Digital image processing (2nd ed.)
Digital image processing (2nd ed.)
Morphological systems: slope transforms and max-min difference and differential equations
Signal Processing - Special issue on mathematical morphology and its applications to signal processing
Morphological signal processing and the slope transform
Signal Processing - Special issue on mathematical morphology and its applications to signal processing
Scale-Space Properties of the Multiscale Morphological Dilation-Erosion
IEEE Transactions on Pattern Analysis and Machine Intelligence
Digital image processing
Discrete-time signal processing (2nd ed.)
Discrete-time signal processing (2nd ed.)
Journal of Mathematical Imaging and Vision
Scale-Space Theory in Computer Vision
Scale-Space Theory in Computer Vision
Pseudo-Linear Scale-Space Theory
International Journal of Computer Vision
Linear Scale-Space has First been Proposed in Japan
Journal of Mathematical Imaging and Vision
Scale-Spaces, PDE's, and Scale-Invariance
Scale-Space '01 Proceedings of the Third International Conference on Scale-Space and Morphology in Computer Vision
Morphological Image Analysis: Principles and Applications
Morphological Image Analysis: Principles and Applications
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
An explanation for the logarithmic connection between linear and morphological systems
Scale Space'03 Proceedings of the 4th international conference on Scale space methods in computer vision
Families of generalised morphological scale spaces
Scale Space'03 Proceedings of the 4th international conference on Scale space methods in computer vision
Gradient watersheds in morphological scale-space
IEEE Transactions on Image Processing
Differential morphology and image processing
IEEE Transactions on Image Processing
Morphological bilateral filtering and spatially-variant adaptive structuring functions
ISMM'11 Proceedings of the 10th international conference on Mathematical morphology and its applications to image and signal processing
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Dorst/van den Boomgaard and Maragos introduced the slope transform as the morphological equivalent of the Fourier transform. Generalising the conjugacy operation from convex analysis it formed the basis of a morphological system theory that bears an almost logarithmic relation to linear system theory; a connection that has not been fully understood so far. Our article provides an explanation by disclosing that morphology in essence is linear system theory in specific algebras. While linear system theory uses the standard plus-prod algebra, morphological system theory is based on the max-plus algebra and the min-plus algebra. We identify the nonlinear operations of erosion and dilation as linear convolutions in the latter algebras. The logarithmic Laplace transform makes a natural appearance as it corresponds to the conjugacy operation in the max-plus algebra. Its conjugate is given by the so-called Cramer transform. Originating from stochastics, the Cramer transform maps Gaussians to quadratic functions and relates standard convolution to erosion. This fundamental transform relies on the logarithm and constitutes the direct link between linear and morphological system theory. Many numerical examples are presented that illustrate the convexifying and smoothing properties of the Cramer transform.