Digital image warping

value of the unit triangle as it crosses the input while it is centered on the output position

x

0.00

0.50

1.00

1.50

2.00

150

90

150

050)(.50) + (78)(.50) = 114

(150)(.25) + (78)(.75) = 96

78

(78)(.50) + (90)(.50) = 84

(78)(.25) + (90)(.75) = 87

90

In general, we can always interpolate the input data as long as the centered convolu-

tion kernel passes through zero at all the input sample positions but one. Thus, when the

kernel is situated on an input sample it will use that data alone to determine the output

value for that point. The unit triangle impulse response function complies with this inter-

polation condition: it has unity value at the center from which it linearly falls to zero over

i single pixel interval.

The Gaussian function shown in Fig. 2.4a does not satisfy this interpolation condi-

tion. Consequently, convolving with this kernel yields an approximating function that

passes near, but not necessarily through, the input data. The extent to which the impulse

response function blurs the input data is determined by its region of support. Wider ker-

nels can potentially cause more blurring. In order to normalize the convolution, the scale

factor reflecting the kemel's region of support is incorporated directly into the kernel.

Therefore, broader kernels are also shorter, i.e., scaled down in amplitude.

(a) (b)

Figure 2.4: Convolution with a Gaussian filter. (a) Input; (b) Output.