Morphology on Convolution Lattices with Applications to the Slope Transform and Random Set Theory
Journal of Mathematical Imaging and Vision
Efficient Dilation, Erosion, Opening, and Closing Algorithms
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Note on Two Classical Shock Filters and Their Asymptotics
Scale-Space '01 Proceedings of the Third International Conference on Scale-Space and Morphology in Computer Vision
Conjugate network calculus: a dual approach applying the Legendre transform
Computer Networks: The International Journal of Computer and Telecommunications Networking - Selected papers from the 3rd international workshop on QoS in multiservice IP networks (QoS-IP 2005)
Computing rank-convolutions with a mask
ACM Transactions on Algorithms (TALG)
Conjugate network calculus: A dual approach applying the Legendre transform
Computer Networks: The International Journal of Computer and Telecommunications Networking - Selected papers from the 3rd international workshop on QoS in multiservice IP networks (QoS-IP 2005)
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
A dual approach to network calculus applying the legendre transform
QoS-IP'05 Proceedings of the Third international conference on Quality of Service in Multiservice IP Networks
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Fourier transforms are among the most useful linear signal transformations for quantifying the frequency content of signals and for analyzing their processing by linear time-invariant systems. Some nonlinear signal transforms are developed that can provide information about the slope content of signals and are useful analytic tools for large classes of nonlinear systems. Many of their theoretical properties are examined, showing a striking conceptual resemblance to Fourier transforms and their application to linear systems. These novel transforms, called slope transforms, are originally derived from the eigenvalues of morphological dilation and erosion systems, where the corresponding eigenfunctions are lines αt+b parameterized by their slope α. They obey a nonlinear superposition principle of the supremum- or infimum-of-sums type. Applied to the impulse response of dilation or erosion systems, the slope transforms provide a slope response function for these systems, which allows their analysis and design in a transform domain, the slope domain. Applied to arbitrary signals, the slope transforms provide information about upper or lower tangents to the signal's graph at varying slopes. The upper or lower envelopes of the signal can be obtained from the inverse transforms. Overall, the slope transforms provide a new transform domain for signals and morphological systems where time lines become slope impulses, time cones become slope bandpass filters, and time dilation/erosion transform into addition of slope transforms. Their application to the design of slope-selective filters is also presented