Coherence Factor Weighting: Foundations & Applications

• Coherence factor weighting is a set of strategies that quantify constructive interference to enhance resolution and suppress noise.
• It is applied across imaging, beamforming, quantum information, and compressed sensing to optimize reconstruction and inference outcomes.
• Adaptive methods like MV, DMAS, and joint spatio-angular coherence deliver significant improvements in SNR, contrast, and artifact suppression.

Coherence factor weighting encompasses a diverse set of methodological strategies that utilize spatial, temporal, or structural coherence as a multiplicative weight or quality metric within signal processing, imaging, quantum information, and compressed sensing. Across these fields, coherence factor computations serve to suppress noise, enhance resolution or contrast, and adaptively control the influence of specific data or state components based on their estimated alignment, phase, or resource character. The underlying principle is to measure the degree of constructive interference, spatial/angular/frequency agreement, or quantum resource content and then adaptively weight results to optimize reconstruction, inference, or quantification outcomes.

1. Mathematical Foundations of Coherence Factor Weighting

Coherence factor weighting is mathematically realized as a local, per-sample, or per-state quantity $w$ or $\mathrm{CF}$ that modulates either the pixel value, measurement contribution, or state amplitude according to a rational functional of “coherent” versus “incoherent” power, or maximal resource fraction. Several canonical forms include:

• Classical Beamforming: For a set of delayed signals $s_i(k)$, the standard coherence factor is

$$\mathrm{CF}(k) = \frac{ \bigl|\sum_{i=1}^M s_i(k)\bigr|^2 }{ M \sum_{i=1}^M |s_i(k)|^2 }$$

where the numerator is the coherent sum (maximum for in-phase signals) and the denominator quantifies the total energy (Mozaffarzadeh et al., 2017, Mozaffarzadeh et al., 2017, Li et al., 2019).

• Quantum Information: The “coherence weight” $C_w(\rho)$ is the minimal $s \in [0,1]$ such that $\rho = (1-s)\sigma + s\tau$ with $\sigma$ incoherent, providing an operationally motivated convex resource monotone (Bu et al., 2017, Yao et al., 2020).
• General Imaging/Processing: The weighting may be derived from a localized spectral, spatial, or frequency decomposition (e.g., ratio of main-lobe energy to total energy in filtered IQ data (Shen et al., 2023), or joint spatio-angular coherence terms (Khetan et al., 18 Apr 2024)).

These weights $w$ serve as multiplicative quality factors, adapting signal contributions or state amplitudes according to the measured or computed coherence.

2. Coherence Factor Weighting in Imaging and Signal Processing

Coherence factor weighting is foundational in suppression of artifacts and enhancement of target detectability in array imaging, radar, ultrasound, and related modalities.

Photoacoustic and Ultrasound Beamforming:

• The classical CF approach is used to downweight incoherent or off-axis energy, significantly reducing sidelobes and clutter in delay-and-sum (DAS) reconstructions (Mozaffarzadeh et al., 2017, Mozaffarzadeh et al., 2017).
• High-resolution or modified variants, such as replacing the numerator with a minimum-variance (MV) (Mozaffarzadeh et al., 2017) or delay-multiply-and-sum (DMAS) (Mozaffarzadeh et al., 2017) output, lead to high-resolution coherence factor (HRCF) or modified CF (MCF) methods. These increase resolution and contrast; for example, MCF yields ≈45% SNR and ≈40% FWHM improvements over standard CF (Mozaffarzadeh et al., 2017), while HRCF yields up to 91% FWHM and 40% SNR gain over DAS+CF (Mozaffarzadeh et al., 2017).
• Multi-dimensional coherence factor extensions account for both spatial and frequency diversity, e.g., 2-D CF in radar imaging jointly suppresses sidelobes in range and cross-range (Li et al., 2019).

Joint Spatio-Angular Coherence in Ultrafast Imaging:

• In plane-wave compound imaging, the joint coherence factor (JCF) computes a fine-grained weight for each transmit-receive trajectory, based on cross-coherence between signal directions. This adaptive weighting significantly suppresses speckle and clutter, yielding images with lower background noise and enhanced structural delineation (Khetan et al., 18 Apr 2024).

Adaptive Filtering and Aberration Correction:

• In PSF restoration for ultrasound, a “coherence index” quantifies the fraction of signal energy in the main lobe after filtering and is used to enhance contrast and further suppress side-lobes when combined with neural network approaches (Shen et al., 2023).

The general framework is robust: the weighting is applied post-acquisition, is local (pixel/trajectory specific), and may be interpreted as a statistical consistency or spatial-spectral agreement measure.

3. Coherence Weighting in Quantum Resource Theories

The “coherence weight” $C_w(\rho)$ is a prominent convex geometric quantifier of quantum resource content, with direct operational and witness interpretations (Bu et al., 2017, Yao et al., 2020). For a quantum state $\rho$ in Hilbert space $\mathcal{H}$ with a fixed incoherent basis, the coherence weight is

$$C_w(\rho) = \min \left\{ s \geq 0 : \exists \sigma \in \mathcal{I},\, \rho \geq (1-s)\sigma \right\}$$

where $\mathcal{I}$ is the set of incoherent diagonal density matrices. The measure satisfies:

• Convexity and strong monotonicity under incoherent operations.
• Characterization via semidefinite programming (primal and dual), where optimal witnesses $W$ can be constructed with $\Delta(W) \leq 0$, $W \leq I$.
• For Werner states and a class of $X$-states, $C_w(\rho)$ coincides exactly with robustness and $l_1$-coherence measures.
• The set of “mixed maximally coherent states” (MMCS) is nontrivial—mixed states with $C_w(\rho) = 1$ can be constructed in any dimension, reflecting the subtle geometry of the convex set of states.

This framework establishes a strict analogy to best-separable approximations in entanglement theory, positioning “coherence factor weighting” as a unifying instrument for operational resource quantification, with clear geometric and witness-theoretic underpinnings (Yao et al., 2020).

4. Statistical Signal Processing and Compressed Sensing

In high-dimensional estimation and dictionary learning, coherence factor weighting is employed in the context of mutual coherence of synthesis frames (Gavruta et al., 2014). The key insights are:

• For a frame $F = \{f_i\}$ with non-normalized columns, weights $w_i = \|f_i\|_2$ are employed to rescale coefficients in $\ell_{1,w}$-minimization,

$$\min_c \|c\|_{1,w} \quad \text{s.t.} \quad \|y - Tc\|_2 \leq \eta$$

yielding error bounds and sparse recovery guarantees strictly tighter than those in the normalized case, as the effective mutual coherence is now properly scaled.

• This weighting restores the theoretical guarantees of incoherence-based compressed sensing to arbitrary frames or overcomplete dictionaries, and supports extensions where specific prior support is favored.

This approach to coherence factor weighting has become fundamental for compressed sensing with general, unnormalized or random dictionaries.

5. Coherence Factor Weighting in Physical and Spectral Theories

In condensed matter and superconductivity, coherence factors are critical weighting terms in matrix elements between exact eigenstates, dictating transition rates and spectral properties.

• For general eigenstates of the reduced BCS or pairing Hamiltonian, the squared coherence factor between “in” and “out” states includes an elliptic/trigonometric prefactor determined by the Abel map on the Riemann surface associated with the distribution of rapidities:

$$|\langle \text{out}; l,p | c_{m\sigma}^\dagger | \text{in} \rangle|^2 = \frac{\pi^2 l N_{l,\sigma} \sin^{-2}(u(\varepsilon_m) + l(p\tau - u_\infty))}{2\omega^2 R_4(\varepsilon_m)}$$

(Gorohovsky et al., 2013, Gorohovsky et al., 2013).

• In equilibrium, the standard BCS coherence factors $u_k^2$ and $v_k^2$ are recovered; out of equilibrium, the full structure encodes the partitioning of pairing fluctuations and macroscopic arc motion in the complex energy plane.

Coherence factor weighting thereby controls observable transition amplitudes, spectral line shapes, and many-body dynamical phenomena.

6. Implementation, Performance, and Limitations

Coherence factor weighting is computationally lightweight in most imaging applications, since the normalization and power-ratio computations can be implemented in closed form and fused efficiently within sliding or beamforming loops. Notable advantages and constraints include:

• Imaging: Sidelobe, ghost, and clutter suppression with minimal impact on main-lobe resolution, especially in 2-D or spatio-angular generalized CF schemes (Mozaffarzadeh et al., 2017, Li et al., 2019, Khetan et al., 18 Apr 2024). Principal drawbacks include sensitivity to small sample supports and increased computational cost for adaptive (e.g., MV, DMAS, JCF) variants.
• Quantum and Statistical Domains: Coherence weight evaluation reduces to efficient semidefinite programs for low to moderate dimension but becomes numerically challenging for large-scale mixed states, motivating polytope refinement and witness-construction algorithms (Yao et al., 2020).
• Signal Processing: Coherence-based gain control provides flexible tunability between artifact suppression and fidelity, with weighting parameters often user-adjustable and interpretable as direct trade-offs (e.g., noise suppression vs. distortion) (Shankar et al., 2020).

In all contexts, the strategy remains robust as long as the underlying coherence measure faithfully reflects the desired property (physical coherence, spatial/angular alignment, or resource purity).

7. Comparative Summary and Research Directions

Field/Domain	Core Quantity	Role of CF Weighting
Imaging (acoustic, EM)	Signal power ratio, spatial/angular/freq coherence	Suppress sidelobes, enhance resolution/contrast
Quantum information	Convex resource fraction $C_w(\rho)$	Quantify minimal “coherent ingredient” in a state
Compressed sensing	Frame-norm weighted $\ell_1$ terms	Normalize dictionary, improve sparse recovery
Condensed matter (BCS, pairing)	Matrix element prefactor between eigenstates	Weight transition probabilities, spectral functions

A plausible implication is that coherence factor weighting serves as a unifying formalism, providing a local (data, signal, or state-specific) indicator of statistical or structural quality, and enabling adaptive, robust processing or resource discrimination. Future research directions include further generalizations to multidimensional coherence landscapes, unified convex optimization algorithms for large-scale quantum resource evaluation, and robust hardware implementations for adaptive real-time imaging and sensing.