Deriche's Edge Detector : Advanced Image Processing

The approach is to approximate the Gaussian filters by recursive filters, which effectively allows  the filtering to be done with a fixed  number of coefficients regardless of the standard deviation of the Gaussian.

This has been done by Prof. Deriche using  a sum of one causal and one anti-causal filter. 


The Gaussian filter is very compute-intensive, as the number of operations per output pixel grows proportionally with .

 However, the IIR Gaussian filter and its derivatives recursively solve a difference equation which is independent of, so the number of operations per output pixel are fixed and not related to . 

The IIR equation is:

The IIR Gaussian filter processes each pixel horizontally and vertically. It is a separable filter; that means the filter can be applied in any order, i.e., horizontally first or vertically first.

So the implementation constitutes from 2 phase . It is possible to transpose the matrix and provide twice execution only horizontal or only vertical manner. But Currently i got an error of ‘use permutation instead of transpose’ . Then I implemented the both horizontal and vertical manner.

Prof. Deriche uses Infinite impulse response filter form; 

The filter optimizes the Canny criteria. As is evident from the preceding formula, the most effective filter is obtained when the value of omega approaches 0. Such filter then uses  the formula:

Some useful informations :

Coefficients : 

    

Formula  for first derivative ;

It’s just the difference between subsequent values and measures the rate of change of  the function.

ex: 

The formula for the 2nd derivative;

Simply takes into account the values both before and after the current value. 

 ex: 

The formula for the Laplace;

   

where 1st partial derivative  in x direction is defined as ; 

where 1st partial derivative  in y direction is defined as ; 

Gradient of an image is defined as ; 

CODE:

First Stage [Horizontal];

Second Stage [Vertical];