)

8.3.2 Sine Transform

The sine transform, introduced by Jain (17), as a fast algorithmic substitute for the

Karhunen–Loeve transform of a Markov process is defined in one-dimensional form

by the basis functions

+( )

+( )

+( )

+( )

COSINE, SINE, AND HARTLEY TRANSFORMS 197

Eq 8.3-2, inserted as the elements of a unitary matrix A, diagonalize the matrix T in

the sense that

(8.3-4)

Matrix D is a diagonal matrix composed of the terms

(8.3-5)

for k = 1, 2, , N Jain (17) has shown that the cosine and sine transforms are

interre-lated in that they diagonalize a family of tridiagonal matrices

FIGURE 8.3-2 Cosine, sine, and Hartley transforms of the smpte_girl_luma image,

log magnitude displays

-=

COSINE, SINE, AND HARTLEY TRANSFORMS 199

The two-dimensional sine transform is defined as

(8.3-6)

Its inverse is of identical form

Sine transform basis functions are plotted in Figure 8.3-3 for N = 15 Figure 8.3-2b is a photograph of the sine transform of the test image The sine transform

can also be computed directly from Eq 8.3-10, or efficiently with a Fourier form algorithm (17)

trans-FIGURE 8.3-3 Sine transform basis functions, N = 15.

8.3.3 Hartley Transform

Bracewell (19,20) has proposed a discrete real-valued unitary transform, called the

Hartley transform, as a substitute for the Fourier transform in many filtering

appli-cations The name derives from the continuous integral version introduced by ley in 1942 (21) The discrete two-dimensional Hartley transform is defined by thetransform pair

structural properties of the discrete Fourier transform (20) Figure 8.3-2c is a

photo-graph of the Hartley transform of the test image

The Hartley transform can be computed efficiently by a FFT-like algorithm (20).The choice between the Fourier and Hartley transforms for a given application isusually based on computational efficiency In some computing structures, the Hart-ley transform may be more efficiently computed, while in other computing environ-ments, the Fourier transform may be computationally superior

8.4 HADAMARD, HAAR, AND DAUBECHIES TRANSFORMS

The Hadamard, Haar, and Daubechies transforms are related members of a family ofnonsinusoidal transforms

8.4.1 Hadamard Transform

The Hadamard transform (22,23) is based on the Hadamard matrix (24), which is a

square array of plus and minus 1s whose rows and columns are orthogonal A malized Hadamard matrix satisfies the relation

HADAMARD, HAAR, AND DAUBECHIES TRANSFORMS 201

(8.4-2)

It is known that if a Hadamard matrix of size N exists (N > 2), then N = 0 modulo 4 (22) The existence of a Hadamard matrix for every value of N satisfying this

requirement has not been shown, but constructions are available for nearly all

per-missible values of N up to 200 The simplest construction is for a Hadamard matrix

of size N = 2n, where n is an integer In this case, if is a Hadamard matrix of size

N, the matrix

(8.4-3)

is a Hadamard matrix of size 2N Figure 8.4-1 shows Hadamard matrices of size 4

and 8 obtained by the construction of Eq 8.4-3

Harmuth (25) has suggested a frequency interpretation for the Hadamard matrixgenerated from the core matrix of Eq 8.4-3; the number of sign changes along each

row of the Hadamard matrix divided by 2 is called the sequency of the row It is

pos-sible to construct a Hadamard matrix of order whose number of sign

changes per row increases from 0 to N – 1 This attribute is called the sequency property of the unitary matrix.

FIGURE 8.4-1 Nonordered Hadamard matrices of size 4 and 8.

H2

12

The rows of the Hadamard matrix of Eq 8.4-3 can be considered to be samples

of rectangular waves with a subperiod of 1/N units These continuous functions are called Walsh functions (26) In this context, the Hadamard matrix merely performs

the decomposition of a function by a set of rectangular waveforms rather than thesine–cosine waveforms with the Fourier transform A series formulation exists forthe Hadamard transform (23)

Hadamard transform basis functions for the ordered transform with N = 16 are

shown in Figure 8.4-2 The ordered Hadamard transform of the test image in shown

in Figure 8.4-3a

FIGURE 8.4-2 Hadamard transform basis functions, N = 16.