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I am reading the implementation of IIR Gaussian blur filter from this link.

In the source code, the coefficients are generated by the following code

/* Calcualte Gaussian Blur Filter Coefficiens
 *  alpha -> smooting gradient depends on sigma
 *  k = ((1-e^-alpha)^2)/(1+2*alpha*e^-alpha - e^-2alpha)
 *  a0 = k; a1 = k*(alpha-1)*e^-alpha; a2 = k*(alpha+1)*e^-alpha; 
 *  a3 = -k*e^(-2*alpha)
 *  b1 = -2*e^-alpha; b2 = e^(-2*alpha)
 */
 void calGaussianCoeff( float sigma,  float *a0, float *a1, float *a2, 
   float *a3, float *b1, float *b2, float *cprev, float *cnext)
{
  float alpha, lamma,  k; 
  // defensive check
  if (sigma < 0.5f)
     sigma = 0.5f;

  alpha = (float) exp((0.726)*(0.726)) / sigma;
  lamma = (float)exp(-alpha);
  *b2 = (float)exp(-2*alpha);
  k = (1-lamma)*(1-lamma)/(1+2*alpha*lamma- (*b2));
  *a0 = k;
  *a1 = k*(alpha-1)*lamma;
  *a2 = k*(alpha+1)*lamma;
  *a3 = -k* (*b2);
  *b1 = -2*lamma;
  *cprev = (*a0 + *a1)/(1+ *b1 + *b2);
  *cnext = (*a2 + *a3)/(1+ *b1 + *b2);
}

Does anyone know where the equation is from?

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2 Answers 2

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@Peter K.'s answer is right to the point. You can also have most proposed implementations side-by-side, with comparisons, in Pascal Getreuer, A Survey of Gaussian Convolution Algorithms, Image Processing On Line, 2013, with alternative source code.

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Did you try reading the references?

enter image description here

The paper by Deriche is very well-known

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    $\begingroup$ Thanks. From the implementation, I think the author used the algorithm proposed by Ian T. Yong because it operates input image horizontally first and operates previous result vertically. I will read the paper by Deriche carefully. I have another question about initial value of the output. If I use the input value as initial value of the output, artifacts will appear around the boundaries. $\endgroup$
    – Jogging Song
    Sep 4, 2015 at 9:28

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