A Sinusoidal Image Model Derived from the Circular Harmonic Vector

Abstract

We introduce a sinusoidal image model consisting of an oriented sinusoid plus a residual component. The model parameters are derived from the circular harmonic vector, a representation of local image structure consisting of the responses to the higher-order Riesz transforms of an isotropic wavelet. The vector is split into sinusoidal and residual components. The sinusoidal component gives a phase-based description of the dominant local linear symmetry, with improved orientation estimation compared to previous sinusoidal models. The residual component describes the remaining parts of the local structure, from which a complex-valued representation of intrinsic dimension is derived. The usefulness of the model is demonstrated for corner and junction detection and parameter-driven image reconstruction.

On the realization and analysis of circular harmonic transforms for feature detection

“…The Laplacian of Gaussian [8], triangular window [9], radial Gaussians [10], and various trigonometric forms on the unit disk [11], are examples of radial profiles defined in the spatial domain. The first scale of the Meyer-type profile [8], Simoncelli's bio-inspired isotropic wavelet [6], and Log-Gabor responses with an infinite number of vanishing moments for multiscale analysis [12], [13], are examples of radial profiles defined in the frequency domain. The responses are bandlimited in all cases.…”

“…Despite their apparent diversity, all rotationally-invariant image-analysis methods, e.g. multiscale circular-harmonics [4], [5], Fourier histograms of oriented gradients [9], steerable wavelets [6], [8], and high-order Riesz transforms [12], are fundamentally similar [14], [15].…”

“…the square of the error norm) for a patch that is regressed against a template is a measure of dissimilarity and it may be more robust in the presence of structured noise. It is also used to match multi-scale circular-harmonic spectra [12]. Other quadratic similarity and dissimilarity measures for matching angular spectra are analyzed in [20] and a sparsity-based method for texture recognition is used in [15].…”

“…The achieved 𝑃 𝐹 and 𝑃 𝐷 for a range of threshold settings was computed, from 10,000 random instantiations of each angle scenario. For a given wedge width 2𝜙 Δ and threshold 𝜆 Δ combination, any detection for a wedge with 𝑟̃Δ < 2 and 2𝜙 ̃Δ > 2𝜙 Δ − 𝜋 12 ⁄ and 2𝜙 ̃Δ < 2𝜙 Δ + 𝜋 12 ⁄ is deemed to be true; otherwise it is false. The 𝜃 Δ grid was quantized using steps of 𝜋 12 ⁄ (for Dets.…”

A Convex Approach to Superresolution and Regularization of Lines in Images

“…in microscopy, of elongated structures like filaments, neurons, veins, which are deteriorated when reconstructed with classical penalties. Some related works are dedicated to the recovery of curve-like singularities, by variational methods [1,88], Riesz-based models [53,54] or so-called finite rate of innovation methods [52,75,20,61,42,59,61]. The originality of our method is that we reduce the minimization over an infinite dictionary of lines to a semidefinite programming problem, taking advantage of the line structure in both directions of the grid.…”