Digital image warping

where x may continue to be a continuous variable, but . now takes on only integer

values. In practice, we use the discrete convolution in Eq. (2.1.12) to compute the output

for our discrete input f (x) and impulse response h (x) at only a limited set of values for

If the impulse response is itself an impulse, then the filter is ideal and the input will

be umampered at the output. That is, the convolution integral in Eq. (2.1.11) reduces to

the sifdng integral in Eq. (2.1.10) with h(x) being replaced by (x). In general, though,

the impulse response extends over neighboring samples; thus several scaled values may

overlap. When these are added together, the series of sums forms the new filtered signal

values. Thus, the output of any linear, space-invariant filter is related to its input by con-

volution.

Convolution can best be understood graphically. For instance, consider the samples

shown in Fig. 2.3a. Each sample is treated as an impulse by the filter. Since the filter is

linear and space-invariant, the input samples are replaced with properly scaled impulse

response functions. In Fig. 2.3b, a triangular impulse response is used to generate the

output signal. Note that the impulse responses are depicted as thin lines, and the output

(summation of scaled and superpositioned triangles) is drawn in boldface. The reader

will notice that this choice for the impulse response is tantamount to linear interpolation.

Although the impulse response function can take on many different forms, we shall gen-

erally be interested in symmetric kernels of finite extent. Various kernels useful for

image reconstruction are discussed in Chapter 5.

(a) (b)

ß It is apparent from this example that convolution is useful to derive continuous

functions from a set of discrete samples. This process, known as reconstruction, is fun-

damental to image warping because it is often necessary to determine image values at

noninteger positions, i.e., locations for which no input was supplied. As an example,

consider the problem of magnification. Given a unit triangle function for the impulse

response, the output g (x) for the input f (x) is derived below in Table 2.1. The table uses

a scale factor of four, thereby accounting for the .25 increments used to index the input.

Note that f (x) is only supplied for integer values of x, and the interpolation makes use of

the two adjacent input values. The weights applied to the input are derived from the