Significant edges in the case of non-stationary Gaussian noise

In this paper, we propose an edge detection technique based on some local smoothing of the image followed by a statistical hypothesis testing on the gradient. An edge point being defined as a zero-crossing of the Laplacian, it is said to be a significant edge point if the gradient at this point is larger than a threshold s(ε) defined by: if the image I is pure noise, then the probability of ‖∇I(x)‖ ≥ s(ε) conditionally on ∆I(x) = 0 is less than ε. In other words, a significant edge is an edge which has a very low probability to be there because of noise. We will show that the threshold s(ε) can be explicitly computed in the case of a stationary Gaussian noise. In the images we are interested in, which are obtained by tomographic reconstruction from a radiograph, this method fails since the Gaussian noise is not stationary anymore. Nevertheless, we are still able to give the law of the gradient conditionally on the zero-crossing of the Laplacian, and thus compute the threshold s(ε). We will end this paper with some experiments and compare the results with those obtained with other edge detection methods.