An explicit spectral decomposition of the ADRT

TL;DR

This work addresses the inverse problem for the approximate discrete Radon transform (ADRT) by developing a spectral decomposition of its factorized form. The ADRT is written as a product $R^n = S^{n}_{(n)} \cdots S^{n}_{(1)}$, and each factor $S^{n}_{(m)}$ (for $m>1$) is decomposed explicitly into $U^{n}_{(m)} \Sigma^{n}_{(m)} V^{n\top}_{(m)}$, while the cross-quadrant level $S^{n}_{(1)}$ is handled with a tailored 2D block-sinusoidal basis to obtain $S^{n}_{(1)} = U^{n}_{(1)} \Sigma^{n}_{(1)} V^{n\top}_{(1)}$. This spectral structure enables the Spectral Pseudo-Inverse, Fast and Explicit (SPIFE) to compute the Moore–Penrose inverse of $R^n$ in $\mathcal{O}(N^2 \log^2 N)$ operations when the data $b$ lies in the range of $R^n$, and to do so with competitive accuracy relative to iterative methods for moderately sized images. Numerical experiments show SPIFE achieving very high accuracy for in-range data (e.g., $10^{-15}$ for $16\times16$ and $10^{-7}$ for $128\times128$) and highlight stability challenges when data are perturbed away from the range, motivating future work on fast range projection and stabilization. Overall, the approach provides a principled, explicit inverse pathway for the ADRT with provable complexity advantages under the range hypothesis and opens avenues for robust extensions to general data.

Abstract

The approximate discrete Radon transform (ADRT) is a hierarchical multiscale approximation of the Radon transform. In this paper, we factor the ADRT into a product of linear transforms that resemble convolutions and derive an explicit spectral decomposition of each factor. We further show that this implies -- for data lying in the range of the ADRT -- that the transform of an $N \times N$ image can be formally inverted with complexity $\mathcal{O}(N^2 \log^2 N)$. We numerically test the accuracy of the inverse on images of moderate size and find that it is competitive with existing iterative algorithms in this special regime.

Figures (11)

• Figure 1: (a) A depiction of two digital lines in $8 \times 8$ images. $D_8(4,2)$ and $D_8(0,3)$ are shown, (b) full ADRT plot stitching four single-quadrant ADRTs applied to rotated and flipped images, and (c) An example of the single-quadrant ADRT of an image taking on values of $0$s and $1$s. The red dashed line denotes the extent of the square image $u \in \mathbb{R}^{16 \times 16}$. $R^n_{(m)} [u]$ is shown for $m = 0, 1, ...\,, 4$. One single-quadrant ADRT is illustrated, for lines with the angles in $[-\frac{\pi}{2}, -\frac{\pi}{4}]$.
• Figure 2: An example of the vectors $f^n_{m, \ell, t}$, illustrating two different instances of these vectors and labeling the index sets for each of the four sections of the ADRT.
• Figure 3: The orientation of the four quadrants. The four corners are numbered to indicate the precise orientation, and the orientation of quadrant $\textup{III}$ matches that of the original image, that is, $T^n_ \textup{III} = \mathop{\mathrm{Id}}\nolimits^n$\ref{['eq:permute']}.
• Figure 4: (a) The blocks of matrix $Z_{r,s}$\ref{['eq:Zrs']}, and (b) an illustration of $[u, v]_t$ for $u, v \in \mathbb{R}^r$ according to Def. \ref{['def:shiftconcat']}.
• Figure 5: Diagram depicting the flattening permutation $P^n_m$ given in \ref{['eq:P']} (top), and diagram depicting the unflattening permutation $Q^n_m$ given in \ref{['eq:Q']} (bottom).

Theorems & Definitions (13)

• Definition 2.1
• Lemma 3.1
• proof
• Corollary 3.2
• proof
• Definition 3.3: Alternating concatenation with shift
• Lemma 3.4: Block convolution form
• Theorem 3.5: SVD of $S^n_{(m)}$ for $m > 1$
• Theorem 4.1: Spectral decomposition of ${S}^{n}_{(1)}$
• proof