Let us briefly show some application of the normalized convolution as means of reconstruction an image from a set of its samples taken at irregular and random positions on the 2-D lattice.
We present results on the test image ``Lena'' (see Figure 2 for the original image).
These results were obtained using the Knutsson and Westin algorithm implemented by Mr Jeffrey Ng and Dr Anil Bharath from the Department of Bioengineering at Imperial College in London (U.K.).
The original image is randomly sampled. The sampled image, shown on Figure 3, contains only of the original number of pixels.
If one tries to reconstruct the image by a conventional convolution with a Gaussian kernel, the result is the one shown in Figure 4. If one uses the normalized convolution the reconstruction is far superior in quality as it is possible to see from Figure 5.
Figure 2: Test image ``Lena''.
Figure 3: Random sampling of the image ( of original number of pixels is kept).
Figure 4: Reconstruction obtained with conventional convolution
Figure 5: Reconstruction obtained with normalized convolution