# 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.