Walsh–Hadamard Angle Transform

• Walsh–Hadamard Angle Transform is a discrete integral transform that decomposes angular signals into binary square-wave components using Walsh–Hadamard basis functions.
• It leverages a hybrid classical–quantum framework to achieve O(N) computational complexity, significantly improving processing speed over classical methods.
• The transform employs precise polar discretization with uniform-area and uniform-radial subdivisions to maintain orthogonality and enable effective image artifact removal.

The Walsh–Hadamard Angle Transform (WAT) is a discrete integral transform operating on angular coordinate samples using Walsh–Hadamard basis functions. Developed as part of a hybrid classical–quantum framework for image processing in polar coordinates, WAT decomposes angular functions into binary-valued “square-wave” components of varying sequency. The transform’s capabilities include the denoising of artifacts such as circular and azimuthal banding in images and applications to optical system diagnostics, medical imaging, and radar/sonar polar mappings. Efficient quantum algorithms, building on earlier Walsh–Hadamard transform work, offer a computational complexity of $\mathcal{O}(N)$, improving upon the classical fast Walsh–Hadamard transform ($\mathcal{O}(N \log_2 N)$) (Rohida et al., 2024).

1. Polar Walsh Basis Functions and Discretization

Walsh basis functions in polar coordinates are constructed on a disk of radius $r_{\max}$, partitioned into $N_r = 2^{n_r}$ radial divisions and $N_\theta = 2^{n_\theta}$ angular sectors. Two subdivision schemes are defined:

• Uniform-area subdivision ($f = \tfrac{1}{2}$): $r_k = r_{\max} \left[(k+1)/N_r \right]^{1/2}$
• Uniform-radial subdivision ($f = 1$): $r_k = r_{\max} \left[(k+1)/N_r \right]$

Angular nodes are similarly given by $\theta_\ell = 2\pi (\ell+1/2)/N_\theta$ for $\mathcal{O}(N \log_2 N)$0. Each annular–sector cell $\mathcal{O}(N \log_2 N)$1 contains a constant-valued polar Walsh function. Two function orders are defined:

• Natural-order: $\mathcal{O}(N \log_2 N)$2
• Sequency-order: $\mathcal{O}(N \log_2 N)$3

The functions factorize into separate radial and angular components: $\mathcal{O}(N \log_2 N)$4, with $\mathcal{O}(N \log_2 N)$5 and $\mathcal{O}(N \log_2 N)$6 in natural order (Rohida et al., 2024).

2. Orthogonality and Normalization

Orthogonality of the Walsh basis holds both discretely and approximately in the continuous limit:

• Discrete orthogonality:

$\mathcal{O}(N \log_2 N)$7

The orthonormal set is $\mathcal{O}(N \log_2 N)$8.

• Continuous approximation:

$\mathcal{O}(N \log_2 N)$9

Appropriate $r_{\max}$0-weighting may be introduced depending on subdivision choice ($r_{\max}$1).

This rigorous structure underpins transform invertibility and spectrum sparsity properties (Rohida et al., 2024).

3. Walsh–Hadamard Angle Transform Kernel

The 1D Walsh–Hadamard transform matrix $r_{\max}$2 of size $r_{\max}$3 is defined via entries

$r_{\max}$4

where $r_{\max}$5.

Given an angular signal $r_{\max}$6, for instance a pixel value at fixed radius versus angle, its WAT is

$r_{\max}$7

for $r_{\max}$8, or in vector notation $r_{\max}$9. The inverse is $N_r = 2^{n_r}$0 since $N_r = 2^{n_r}$1. In sequency order, the bit transform of $N_r = 2^{n_r}$2 is utilized (Rohida et al., 2024).

The transform decomposes $N_r = 2^{n_r}$3 into angular “square-wave” functions of increasing sequency, with low $N_r = 2^{n_r}$4 indexing smooth angular features and high $N_r = 2^{n_r}$5 encoding rapid oscillations.

4. Hybrid Classical–Quantum Implementation and Complexity

A two-dimensional WAT (across both radius and angle) is realized as $N_r = 2^{n_r}$6 for an image $N_r = 2^{n_r}$7 of size $N_r = 2^{n_r}$8. A hybrid classical–quantum algorithm leverages quantum Hadamard operations to achieve cost $N_r = 2^{n_r}$9 for $N_\theta = 2^{n_\theta}$0, outperforming the classical fast Walsh–Hadamard transform’s $N_\theta = 2^{n_\theta}$1.

PolarQWHT(X) Procedure

1. Apply 1D Walsh–Hadamard transform to each angular column at fixed radius.
2. Apply 1D Walsh–Hadamard transform to each row (angularized) on the result.
3. Each 1D transform QWHT_1d is implemented by:
   • State preparation (classical, $N_\theta = 2^{n_\theta}$2)
   • Hadamard gate layer (quantum, depth 1, $N_\theta = 2^{n_\theta}$3)
   • Measurement, sign correction, and Walsh spectrum reconstruction The sign correction uses the classical “delta-shift trick” (Rohida et al., 2024).

Quantum circuit steps include $N_\theta = 2^{n_\theta}$4 classical preprocessing/postprocessing, parallel Hadamard gates, $N_\theta = 2^{n_\theta}$5 shots for output probability estimation, and classical sign correction.

5. Practical Applications: Image Denoising and Diagnostic Imaging

The WAT framework enables efficient removal of structured noise in polar images. For example, azimuthal banding artifacts can be suppressed by:

• Transforming a noisy angular profile $N_\theta = 2^{n_\theta}$6 noise using $N_\theta = 2^{n_\theta}$7
• Suppressing components at specific sequency $N_\theta = 2^{n_\theta}$8, then reconstructing the cleansed angular profile by inverse transform

Full-image artifact removal involves Cartesian-to-polar mapping, PolarQWHT application, selective H-domain filtering, inverse transform, and interpolation back to Cartesian coordinates. The spectrum before and after demonstrates targeted suppression of banding noise (Rohida et al., 2024).

Applications span:

• Optical system diagnostics (e.g., Airy pattern removal)
• Astronomical imaging
• Medical CT ring artifact reduction
• Sonar and radar polar mapping

6. Practical Considerations and Limitations

Key consideration include:

• Interpolation challenges: Cartesian-to-polar conversion requires averaging or bilinear interpolation, balancing smoothing with aliasing risk.
• Sampling uniformity: Uniform-area subdivision ($N_\theta = 2^{n_\theta}$9) equalizes pixel density across annuli, stabilizing sampling.
• Quantum measurement noise: Obtaining accurate WHT coefficients necessitates many shots and error mitigation.
• Restrictive grid sizes: Both $f = \tfrac{1}{2}$0 and $f = \tfrac{1}{2}$1 must be powers of two.
• Polar grid artifacts: Non-injective mapping leads to edge artifacts, especially near the disk center and perimeter.
• Current hybrid limitations: There is an $f = \tfrac{1}{2}$2 classical overhead for quantum state preparation and sign correction (Rohida et al., 2024).

7. Extensions and Open Research Questions

Proposed extensions and outstanding problems include:

• Development of continuous, analytic polar Walsh–Hadamard transforms amendable to fast hardware
• Explicit error bounds for interpolation and QWHT sign recovery in the presence of realistic noise
• Generalization to non-uniform angular and radial grids
• Exploration of richer radial bases and non-binary subdivisions
• Hybrid transforms blending Fourier and Walsh–Hadamard methods

A plausible implication is that such developments could expand applicability and further reduce computational overheads for large-scale polar image processing tasks (Rohida et al., 2024).