Filtering using F-transform

Goal

This tutorial demonstrates to you how to use F-transform for image filtering. You will see:

• basic theory behind,
• illustration of different settings.

Fuzzy transform application

As I shown in previous tutorial, F-transform is a tool of fuzzy mathematics highly usable in image processing. Let me rewrite the formula using kernel \(g\) introduced before as well:

\[ F^0_{kl}=\frac{\sum_{x=0}^{2h+1}\sum_{y=0}^{2h+1} \iota_{kl}(x,y) g(x,y)}{\sum_{x=0}^{2h+1}\sum_{y=0}^{2h+1} g(x,y)}, \]

where \(\iota_{kl} \subset I\) centered to pixel \((k \cdot h,l \cdot h)\) and \(g\) is a kernel. More details can be found in related papers.

Code

/* Sample - Filtering
 * Target is to apply filtering using F-transform
 * on the image "input.png". Two different kernels
 * are used, where bigger radius (100 in this case)
 * means higher level of blurriness.
 *
 * Image "output1_filter.png" is created from "input.png"
 * using "kernel1" with radius 3.
 *
 * Image "output2_filter.png" is created from "input.png"
 * using "kernel2" with radius 100.
 *
 * Both kernels are created from linear function, using
 * linear interpolation (parameter ft:LINEAR).
 */
 
#include "opencv2/core.hpp"
#include "opencv2/highgui.hpp"
#include "opencv2/fuzzy.hpp"
 
using namespace std;
using namespace cv;
 
int main(void)
{
    // Input image
    Mat I = imread("input.png");
 
    // Kernel creation
    Mat kernel1, kernel2;
 
    ft::createKernel(ft::LINEAR, 3, kernel1, 3);
    ft::createKernel(ft::LINEAR, 100, kernel2, 3);
 
    // Filtering
    Mat output1, output2;
 
    ft::filter(I, kernel1, output1);
    ft::filter(I, kernel2, output2);
 
    // Save output
 
    imwrite("output1_filter.png", output1);
    imwrite("output2_filter.png", output2);
 
    return 0;
}

Explanation

Image filtering changes input in a defined way to enhance or simply change some concrete feature. Let me demonstrate some simple blur.

As a first step, we load input image.

// Input image
Mat I = imread("input.png");

Following the F-transform formula, we must specify a kernel.

// Kernel cretion
Mat kernel1, kernel2;
 
ft::createKernel(ft::LINEAR, 3, kernel1, 3);
ft::createKernel(ft::LINEAR, 100, kernel2, 3);

‍So now, we have two kernels that differ in radius. Bigger radius leads to bigger blur.

The filtering itself is applied as shown below.

// Filtering
Mat output1, output2;
 
ft::filter(I, kernel1, output1);
ft::filter(I, kernel2, output2);

Output images look as follows.

input, output1 (radius 3), output2 (radius 100)