Statistical performance theory for the wavelet-domain monogenic AM/FM estimators of the Riesz–Laplace transform: bias and variance under noise and finite sampling

The Riesz–Laplace monogenic wavelet transform (Unser, Sage, and Van De Ville, IEEE TIP 2009) extracts local orientation, amplitude, phase, and instantaneous frequency (wavenumber) from 2-D images at multiple scales, but characterized the reliability of these estimators only qualitatively: the directional quadrature response falls off as the cosine of the orientation mismatch, wavenumber estimates degrade in low-coherency neighborhoods and far from the analysis band's center frequency, and no noise model or finite-sampling analysis of any estimator was given. Later literature resolved the higher-dimensional and frame-theoretic generalizations and produced second-order statistics of the random monogenic signal for orientation and directionality detection, deterministic demodulation guarantees, and empirical multiscale noise mitigation, but no estimator-level statistical theory. The open problem is to establish a quantitative bias/variance theory for the transform's AM/FM estimators under explicit noise and sampling models, including the propagation of orientation error through the quadrature attenuation.

Background

For a one-dimensional real signal, the analytic signal is formed by adding the Hilbert transform as an imaginary part; its polar form yields a local amplitude and phase, and the spatial derivative of the phase is the instantaneous frequency. For images, the accepted counterpart is the monogenic signal introduced by Felsberg and Sommer: the image is paired with the two components of the Riesz transform, the multidimensional analogue of the Hilbert transform whose Fourier multipliers are $\omega_1/\|\boldsymbol{\omega}\|$ and $\omega_2/\|\boldsymbol{\omega}\|$ (an all-pass, vector-valued filter that is singular at the origin and hence suppresses the DC, i.e., zero-frequency, component). This yields local amplitude, phase, and orientation for patterns that are locally one-dimensional, and the Riesz filter is steerable: its directional response along any orientation can be computed exactly from a fixed finite set of measured responses. AM/FM stands for amplitude-modulation/frequency-modulation and denotes image models that are locally a spatially varying sinusoidal carrier, $A(\boldsymbol{x})\cos(\varphi(\boldsymbol{x}))$, whose amplitude, phase, and instantaneous wavenumber (the gradient of the phase) are the quantities to be estimated. The original monogenic signal operates at a single scale, so Unser, Sage, and Van De Ville (2009) transposed it to the wavelet domain: they paired a real, quasi-isotropic polyharmonic-spline wavelet that behaves like a fractional Laplacian (a bandpass differential operator acting at dyadic scales) with its complex Riesz transform, producing a multiresolution monogenic analysis implemented by perfect-reconstruction filterbanks, with redundancy factor 3 in the critically-sampled configuration and factor 4 in a nonsubsampled configuration with improved shift-invariance. A scaling function is the low-resolution generator of a wavelet multiresolution analysis; the authors note that the Riesz operator does not map scaling functions into scaling functions, destroying the partition of unity and delocalizing the approximation side because of its DC null — an inherent property, since it follows from the Riesz multiplier's singularity at the origin. From the transform coefficients $(w_i,r_{1,i},r_{2,i})$ at each scale and location, the paper extracts: a tensor-based orientation, obtained by aggregating the Riesz coefficients into a local $2\times2$ structure tensor (a symmetric matrix of windowed outer-product averages) whose dominant eigenvector gives the orientation and whose eigenvalue ratio defines a coherency index $\chi\in[0,1]$ measuring local directionality; a steered amplitude and phase from the directional analytic pair $w_i+jq_i$ along the estimated orientation $\theta$; and an arctan-free instantaneous-wavenumber estimator combining the coefficients and their spatial derivatives (computed exactly by auxiliary derivative wavelets), which is exact within the wavelet model. The paper documents the estimator's reliability limits only qualitatively: the directional Hilbert behavior is ideal only along the true orientation and falls off as $\cos(\theta-\theta_0)$ under orientation mismatch, so amplitude, phase, and frequency estimates inherit orientation errors; wavenumber estimates are most reliable near the wavelet band's center frequency and fail where highpass attenuation dominates (the center of their zoneplate test); low-coherency neighborhoods are unreliable; and no noise or finite-sampling statistics of any estimator are provided. The paper also provides test models for these estimators — amplitude-modulated carriers and a zoneplate — along with publicly available implementations of the transform. Subsequent work covered adjacent ground. Held, Storath, Massopust, and Forster (2010) independently constructed steerable wavelet frames based on the Riesz transform and a hypercomplex monogenic generalization in dimensions higher than two; Unser and Van De Ville (2010) developed higher-order Riesz transforms for steerability; Unser, Chenouard, and Van De Ville (2011) built tight steerable frames in $L_2(\mathbb{R}^d)$ for arbitrary dimension $d$; and Chenouard and Unser (2012) demonstrated 3-D steerable wavelets in practice — together resolving the dimension-generalization directions. On the stochastic side, Olhede, Ramirez, and Schreier (2014) derived the second-order statistics of the random monogenic signal for isotropic, anisotropic, and unidirectional Gaussian random fields and a directionality detection threshold, and Polisano, Clausel, Perrier, and Condat (2021) defined orientation intrinsically for localizable Gaussian fields via Riesz-based structure tensors — both treat the orientation of random fields under signal models, not the noise performance of the wavelet-domain estimators. Clausel, Oberlin, and Perrier (2015) gave deterministic separation and reconstruction guarantees for AM–FM images via a monogenic synchrosqueezed wavelet transform; Schmitt, Pustelnik, Borgnat, Flandrin, and Condat (2014) proposed the 2-D Prony–Huang transform as another tool for local spectral analysis of nonstationary images, an area they present as actively unresolved; and Kaseb, Mercère, Biermé, Brémand, and Carré (2019) analyzed monogenic phase estimation for interference fringes theoretically only in the ideal cosine case, naming noise as a weakness handled by an empirical multiscale combination strategy. Knight and Saito (2024) proposed structure-multivector phase estimation with an empirical orientation-variance quality map, again without estimator statistics. No located work derives the estimator-level bias/variance theory for the Riesz–Laplace wavelet AM/FM estimators.

Problem Statement

For the monogenic Riesz–Laplace wavelet transform of Unser, Sage, and Van De Ville (2009), in both its critically-sampled and nonsubsampled configurations, establish a quantitative statistical performance theory for its wavelet-domain AM/FM feature estimators — the structure-tensor orientation and coherency, the steered amplitude and phase, and the arctan-free instantaneous-wavenumber estimator — under explicit observation models (at minimum, additive white or colored noise on the sampled image, together with finite sampling on the analysis lattice). The theory should deliver, with each requested result serving the single objective of replacing the source's qualitative reliability caveats by predictive statistics: (i) the bias and variance of each estimator, as closed-form expressions, controlled asymptotic approximations, or certified bounds, as functions of scale, orientation, coherency, and the offset of the true instantaneous wavenumber from the analysis band's center frequency; and (ii) the propagation law by which orientation-estimation error enters the directional quadrature through the $\cos(\theta-\theta_0)$ attenuation and from there the amplitude, phase, and wavenumber estimates. The theory is to cover the estimator set and both coupling mechanisms (i) and (ii); a characterization restricted to a single estimator or to noiseless inputs is not the question's target.

References

1. M. Unser, D. Sage, D. Van De Ville, 'Multiresolution Monogenic Signal Analysis Using the Riesz–Laplace Wavelet Transform', IEEE Transactions on Image Processing 18(11):2402–2418, 2009. DOI 10.1109/TIP.2009.2027628, https://doi.org/10.1109/TIP.2009.2027628
2. M. Felsberg, G. Sommer, 'The monogenic signal', IEEE Transactions on Signal Processing 49(12):3136–3144, 2001. DOI 10.1109/78.969520, https://doi.org/10.1109/78.969520
3. S. Held, M. Storath, P. Massopust, B. Forster, 'Steerable Wavelet Frames Based on the Riesz Transform', IEEE Transactions on Image Processing 19(3):653–667, 2010. DOI 10.1109/TIP.2009.2036713, https://doi.org/10.1109/TIP.2009.2036713
4. M. Unser, D. Van De Ville, 'Wavelet Steerability and the Higher-Order Riesz Transform', IEEE Transactions on Image Processing 19(3):636–652, 2010. DOI 10.1109/TIP.2009.2038832, https://doi.org/10.1109/TIP.2009.2038832
5. M. Unser, N. Chenouard, D. Van De Ville, 'Steerable Pyramids and Tight Wavelet Frames in $L_2(\mathbb{R}^d)$', IEEE Transactions on Image Processing 20(10):2705–2721, 2011. DOI 10.1109/TIP.2011.2138147, https://doi.org/10.1109/TIP.2011.2138147
6. N. Chenouard, M. Unser, '3D Steerable Wavelets in Practice', IEEE Transactions on Image Processing 21(11):4522–4533, 2012. DOI 10.1109/TIP.2012.2206044, https://doi.org/10.1109/TIP.2012.2206044
7. S. C. Olhede, D. Ramirez, P. J. Schreier, 'Detecting Directionality in Random Fields Using the Monogenic Signal', IEEE Transactions on Information Theory 60(10):6491–6510, 2014. DOI 10.1109/TIT.2014.2342734, https://doi.org/10.1109/TIT.2014.2342734 (arXiv:1304.2998, https://arxiv.org/abs/1304.2998)
8. M. Clausel, T. Oberlin, V. Perrier, 'The monogenic synchrosqueezed wavelet transform: a tool for the decomposition/demodulation of AM–FM images', Applied and Computational Harmonic Analysis 39(3):450–486, 2015. DOI 10.1016/j.acha.2014.10.003, https://doi.org/10.1016/j.acha.2014.10.003 (arXiv:1211.5082, https://arxiv.org/abs/1211.5082)
9. J. Schmitt, N. Pustelnik, P. Borgnat, P. Flandrin, L. Condat, '2D Prony-Huang Transform: A New Tool for 2D Spectral Analysis', IEEE Transactions on Image Processing 23(12):5233–5248, 2014. DOI 10.1109/TIP.2014.2363000, https://doi.org/10.1109/TIP.2014.2363000
10. M. Kaseb, G. Mercère, H. Biermé, F. Brémand, P. Carré, 'Phase estimation of a 2D fringe pattern using a monogenic-based multiscale analysis', Journal of the Optical Society of America A 36(11):C143–C153, 2019. DOI 10.1364/JOSAA.36.00C143, https://doi.org/10.1364/JOSAA.36.00C143
11. K. Polisano, M. Clausel, V. Perrier, L. Condat, 'Riesz-based orientation of localizable Gaussian fields', Applied and Computational Harmonic Analysis 50:353–385, 2021. DOI 10.1016/j.acha.2019.08.007, https://doi.org/10.1016/j.acha.2019.08.007
12. B. Knight, N. Saito, 'A Structurally Coherent Spatial Phase Estimate', arXiv:2412.08070, 2024. https://arxiv.org/abs/2412.08070