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185 8 UNITARY TRANSFORMS Two-dimensional unitary transforms have found two major applications in image processing. Transforms have been utilized to extract features from images. For example, with the Fourier transform, the average value or dc term is proportional to the average image amplitude, and the high-frequency terms (ac term) give an indica- tion of the amplitude and orientation of edges within an image. Dimensionality reduction in computation is a second image processing application. Stated simply, those transform coefficients that are small may be excluded from processing opera- tions, such as filtering, without much loss in processing accuracy. Another applica- tion in the field of image coding is transform image coding, in which a bandwidth reduction is achieved by discarding or grossly quantizing low-magnitude transform coefficients. In this chapter we consider the properties of unitary transforms com- monly used in image processing. 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 cer- tain orthogonality conditions (1,2). The forward unitary transform of the image array results in a transformed image array as defined by (8.1-1) N 1 N 2 × Fn 1 n 2 ,() N 1 N 2 × F m 1 m 2 ,() Fn 1 n 2 ,()An 1 n 2 m 1 m 2 ,;,() n 2 1 = N 2 ∑ n 1 1 = N 1 ∑ = 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) 186 UNITARY TRANSFORMS where represents the forward transform kernel. A reverse or inverse transformation provides a mapping from the transform domain to the image space as given by (8.1-2) where denotes the inverse transform kernel. The transformation is unitary if the following orthonormality conditions are met: (8.1-3a) (8.1-3b) (8.1-3c) (8.1-3c) The transformation is said to be separable if its kernels can be written in the form (8.1-4a) (8.1-4b) where the kernel subscripts indicate row and column one-dimensional transform operations. A separable two-dimensional unitary transform can be computed in two steps. First, a one-dimensional transform is taken along each column of the image, yielding (8.1-5) Next, a second one-dimensional unitary transform is taken along each row of , giving (8.1-6) An 1 n 2 m 1 m 2 ,;,() Fn 1 n 2 ,() F m 1 m 2 ,()Bn 1 n 2 m 1 m 2 ,;,() m 2 1 = N 2 ∑ m 1 1 = N 1 ∑ = Bn 1 n 2 m 1 m 2 ,;,() An 1 n 2 m 1 m 2 ,;,()A ∗ j 1 j 2 m 1 m 2 ,;,() m 2 ∑ m 1 ∑ δ n 1 j 1 – n 2 j 2 –,()= Bn 1 n 2 m 1 m 2 ,;,()B ∗ j 1 j 2 m 1 m 2 ,;,() m 2 ∑ m 1 ∑ δ n 1 j 1 – n 2 j 2 –,()= An 1 n 2 m 1 m 2 ,;,()A ∗ n 1 n 2 k 1 k 2 ,;,() n 2 ∑ n 1 ∑ δ m 1 k 1 – m 2 k 2 –,()= Bn 1 n 2 m 1 m 2 ,;,()B ∗ n 1 n 2 k 1 k 2 ,;,() n 2 ∑ n 1 ∑ δ m 1 k 1 – m 2 k 2 –,()= An 1 n 2 m 1 m 2 ,;,()A C n 1 m 1 ,()A R n 2 m 2 ,()= Bn 1 n 2 m 1 m 2 ,;,()B C n 1 m 1 ,()B R n 2 m 2 ,()= Pm 1 n 2 ,() Fn 1 n 2 ,()A C n 1 m 1 ,() n 1 1 = N 1 ∑ = Pm 1 n 2 ,() F m 1 m 2 ,() Pm 1 n 2 ,()A R n 2 m 2 ,() n 2 1 = N 2 ∑ = 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 two-dimen- sional unitary transform written in vector form is given by (8.1-7) where A is the forward transformation matrix. The reverse transform is (8.1-8) where B represents the inverse transformation matrix. It is obvious then that (8.1-9) For a unitary transformation, the matrix inverse 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, (8.1-11) If the transform kernels are separable such that (8.1-12) where and are row and column unitary transform matrices, then the trans- formed image matrix can be obtained from the image matrix by (8.1-13a) The inverse transformation is given by (8.1-13b) F f ff f f ff f Af= fBf ff f= BA 1 – = A 1 – A ∗ T = A 1 – A T = AA C A R ⊗= A R A C F A C FA R T = F B C F B R T = 188 UNITARY TRANSFORMS where and . Separable unitary transforms can also be expressed in a hybrid series–vector space form as a sum of vector outer products. Let and represent rows n 1 and n 2 of the unitary matrices A R and A R , respectively. Then, it is easily verified that (8.1-14a) Similarly, (8.1-14b) where and denote rows m 1 and m 2 of the unitary matrices B C and B R , 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. An image transformation can be interpreted as a decomposition of the image data into a generalized two-dimensional spectrum (4). Each spectral component in the trans- form domain corresponds to the amount of energy of the spectral function within the original image. In this context, the concept of frequency may now be generalized to include transformations by functions other than sine and cosine waveforms. This type 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 multidimensional rotation of coordinates. One of the major properties of a unitary transformation is that measure is preserved. For example, the mean-square difference between two images is equal to the mean-square difference between the unitary transforms of the images. 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 mathe- matical functions for a fixed transform domain coordinate . In this interpretation, the kernel is called a two-dimen- sional basis function and the transform coefficient is the amplitude of the basis function required in the synthesis of the image. 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. 8.1-1 and 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 B C A C 1 – =B R A R 1 – = a C n 1 () a R n 2 () F Fn 1 n 2 ,()a C n 1 ()a R T n 2 () n 2 1 = N 2 ∑ n 1 1 = N 1 ∑ = F F m 1 m 2 ,()b C m 1 ()b R T m 2 () m 2 1 = N 2 ∑ m 1 1 = N 1 ∑ = b C m 1 () b R m 2 () Bn 1 n 2 m 1 m 2 ,;,() m 1 m 2 ,() Bn 1 n 2 m 1 m 2 ,;,() F m 1 m 2 ,() FOURIER TRANSFORM 189 (8.1-15a) and the inverse transform is (8.1-15b) 8.2. FOURIER TRANSFORM The discrete two-dimensional Fourier transform of an image array is defined in series form as (5–10) (8.2-1a) where , and the discrete inverse transform is given by (8.2-1b) 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 uni- versally accepted by all authors; some prefer to place all scaling constants in the inverse transform equation, while still others employ a reversal in the sign of the kernels. Because the transform kernels are separable and symmetric, the two dimensional transforms can be computed as sequential row and column one-dimensional trans- forms. The basis functions of the transform are complex exponentials that may be decomposed into sine and cosine components. The resulting Fourier transform pairs then 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 F uv,() Fjk,()Ajkuv,;,() k 0 = N 1 – ∑ j 0 = N 1 – ∑ = Fjk,() F uv,()Bjkuv,;,() v 0 = N 1 – ∑ u 0 = N 1 – ∑ = F uv,() 1 N ---- Fjk,() 2πi– N ----------- uj vk+()    exp k 0 = N 1 – ∑ j 0 = N 1 – ∑ = i 1–= Fjk,() 1 N ---- F uv,() 2πi N -------- uj vk+()    exp v 0 = N 1 – ∑ u 0 = N 1 – ∑ = Ajk uv,;,() 2πi– N ----------- uj vk+()    exp 2π N ------ uj vk+()    cos i 2π N ------ uj vk+()    sin–== Bjkuv,;,() 2πi N -------- uj vk+()    exp 2π N ------ uj vk+()    cos i 2π N ------ uj vk+()    sin+== 190 UNITARY TRANSFORMS highest-frequency basis function is a square wave. Also, there are obvious redun- dancies between the sine and cosine components. 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. F 00,() 1 N ---- Fjk,() k 0 = N 1 – ∑ j 0 = N 1 – ∑ = uumN+= vvnN+= 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 Figure 8.2-2a, (8.2-5) for . The two-dimensional Fourier transform of an image is essentially a Fourier series representation of a two-dimensional field. For the Fourier series representation to be 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 the image 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 these transitions. If the image array represents a luminance field, will be a real positive function. However, its Fourier transform will, in general, be complex. Because the transform domain contains components, the real and imaginary, or phase and magnitude components, of each coefficient, it might be thought that the Fourier transformation causes an increase in dimensionality. This, however, is not the case because 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. F umN+ vnN+,() 1 N ---- Fjk,() 2πi– N ----------- uj vk+()    exp 2πi– mj nk+(){}exp k 0 = N 1 – ∑ j 0 = N 1 – ∑ = F umN+ vnN+,()F uv,()= mn, 012…,±,±,= Fjk,() 2N 2 F uv,() 192 UNITARY TRANSFORMS (8.2-6) By the substitution and it can be shown that (8.2-7) for . As a result of the conjugate symmetry property, almost one- half of the transform domain samples are redundant; that is, they can be generated from other transform samples. Figure 8.2-3 shows the transform plane with a set of redundant components crosshatched. It is possible, of course, to choose the left half- plane samples rather than the upper plane samples as the nonredundant set. 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 is much 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 a unit range display array can be obtained by clipping large-magnitude values according to the relation FIGURE 8.2-3. Fourier transform frequency domain. F * umN+ vnN+,() 1 N ---- Fjk,() 2πi– N ----------- uj vk+()    exp k 0 = N 1 – ∑ j 0 = N 1 – ∑ = uu–= vv–= F uv,()F * u– mN+ v– nN+,()= n 012…,±,±,= 0.0 Fjk,()1.0≤≤ D uv,() FOURIER TRANSFORM 193 if (8.2-8a) if (8.2-8b) where is the clipping factor and is the maximum coefficient magnitude. Another form of amplitude compression is to take the logarithm of each component as given by (8.2-9) 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 D uv,() 1.0 F uv,() c F max ---------------------      = F uv,()c F max ≥ F uv,()c F max < 0.0 c< 1.0≤ F max D uv,() abF uv,()+{}log abF max +{}log -------------------------------------------------= 194 UNITARY TRANSFORMS 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 transform domain 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. A computer-generated two-dimensional discrete Fourier transform with its origin at its center 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 the photograph. The Fourier transform written in series form in Eq. 8.2-1 may be redefined in vector-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) Fxy,() 1–() jk + mn 1 2 --- == iπ jk+(){}exp 1–() jk + = F uN2⁄+ vN2⁄+,() 1 N ---- Fjk,()1–() jk + 2πi– N ----------- uj vk+()    exp k 0 = N 1 – ∑ j 0 = N 1 – ∑ = f ff f Af= fA ∗ T f ff f= f ff f AA C A R ⊗=