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 $\ell=0,\ldots,N_\theta-1$. Each annular–sector cell $S(k,\ell)$ contains a constant-valued polar Walsh function. Two function orders are defined:

• Natural-order: $W_{n,m}(r,\theta) = (-1)^{n\cdot k + m \cdot \ell}$
• Sequency-order: $$W^s_{n,m}(r,\theta) = (-1)^{\sum_{i=0}^{n_r-1} (n_{n_r-1-i} \oplus n_{n_r-i}) k_i + \sum_{j=0}^{n_\theta-1} (m_{n_\theta-1-j} \oplus m_{n_\theta-j}) \ell_j}$$

The functions factorize into separate radial and angular components: $$W_{n,m}(r_k,\theta_\ell) = R_n(r_k) \cdot A_m(\theta_\ell)$$, with $R_n(r_k) = (-1)^{n \cdot k}$ and $A_m(\theta_\ell) = (-1)^{m \cdot \ell}$ 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:

$$\sum_{k=0}^{N_r-1} \sum_{\ell=0}^{N_\theta-1} W_{n,m}(r_k, \theta_\ell) W_{n',m'}(r_k, \theta_\ell) = N_r N_\theta \delta_{n,n'} \delta_{m,m'}$$

The orthonormal set is $$\psi_{n,m}(r, \theta) = W_{n,m}(r, \theta)/\sqrt{N_r N_\theta}$$.

• Continuous approximation:

$$\int_0^{r_{\max}} \int_0^{2\pi} W_{n,m}(r,\theta) W_{n',m'}(r,\theta) r\,dr\,d\theta \approx \pi r_{\max}^2 \delta_{n,n'} \delta_{m,m'}$$

Appropriate $r$-weighting may be introduced depending on subdivision choice ($f$).

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 $H^{(N_\theta)}$ of size $N_\theta \times N_\theta$ is defined via entries

$H_{p,q} = (-1)^{p \cdot q}/\sqrt{N_\theta}$

where $p, q \in \{0, ..., N_\theta-1\}$.

Given an angular signal $f(\theta_\ell)$, for instance a pixel value at fixed radius versus angle, its WAT is

$F_p = \sum_{q=0}^{N_\theta-1} H_{p,q} f(\theta_q)$

for $p = 0, ..., N_\theta-1$, or in vector notation $F = H f$. The inverse is $f = H F$ since $H^2 = I$. In sequency order, the bit transform of $p \cdot q$ is utilized (Rohida et al., 2024).

The transform decomposes $f(\theta)$ into angular “square-wave” functions of increasing sequency, with low $p$ indexing smooth angular features and high $p$ encoding rapid oscillations.

4. Hybrid Classical–Quantum Implementation and Complexity

A two-dimensional WAT (across both radius and angle) is realized as $Y = H_r X H_\theta^T$ for an image $X$ of size $N_r \times N_\theta$. A hybrid classical–quantum algorithm leverages quantum Hadamard operations to achieve cost $\mathcal{O}(N)$ for $N = N_r N_\theta$, outperforming the classical fast Walsh–Hadamard transform’s $\mathcal{O}(N \log_2 N)$.

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, $\mathcal{O}(N)$)
   • Hadamard gate layer (quantum, depth 1, $n=\log_2 N$)
   • Measurement, sign correction, and Walsh spectrum reconstruction The sign correction uses the classical “delta-shift trick” (Rohida et al., 2024).

Quantum circuit steps include $O(N)$ classical preprocessing/postprocessing, parallel Hadamard gates, $N$ 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 $f(\theta_\ell) = \sin(4\ell \pi/N_\theta) +$ noise using $F_p = \sum H_{p,\ell} f(\theta_\ell)$
• Suppressing components at specific sequency $p$, 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 ($f = \tfrac{1}{2}$) equalizes pixel density across annuli, stabilizing sampling.
• Quantum measurement noise: Obtaining accurate WHT coefficients necessitates many shots and error mitigation.
• Restrictive grid sizes: Both $N_r$ and $N_\theta$ 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 $\mathcal{O}(N)$ 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).