Calculating offset of linear gaussian blur based on variable amount of weights - glsl

Question

I've recently been looking into optimising a gaussian blur shader by using a linear sampling method instead of a discrete method.
I read a very informative article:
Efficient Gaussian Blur With Linear Sampling
In case of such a merge of two texels we have to adjust the coordinates that the distance of the determined coordinate from the texel #1 center should be equal to the weight of texel #2 divided by the sum of the two weights. In the same style, the distance of the determined coordinate from the texel #2 center should be equal to the weight of texel #1 divided by the sum of the two weights.
While I understand the logic behind this, I'm not sure how they arrived at the figures for the offset with the given weights. Would anyone be kind enough to shed more light on this for me and to also explain how, given uniform weight variables we could calculate correct offsets?
Regarding non hard coded offsets, I found another post which recommended a method of calculating the offsets, however no solution was posted for a variable amount of samples. How could I achieve that?
vec2 offsets[3];
offsets[0] = vec2(0.0, 0.0);
offsets[1] = vec2(dFdx(gl_TexCoord[0].s), dFdy(gl_TexCoord[0].t));
offsets[2] = offsets[1] + offsets[1];

Answer 1

I just came across the same article, and found it tremendously useful as well. The formula to calculate the weights and offsets is given in it:
(source: rastergrid.com)
The author arrived at the weights by using the 12th row in the pascal triangle. So for example the second offset is calculated by:
1.3846153846 = (1 * 792 + 2 * 495) / (792 + 495)
The second weight is calculated by:
0.1945945946 = (792 + 495) / 4070
I'm not sure what you mean by calculating the offsets given uniform weight variables but if it's of help I've included a C++ program at the end of this post that outputs the offsets and weights for an arbitrary row in the pascal triangle.
If I understand your question about non hardcoded offsets, then you want to be able to calculate the offsets on the fly in GLSL? You could do that by porting the program below, but you'll still need to hardcode the binomial coefficients, or calculate those on the fly as well. However, that will be expensive since it will have to be done for every pixel. I think a better alternative is to precalculate the offsets and weights in C (or whatever programming language you're using), and then bind them to a uniform array value in GLSL. Here's the GLSL snippet for what I mean:
uniform float offset[5];
uniform float weight[5];"
uniform int numOffsets;
You'll want to replace "5" with the maximum number of offsets/weights you plan to use, and set numOffsets to the number you're using for a particular operation.
Here's the program that outputs the weights and offsets. "coeffs" should be replaced with the binomial coefficients of the row you want in the pascal table. The one included here is from the 22nd row
#include <iostream>
#include <vector>
using namespace std;
int main(int argc, char* argv[])
{
float coeffs[] = { 705432, 646646, 497420, 319770, 170544, 74613, 26334, 7315, 1540, 231 };
double total = coeffs[0];
for (int i = 1; i < sizeof(coeffs) / sizeof(float); i++)
{
total += 2 * coeffs[i];
}
vector<float> offsets;
vector<float> weights;
offsets.push_back(0);
weights.push_back(coeffs[0] / total);
for (int i = 1; i <= (sizeof(coeffs) / sizeof(float) - 1) / 2; i++)
{
int index = (i - 1) * 2 + 1;
float weight = coeffs[index] + coeffs[index + 1];
offsets.push_back((coeffs[index] * index + coeffs[index + 1] * (index + 1)) / weight);
weights.push_back(weight / total);
}
for (int i = 0; i < offsets.size(); i++)
{
cout << offsets[i] << ", ";
}
cout << "\n";
for (int i = 0; i < weights.size(); i++)
{
cout << weights[i] << ", ";
}
cout << "\n";
}