Gaussian Convolution is equivalent to solve the heat equation.
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.