Total Least Square Optimal Analytic Signal by Structure Tensor for N-D images

TL;DR

The paper presents a TLS-based, Structure Tensor–driven framework to construct a true N-D analytic signal by jointly estimating local orientation and scale, enabling adaptive Gabor-like filtering and phase recovery without exhaustive local spectrum sampling. By formulating two STs for direction and scale and deriving a phase $\Phi^{ST}$ from a locally tuned Gabor filter, the method achieves a continuous, isotropic representation $f_A$ with robust singularity detection. Extensive experiments on ground-truth synthetic data demonstrate improved isotropy, phase continuity, and resilience to non-linear contamination compared to Monogenic and spline-wavelet baselines, with practical applications in fingerprint enhancement and fringe-pattern analysis. The approach extends naturally to N-D, offering a principled, data-driven way to sample the spectrum and extract meaningful local frequency content for advanced image analysis tasks.

Abstract

We produce the analytic signal by using the Structure Tensor, which provides Total Least Squares optimal vectors for estimating orientation and scale locally. Together, these vectors represent N-D frequency components that determine adaptive, complex probing filters. The N-D analytic signal is obtained through scalar products of adaptive filters with image neighborhoods. It comprises orientation, scale, phase, and amplitude information of the neighborhood. The ST analytic signal $ f_A $ is continuous and isotropic, and its extension to N-D is straightforward. The phase gradient can be represented as a vector (instantaneous frequency) or as a tensor. Both are continuous and isotropic, while the tensor additionally preserves continuity of orientation and retains the same information as the vector representation. The tensor representation can also be used to detect singularities. Detection with known phase portraits has been demonstrated in 2-D with relevance to fringe pattern processing in wave physics, including optics and fingerprint measurements. To construct adaptive filters we have used Gabor filter family members as probing functions, but other function families can also be used to sample the spectrum, e.g., quadrature filters. A comparison to three baseline alternatives-in representation (Monogenic signal), enhancement (Monogenic signal combined with a spline-wavelet pyramid), and singularity detection (mindtct, a fingerprint minutia detector widely used in numerous studies)-is also reported using images with precisely known ground truths for location, orientation, singularity type (where applicable), and wave period.

Figures (16)

• Figure 1: FM test image with size 512x512. The period changes linearly with radius.
• Figure 2: Singularities are produced by phase gradient superpositions in image with size 758x768. Contamination is replacement noise, i.e. not additive. Al ground-truths of singularity parameters, as well as those of the phase, are known.
• Figure 3: ST phase of clean input (underlying Fig. \ref{['fg:minutia_synt_mm_sp_noise_infull']});
• Figure 4: ST complex signal. Hue represents signal argument and brighntess is its magnitude, in HSV colors.
• Figure 5: Monogenic complex signal. Hue represents signal argument and brighntess is its magnitude, in HSV colors.