Nonlinear image operators for the evaluation of local intrinsic dimensionality

Area of Science:

• Signal Processing
• Image Analysis
• Computational Geometry

Background:

• Local intrinsic dimensionality (LID) is a fundamental signal property not detectable by linear filters.
• LID relates to signal redundancy and multidimensional surface geometry, including Gaussian curvature.
• Linear operators are insufficient for processing LID due to their reliance on superposition.

Purpose of the Study:

• To derive polynomial operators for detecting intrinsically 2-D image features.
• To explore the connection between intrinsic dimensionality, signal redundancy, and Gaussian curvature.
• To develop nonlinear operators capable of exploiting multiplicative relations in frequency components.

Main Methods:

• Derivation of a class of polynomial operators for LID detection.
• Application of frequency design methods to obtain quadratic Volterra operators.
• Adaptation of operators for adjustable "curvature tuning" and orientation selectivity.

Main Results:

• Developed polynomial operators capable of detecting intrinsically 2-D image features like curved edges and junctions.
• Established quadratic Volterra operators selective to intrinsically 2-D signals.
• Demonstrated operator adaptability for "curvature tuning" and orientation selectivity, with some relating to visual cortex neurons.

Conclusions:

• Polynomial operators are essential for processing intrinsic dimensionality, enabling detection of complex 2D features.
• Quadratic Volterra operators offer a flexible approach to signal processing tasks involving intrinsic dimensionality and Gaussian curvature.
• The developed operators show potential applications in image analysis and understanding biological visual systems.