What is the significance of Butterworth filter in image processing?

Question

Filters like mean,median,gaussian,bilateral and adaptive forms of these filters are used commonly in image processing but not Butterworth filter. The reasons I could think of are

a) Butterworth filtering can be done only in frequency domain(AFAIK there is no spatial filtering equivalent for Butterworth filter, correct me if I am wrong)

b) One of the main advantages of Butterworth filter is since there is no sharp cutoff frequency, there won't be any ripples, but since Gaussian also has this property there is no need for Butterworth.

Are there any specific applications Butterworth filter offers advantage over other filters? Do Butterworth have any significance in image processing?

Answer 1

Some important differences:

• In Image Processing (IP), there is no causality like in Signal Processing (SP), hence there is not a tradeoff between filter quality and sampling sequence.

• In IP, the FIR versions of SP are preferred instead of the IIR version (which are rare as you pointed). A possible relevant cause for this is FIR are designed as linear phase, unlike IIR which cannot be linear phase. Linear phase in FIR means the delay is constant (some pixels or fractions of pixels) while nonlinear phase in IIR means distorsion (like a blur) over the image toward the axis in which the filter is applied.

• In IP there is not frequency spectrum, you do not seek to have a cutoff at x Hz and a flat passband. In a Butterworth this is the main design focus.

• A butterworth or any other SP filter IIR version can be readily be extended to IP. The implementation is not a limitation.

• A FIR like the Gaussian and others, dont have ripples in their impulse response. Butterworth, Chebychev, eliptical and ALL other IIR filters have an infinite number of ripples! This is another way to see the distortion which are produced by them.

• The butterworth is ripple-less in their frequency response (the spectrum of the inpulse response) which we already agreed it do not have a direct sense in IP.

• IIR means infinite impulse response: hence if you apply a Butterworth, one convolution affects all other axis pixels.

• Nevertheless, there are frequency figures on IP. You can define and extend 2D noises distribution based on an spatial frequency, and thus define frequency cutoffs such as the SP case. But this normally solved with a FIR filter, without paying the cost of the distortion an IIR implies.

Which is the significance? Most of their good properties in SP are not applicable in IP, with a major demerit added.