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.