# Multiscale Edge Detection

and Reconstruction

As in the one dimensional case, dyadic modulus
maxima are used to dectect edges.

Provided that the two dimensional geometry is
taken into account, these edges can be interpreted as
contours.

A similar algorithm to the one dimensional case
reconstructs a good approximation of an image from its edges.

## Multiscale edges

In images, what is most often perceived as an edge
is a curve across which there is a sharp variation of brightness. To
make things simpler, the image will be assumed to be monochrome.
While the actual concept of an edge is more involved and depends in
particular on a priori knowledge about the featured objects, this
presentation has the advantage of leading to a precise mathematical
definition of an "edge point".

To do so, consider a two dimensioanl wavelet
defined by partial differentiation of a kernel:

The dyadic wavelet transform is defined by

with, for k=1,2,

The two coordinates of the dyadic wavelet
transform are that of the gradient of the convolution of the signal
with the dilated kernel:

The multiscale edge points are the points where
the dyadic transform modulus is locally maximum along this direction.
This corresponds to a **locally sharpest
variation of image intensity orthogonally to the lines of constant
brightness.**

## Examples

A synthetic example analyzes the edges of a
circle.

Another example analyses a classical wavelet
picture.

## Remark

It is rare that an image line has no hole in it.
The brain compensate these defaults using more elaborate image
analysis. Notice that the use of color is useful. As illustration,
here is an optical illusion where joining edges is far from being
obvious:

## Reconstruction

As in the one dimensional
case, the frame inverse operator can
be used to reconstruct a minimum norm image with prescribed values at
the maxima locations. Mean square relative errors of
l0^{-2} can be
obtained.

On an example, one can see that
the reconstruction error is visually neglectible.

## Implementation

The computations are performed with separable
wavelets whose Fourier transforms are

where g is a finite difference filter; the two
wavelets then approximate the partial derivatives of

where f is a scaling function defined
by a finite impulse response filter h. The dyadic wavelet transform
is computed by two dimensional extension of the algorithme à trous.

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