In differential morphology, multiscale dilations and erosions are modeled via nonlinear partial differential equations (PDEs) in scale-space. Curve evolution employs methods of differential geometry to study the differential equations governing the p

The distance transform has found many applications in image analysis. Chamfer distance transforms are a class of discrete algorithms that offer a good approximation to the desired Euclidean distance transform at a lower computational cost. They can also give integer-valued distances that are more suitable for several digital image processing tasks. The local distances used to compute a chamfer distance transform are selected to minimize an approximation error. A new geometric approach is developed to find optimal local distances. This new approach is easier to visualize than the approaches found in previous work, and can be easily extended to chamfer metrics that use large neighborhoods. A new concept of critical local distances is presented which reduces the computational complexity of the chamfer distance transform without increasing the maximum approximation error.

Most traditional methodologies in digital image processing start with a discrete model which is directly related to the subsequent discrete processing of the image data. We review some examples and focus on some details of an emerging new methodology that starts from some continuous models provided by PDEs and proceeds with discrete processing of the image data via the numerical implementation of these PDEs on some discrete grid. We examine the PDEs modeling multiscale morphological image analysis and the related eikonal PDE of optics. The PDE approach is very promising because it provides many new mathematical models and has connections with the physics of imaging.