Image Filtering and Image Enhancement (DEMO page)

In this page, we present some demos on image filtering and image enhancement. We begin by the linear filtering. We notice
that convolution with gaussian filter is equivalent to solve the heat equation, that is.


Gaussian Convolution is equivalent to solve the heat equation.

L.Alvarez and L.Mazorra have developed a recursive approach to the gaussian convolution based on an implicit numerical
discretization of the heat equation, in such a way that the convolution with gaussian filter can be approximated by the recursive
scheme.


Recursive implementation of the Heat Equation

In the next experience we show the result of applying this scheme to filter a real image.

 
            Original Image                       Zero-Crossing of the Laplacian
Click here to see the filtering evolution            Click here to see the
                                                        Zero-Crossing  evolution

In the next  experience, we will use the denoising model developed by L.Alvarez, P.L.Lions and Jean-Michel Morel given by the partial differential ecuación:


Denoising Mathetical Model

where x represents the direction of edges, in the next figure we ilustrate this reference system associated to each
point of the image:


Reference system associated to the edges

In the next image we present the result of applying the above model to the original image.


Original Image                        Image after Restoration



In the second experience that we present here, we use the mathematical model introduced by L.Alvarez and Luis Mazorra
given by the equation


Deblurring Mathematical Model

This model is oriented to restore discontinuities in the image using a shock filter strategy. In the next experience we
show the capabilities of this model to restore discontinuities.
 


Original Image. If you click here, you will see a movie
which ilustrates the deblurring process.