Optimal edge detection using a multiresolution approach

The main aim of the present work is to tackle edge detection in a way which expresses the inherent trade-off between noise rejection and interference, which Canny (1986) and others have solved by the artificial device of a `localisation criterion'. The problem is first examined in 1D. The `edge' signal is generated by a simple 2-state Markov chain, which implies that successive transitions are of opposite sign, but otherwise independent: the intervals between transitions have a geometric distribution. The signal may therefore be said to contain features at all scales. The data are produced from the signal by the addition of independent white Gaussian noise. The problem is simply to detect the transitions in the Markov chain as reliably as possible. It is to be expected that the optimum detection filter will have a scale dependent on both the noise variance and the rate at which signal transitions occur. This has nothing to do with the `localisation' criteria of Canny. It is shown that there is a solution, which is based on a generalisation of the matched filter. The 2D problem is then considered and it is shown that a pyramid algorithm, based on an optimum isotropic lowpass filter, gives an approximation to the ideal transient detector, with small approximation error. Some image edge detection results are given and the paper concludes with a discussion of some of the issues raised