https://fine-arts-museum.academia.edu/FrederikLeen

Filtering in the spatial domain is possible and routinely implemented. See eg. the median filters (that have no equivalence in the frequency domain due to their non-linear nature).

What I'm not happy with: the characterization IIR. In image processing, filters in the spatial domain are typically of the FIR type and quite small (3×3, 5×5). Thus the implementation of a 'Butterworth' filter in image processing is hardly feasible. Butterworth is good for 1D (values over time) filtering. In image processing, most focus is on 2D filtering. Although, when it comes to video processing, the 3rd dimension time re-enters.

Hope this helps

Thank you, Alfredo.

At first, I'm sorry I'm not familiar with MATLAB.

As far as I understand, the function ftrans2 is to transform a 1D FIR filter to a 2D FIR filter, not for a IIR filter.

To Wolfgang Reichel:

Thank you for a great hint.

I thought I must use the value of (x-1,y-1) to calculate the value of (x,y).

I will rethink and make a simulation by extending 1D IIR filter to 2D.

To U. Dreher:

Thank you for an answer.

Yes, the 1D Butterworth filter is easy in spatial domain.

But its 2D version seems very difficult.

Oh, I think there is a slight misunderstanding: my statement about the application of Butterworth filters was meant to cover the time domain - explicitly not image processing!

I can only imagine ONE application where an IIR filter could make sense in image processing: when using a line sensor where the 'scene' moves underneath. But even in this application I cannot imagine any useful application of IIR filtering orthogonal to the direction of movement.

IIRs would give different output depending on the direction of filter movement (left-right resp. top-down and vice versa) - being neither cumulative nor associative.

To U. Dreher:

Thank you for a helpful comment. 

yes, it is possible inn spatial domain, with some drawbacks. some high value will be excluded while implementation in spatial domain, and effect of factored transfer function can not be achieved while using in spatial domain.

Thank you, Shahid Rasheed.

The results of my study are as follows;

1) I could find an algorithm for 1D Butterworth filtering in the spatial domain, i.e. using convolution.

2) But the 2D algorithm is still unknown.

3) For a practical application, I developed a software using FFT.

Yes, it is possible to apply (non-causal recursive) IIR filters in multiple dimensions (e.g. for image and video processing). FIR filters are more common, but IIR filters are more efficient when long impulse responses are required. There are some additional complications with IIR filters (e.g. initialization and decomposition into forward and backward parts) but these issues can be dealt with.      

Thank you, Hugh Lachlan Kennedy.

Is there any published literature explaining the technique?

I just wrote a paper on this. I'll upload the draft soon and send you a link.

Thank you very much, Hugh Lachlan Kennedy.

Done! See ...

Article Optimal digital design of steerable differentiators with the...

Thank you very much, Hugh Lachlan Kennedy.

[1] M. M. Daniel and A. S. Willsky, “Efficient implementations of 2-D noncausal IIR filters,” IEEE Trans. Circuits Syst. II Analog Digit. Signal Process., vol. 44, no. 7, pp. 549–563, Jul. 1997.

[2] D. Gorinevsky and S. Boyd, “Optimization-based design and implementation of multidimensional zero-phase IIR filters,” IEEE Trans. Circuits Syst. I Regul. Pap., vol. 53, no. 2, pp. 372–383, Feb. 2006.

Thank you very much,  Xiangguo Li .