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Relationships between input data, output data, and impulse response arrays for finite-area superposition; upper left corner justified array definition... Relationships between input data

7

SUPERPOSITION AND CONVOLUTION

In Chapter 1, superposition and convolution operations were derived for continuoustwo-dimensional image fields This chapter provides a derivation of these operationsfor discrete two-dimensional images Three types of superposition and convolutionoperators are defined: finite area, sampled image, and circulant area The finite-areaoperator is a linear filtering process performed on a discrete image data array Thesampled image operator is a discrete model of a continuous two-dimensional imagefiltering process The circulant area operator provides a basis for a computationallyefficient means of performing either finite-area or sampled image superposition andconvolution

7.1 FINITE-AREA SUPERPOSITION AND CONVOLUTION

Mathematical expressions for finite-area superposition and convolution are oped below for both series and vector-space formulations

devel-7.1.1 Finite-Area Superposition and Convolution: Series Formulation

Let denote an image array for n1, n2 = 1, 2, , N For notational simplicity,

all arrays in this chapter are assumed square In correspondence with Eq 1.2-6, the

weighted Dirac delta functions by the discrete sifting summation

The term

pro-duces an output image array

(7.1-3)

the sifting summation of Eq 7.1-1,

(7.1-4a)

or

(7.1-4b)

array of the linear operator and is written as

that in the general case, called finite area superposition, the impulse response array

Follow-ing this nomenclature, the finite area superposition operation is defined as

δ m( 1–n1+1,m2–n2+1)

10

The limits of the summation are

(7.1-7)

argu-ments, respectively Examination of the indices of the impulse response array at its

extreme positions indicates that M = N + L - 1, and hence the processed output array

Q is of larger dimension than the input array F Figure 7.1-1 illustrates the geometry

of finite-area superposition If the impulse response array H is spatially invariant,

the superposition operation reduces to the convolution operation

(7.1-8)

Figure 7.1-2 presents a graphical example of convolution with a impulseresponse array

Equation 7.1-6 expresses the finite-area superposition operation in left-justified

form in which the input and output arrays are aligned at their upper left corners It is

often notationally convenient to utilize a definition in which the output array is

cen-tered with respect to the input array This definition of cencen-tered superposition is

given by

FIGURE 7.1-1 Relationships between input data, output data, and impulse response arrays

for finite-area superposition; upper left corner justified array definition

summa-tion are

(7.1-10)

Figure 7.1-3 shows the spatial relationships between the arrays F, H, and Q c for

In digital computers and digital image processors, it is often convenient to restrictthe input and output arrays to be of the same dimension For such systems, Eq 7.1-9

FIGURE 7.1-2 Graphical example of finite-area convolution with a 3 × 3 impulse responsearray; upper left corner justified array definition

array is located on the border of the input array, the product computation of Eq.7.1-9 does not involve all of the elements of the impulse response array This situa-tion is illustrated in Figure 7.1-3, where the impulse response array is in the upperleft corner of the input array The input array pixels “missing” from the computationare shown crosshatched in Figure 7.1-3 Several methods have been proposed todeal with this border effect One method is to perform the computation of all of theimpulse response elements as if the missing pixels are of some constant value If the

constant value is zero, the result is called centered, zero padded superposition A

variant of this method is to regard the missing pixels to be mirror images of the inputarray pixels, as indicated in the lower left corner of Figure 7.1-3 In this case the

centered, reflected boundary superposition definition becomes

(7.1-11)

where the summation limits are

(7.1-12)

FIGURE 7.1-3 Relationships between input data, output data, and impulse response arrays

for finite-area superposition; centered array definition

In many implementations, the superposition computation is limited to the range

to zero In effect, the superposition operation is computed only when the impulseresponse array is fully embedded within the confines of the input array This region

is described by the dashed lines in Figure 7.1-3 This form of superposition is called

centered, zero boundary superposition.

If the impulse response array H is spatially invariant, the centered definition for

convolution becomes

(7.1-14)

is fundamental to many image processing algorithms (1) When the SGK is totallyembedded within the input data array, the general term of the centered convolutionoperation can be expressed explicitly as

(7.1-15)

impulse response arrays can be achieved by sequential convolutions with SGKs.The four different forms of superposition and convolution are each useful in var-ious image processing applications The upper left corner–justified definition isappropriate for computing the correlation function between two images The cen-tered, zero padded and centered, reflected boundary definitions are generallyemployed for image enhancement filtering Finally, the centered, zero boundary def-inition is used for the computation of spatial derivatives in edge detection In thisapplication, the derivatives are not meaningful in the border region

n' i

2 n– i

n i 2N n– i

2 j≤ ≤i N 1–

Figure 7.1-4 shows computer printouts of pixels in the upper left corner of aconvolved image for the four types of convolution boundary conditions In thisexample, the source image is constant of maximum value 1.0 The convolution

7.1.2 Finite-Area Superposition and Convolution: Vector-Space Formulation

vec-tor f and the vector q, respectively, the finite-area superposition operation

can be written as (2)

(7.1-16)

convenient to partition the superposition operator matrix D into submatrices of

(a) Upper left corner justified (b) Centered, zero boundary

(c) Centered, zero padded (d) Centered, reflected

The general nonzero term of D is then given by

(7.1-18)

Thus, it is observed that D is highly structured and quite sparse, with the center band

of submatrices containing stripes of zero-valued elements

If the impulse response is position invariant, the structure of D does not depend

(7.1-19)

As a result, the columns of D are shifted versions of the first column Under these

conditions, the finite-area superposition operator is known as the finite-area

convo-lution operator Figure 7.1-5a contains a notational example of the finite-area

array, and a (L = 3) impulse response array The integer pairs (i, j) at each

more clearly in the larger matrix depicted in Figure 7.l-5b In this example, M = 16,

FIGURE 7.1-5 Finite-area convolution operators: (a) general impulse array, M = 4, N = 2,

L = 3; (b) Gaussian-shaped impulse array, M = 16, N = 8, L = 9.

(b)

11 21 31 0

0 11 21 31

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

12 22 32 0

0 12 22 32

11 21 31 0

0 11 21 31 13

23 33 0

0 13 23 33

0 13 23 33

12 22 32 0

0 12 22 32 13 23 33 0

11 21 31

12 22 32

13 23 33

N = 8, L = 9, and the impulse response has a symmetrical Gaussian shape Note that

D is a 256 × 64 matrix in this example

Following the same technique as that leading to Eq 5.4-7, the matrix form of thesuperposition operator may be written as

(7.1-20)

If the impulse response is spatially invariant and is of separable form such that

(7.1-21)

responses, respectively, then

In vector form, the general finite-area superposition or convolution operator requires

operations if the zero-valued multiplications of D are avoided The separable

7.2 SAMPLED IMAGE SUPERPOSITION AND CONVOLUTION

Many applications in image processing require a discretization of the superpositionintegral relating the input and output continuous fields of a linear system For exam-ple, image blurring by an optical system, sampling with a finite-area aperture orimaging through atmospheric turbulence, may be modeled by the superposition inte-gral equation

(7.2-1a)

system model In this chapter, a tilde over a variable indicates that the spatial indices

of the variable are bipolar; that is, they range from negative to positive spatial limits

In this formulation, the impulse response may change form as a function of its fourindices: the input and output coordinates If the linear system is space invariant, theoutput image field may be described by the convolution integral

(7.2-1b)

For discrete processing, physical image sampling will be performed on the outputimage field Numerical representation of the integral must also be performed inorder to relate the physical samples of the output field to points on the input field.Numerical representation of a superposition or convolution integral is an impor-tant topic because improper representations may lead to gross modeling errors ornumerical instability in an image processing application Also, selection of a numer-ical representation algorithm usually has a significant impact on digital processingcomputational requirements

As a first step in the discretization of the superposition integral, the output image

(7.2-2)

for notational simplicity The effect of finite area sample pulses can easily be

represents the pulse shape of the sampling pulse The delta function may

be brought under the integral sign of the superposition integral of Eq 7.2-la to give

(7.2-3)

G˜ x y( , ) F˜(α β, )J˜ x y α β( , ; , ) αd dβ

∞ –

∞

∫

∞ –

∞

∫

∞ –

∞

∫

=

2J 1+( )×(2J 1+ )

∞

∫

∞ –

∞

∫

=

It should be noted that the physical sampling is performed on the observed image

spatial variables (x, y); physical sampling does not affect the dummy variables of

Truncation of the impulse response is equivalent to multiplying the impulse

zero elsewhere By the Fourier convolution theorem, the Fourier spectrum of G(x, y)

is equivalently convolved with the Fourier transform of V(x, y), which is a dimensional sinc function This distortion of the Fourier spectrum of G(x, y) results

two-in the two-introduction of high-spatial-frequency artifacts (a Gibbs phenomenon) at

shaped window, such as the Bartlett, Blackman, Hamming, or Hanning windows(3), which smooth the sharp cutoff effects of a rectangular window This step isespecially important for image restoration modeling because ill-conditioning of thesuperposition operator may lead to severe amplification of the truncation artifacts

In the next step of the discrete representation, the continuous ideal image array

is represented by mesh points on a rectangular grid of resolution and

an abstract numerical representation whose general term is described by

(7.2-6)

If the ultimate objective is to estimate the continuous ideal image field by cessing the physical observation samples, the mesh spacing should be fineenough to satisfy the Nyquist criterion for the ideal image That is, if the spectrum ofthe ideal image is bandlimited and the limits are known, the mesh spacing should beset at the corresponding Nyquist spacing Ideally, this will permit perfect interpola-

The continuous integration of Eq 7.2-5 can now be approximated by a discretesummation by employing a quadrature integration formula (4) The physical imagesamples may then be expressed as

where is a weighting coefficient for the particular quadrature formulaemployed Usually, a rectangular quadrature formula is used, and the weightingcoefficients are unity In any case, it is notationally convenient to lump the weight-ing coefficient and the impulse response function together so that

(7.2-8)Then,

(7.2-9)

Again, it should be noted that is not spatially discretized; the function is simplyevaluated at its appropriate spatial argument The limits of summation of Eq 7.2-9are

(7.2-10)

Figure 7.2-1 provides an example relating actual physical sample values

exam-ple, the mesh spacing is twice as large as the physical sample spacing In the figure,

FIGURE 7.2-1 Relationship of physical image samples to mesh points on an ideal image

field for numerical representation of a superposition integral

W ˜ k( 1,k2)

H˜ j(1∆S j, 2∆S k; 1∆I k, 2∆I) = W ˜ k( 1,k2)J˜ j(1∆S j, 2∆S k; 1∆I k, 2∆I)

G˜ j(1∆S j, 2 ∆S) F˜ k( 1∆I k, 2∆I )H˜ j(1∆S j, 2∆S k; 1∆I k, 2∆I)

∆I

–

the values of the impulse response function that are utilized in the summation of

Eq 7.2-9 are represented as dots

An important observation should be made about the discrete model of Eq 7.2-9for a sampled superposition integral; the physical area of the ideal image field containing mesh points contributing to physical image samples is larger

samples and mesh points The dimensions of the two image fields, as shown inFigure 7.2-2, are related by

(7.2-11)

to within an accuracy of one sample spacing

At this point in the discussion, a discrete and finite model for the sampled position integral has been obtained in which the physical samples

mathemati-cal superposition operation This discrete superposition is an approximation to tinuous superposition because of the truncation of the impulse response function

and quadrature integration The truncation approximation can, ofcourse, be made arbitrarily small by extending the bounds of definition of theimpulse response, but at the expense of large dimensionality Also, the quadratureintegration approximation can be improved by use of complicated formulas ofquadrature, but again the price paid is computational complexity It should be noted,however, that discrete superposition is a perfect approximation to continuous super-position if the spatial functions of Eq 7.2-1 are all bandlimited and the physical

FIGURE 7.2-2 Relationship between regions of physical samples and mesh points for

numerical representation of a superposition integral

sampling and numerical representation periods are selected to be the correspondingNyquist period (5).

It is often convenient to reformulate Eq 7.2-9 into vector-space form Toward

such that all indices are positive Let

(7.2-13)

Following the techniques outlined in Chapter 5, the vectors g and f may be formed

by column scanning the matrices G and F to obtain

H m( 1∆S m, 2∆S n; 1∆ I n, 2∆I) = H ˜ j( 1∆S j, 2∆S k; 1∆I k, 2∆I)

G m( 1 ∆S m, 2 ∆S) F n( 1∆I n, 2∆I ) H m( 1∆ S m, 2∆S n; 1∆I n, 2∆I)

The general term of B is defined as

(7.2-16)

descrip-tional simplicity, B is called the blur matrix of the superposition integral.

If the impulse response function is translation invariant such that

(7.2-17)

then the discrete superposition operation of Eq 7.2-13 becomes a discrete tion operation of the form

convolu-(7.2-18)

If the physical sample and quadrature mesh spacings are equal, the general term

of the blur matrix assumes the form

H m( 1∆S m, 2∆S n; 1∆I n, 2∆I) = H m( 1∆S n– 1∆I m, 2∆S n– 2∆I)

G m( 1∆S m, 2∆S) F n( 1∆I n, 2∆I )H m( 1∆S n– 1∆I m, 2∆S n– 2∆I)