Filter Echo: Matrix-Based Image Processing

• Filter echo is a generalisation of diffusion echo that represents the filter action through a state transition matrix, enabling a unified view of both linear and nonlinear filters.
• The framework differentiates source and drain echoes with explicit reconstruction formulas, offering clear insights into how input values propagate and contribute to outputs.
• A compression method using truncated SVD reduces storage from N² to 2Nk values, making the practical visualization of complex filter operations feasible across various applications.

Filter echo is a generalisation of the diffusion echo that represents the action of a filter through its state transition matrix. In the formulation introduced for image processing and computer vision, a filtered image is written as $\mathbf{u}=\mathbf{S}\mathbf{f}$, where $\mathbf{f}\in\mathbb{R}^N$ is the input image, $\mathbf{u}\in\mathbb{R}^N$ is the filtered image, and $\mathbf{S}\in\mathbb{R}^{N\times N}$ encodes the complete filter action. The framework is intended for visualising and inspecting echoes from various filters with different applications, including adaptive smoothing, image inpainting, osmosis, and variational optic flow computation, while a compression method is proposed to reduce storage requirements by a factor of 20 to 100 (Gaa et al., 15 Sep 2025).

1. State-transition formulation

The mathematical basis of filter echo is the state transition matrix of a filtering operation. For a discrete image, the filtering process is expressed as

$\mathbf{u}=\mathbf{S}\mathbf{f}.$

Here, $\mathbf{S}$ is the central object: it describes how every input location contributes to every output location. In this representation, the filter echo is not an auxiliary diagnostic detached from the filter itself; it is a re-expression of the filter action in matrix form (Gaa et al., 15 Sep 2025).

The framework is presented as applicable to a broad class of filters, including linear and nonlinear filters as well as local and space-variant filters. The data emphasize that this extends the earlier diffusion-echo viewpoint beyond adaptive smoothing. A plausible implication is that filter echo functions as a common representational layer for comparing filters whose operational forms differ substantially, provided their action can be captured through the matrix $\mathbf{S}$ or through an operator that acts equivalently on vectors (Gaa et al., 15 Sep 2025).

2. Source and drain echoes

Two dual notions structure the framework: the source echo and the drain echo. They are defined directly from the columns and rows of the state transition matrix.

Echo type	Formula	Interpretation
Source echo $\mathbf{s}_i$	$\mathbf{s}_i=\mathbf{S}\mathbf{e}_i$	Distribution of input value at position $i$ through the filter
Drain echo $\mathbf{f}\in\mathbb{R}^N$0	$\mathbf{f}\in\mathbb{R}^N$1	Receptive field for output position $\mathbf{f}\in\mathbb{R}^N$2

In these definitions, $\mathbf{f}\in\mathbb{R}^N$3 and $\mathbf{f}\in\mathbb{R}^N$4 are unit impulse vectors. The source echo is the output produced by a unit impulse at pixel $\mathbf{f}\in\mathbb{R}^N$5, so it is the $\mathbf{f}\in\mathbb{R}^N$6-th column of $\mathbf{f}\in\mathbb{R}^N$7. The drain echo is the $\mathbf{f}\in\mathbb{R}^N$8-th row of $\mathbf{f}\in\mathbb{R}^N$9, interpreted as the local, space-variant filter kernel at output location $\mathbf{u}\in\mathbb{R}^N$0. Their pointwise relation is

$\mathbf{u}\in\mathbb{R}^N$1

When $\mathbf{u}\in\mathbb{R}^N$2 is symmetric, source and drain echoes are identical up to transposition (Gaa et al., 15 Sep 2025).

The framework also provides explicit reconstruction formulas. From the source echoes,

$\mathbf{u}\in\mathbb{R}^N$3

From the drain echoes,

$\mathbf{u}\in\mathbb{R}^N$4

These identities make the interpretive role of echoes exact rather than heuristic: complete knowledge of the echoes yields complete knowledge of the filter action (Gaa et al., 15 Sep 2025).

3. Relation to diffusion echoes

Filter echo is introduced as a generalisation of the diffusion echo. Diffusion echoes were originally proposed for diffusion filters, where the echo at a location is the evolution of a unit impulse under a possibly nonlinear PDE up to a chosen time. The newer formulation addresses two obstacles identified in the literature: first, diffusion echoes were introduced specifically for diffusion filters and could therefore appear too limited; second, diffusion echoes have large storage requirements, which restrict practicality (Gaa et al., 15 Sep 2025).

Within this broader formulation, diffusion echo becomes a special case. The data state that the general framework unifies the characterisation of space-variant linear and nonlinear filters, including bilateral filters, nonlocal means, inpainting, osmosis, and variational models such as optic flow. This suggests that the key shift is from a diffusion-specific impulse-propagation concept to a fully discrete, matrix-based formulation that can be reused across tasks rather than re-derived for each model family (Gaa et al., 15 Sep 2025).

A common misconception is therefore to treat filter echo as merely a renaming of diffusion echo. The relationship is narrower: diffusion echo is presented as the historical precursor, whereas filter echo is intended as the more general construct. Another persistent limitation is storage, which the paper addresses explicitly through compression rather than through restricting the class of filters under study (Gaa et al., 15 Sep 2025).

4. Applications across image processing and computer vision

The framework is demonstrated on multiple task classes. In image smoothing and denoising, linear diffusion yields echoes that correspond to Gaussian kernels, while nonlinear diffusion produces echoes that reflect adaptive smoothing sensitive to image edges. For anisotropic diffusion such as EED, the echoes become elongated along edges, directly visualising edge-preserving behaviour (Gaa et al., 15 Sep 2025).

In image inpainting, the masking operator alters the interpretation of the echoes: source echoes vanish outside known pixels and show how known data propagate into unknown regions. The data further state that visualisation of echoes makes it possible to inspect the information flow during inpainting and to compare how different PDE-based inpainting models reconstruct geometric structures (Gaa et al., 15 Sep 2025).

In osmosis, which is based on drift-diffusion models and is used for applications such as shadow removal and image cloning, the steady-state echo matrix is reported to reduce to rank 1, and all source echoes align with the steady-state solution. In variational optic flow, including models such as Horn–Schunck and Nagel–Enkelmann, the filter echo describes how normal flow information is propagated to estimate dense flow fields (Gaa et al., 15 Sep 2025).

The visualisation framework is also used comparatively. Echo maps at selected or all positions make locality, directionality, and adaptation visible, and comparisons across filters expose behavioural differences that are otherwise difficult to inspect directly. The data additionally note that, for segmentation with strong nonlinear diffusion, source echoes aligned to segments can directly yield segmentation masks (Gaa et al., 15 Sep 2025).

5. Compression, storage, and computational feasibility

A major practical difficulty is that the echo matrix $\mathbf{u}\in\mathbb{R}^N$5 is typically dense and has size $\mathbf{u}\in\mathbb{R}^N$6. The proposed solution is truncated singular value decomposition,

$\mathbf{u}\in\mathbb{R}^N$7

retaining only the largest $\mathbf{u}\in\mathbb{R}^N$8 singular values and vectors. The Eckart–Young–Mirsky theorem is cited as the optimality guarantee in Frobenius norm. The error is tied to the neglected singular values through

$\mathbf{u}\in\mathbb{R}^N$9

This gives an explicit accuracy–storage trade-off (Gaa et al., 15 Sep 2025).

To avoid explicit formation of $\mathbf{S}\in\mathbb{R}^{N\times N}$0, the framework uses randomised SVD methods that apply the filter to batches of vectors or matrices. In the compressed representation, storage drops from $\mathbf{S}\in\mathbb{R}^{N\times N}$1 floating-point values to $\mathbf{S}\in\mathbb{R}^{N\times N}$2 values. The data give the example that $\mathbf{S}\in\mathbb{R}^{N\times N}$3 corresponds to storing 5% of the full rank and reduces storage by a factor of 20, while some cases, such as EED, allow up to $\mathbf{S}\in\mathbb{R}^{N\times N}$4 reduction while maintaining quality (Gaa et al., 15 Sep 2025).

The reconstruction of individual echoes remains simple: $\mathbf{S}\in\mathbb{R}^{N\times N}$5 The paper also proposes an exclusion mechanism for cases with near-impulse echoes, such as rapidly decaying diffusivities like Weickert’s: echoes whose central pixel is almost 1 are stored separately as impulses, and the remainder is compressed by SVD. Experimentally, the data report that on $\mathbf{S}\in\mathbb{R}^{N\times N}$6 images, storage can be reduced from 16GB to a few hundred MB or less, and that key echo structures remain visible even when only 0.5–5% of singular values are kept, except for highly localised echoes (Gaa et al., 15 Sep 2025).

6. Terminological divergence in acoustic echo cancellation

The phrase “filter echo” also appears in a distinct sense in acoustic echo cancellation. In that literature, it is used for the imperfect echo estimate produced by linear adaptive filters and denotes residual echo left after linear cancellation rather than a matrix-based visualisation object. One paper states explicitly that “filter echo” refers to the imperfect echo estimate produced by linear adaptive filters such as MDF or wRLS, motivating a cascaded design in which a neural post-filter suppresses residual non-linear echo components (Zhang et al., 2022).

This usage differs fundamentally from the image-processing concept. In acoustic echo cancellation, hybrid systems combine adaptive filters with neural modules to suppress residual echo, improve double-talk robustness, or accelerate reconvergence. The frequency-domain adaptive Kalman filter is one such linear front end, and later neural Kalman filter variants were investigated specifically to improve covariance estimation, convergence, and double-talk near-end speech preservation in acoustic echo cancellation (Seidel et al., 22 Jan 2025). This suggests that the term is polysemous across signal-processing subfields: in image processing it denotes a visualisation framework based on state-transition structure, whereas in acoustic echo cancellation it denotes an artifact or residual left by imperfect filtering.

The distinction matters because the two usages imply different mathematical objects, different evaluation criteria, and different research goals. The former seeks interpretability and inspection of filter action; the latter seeks suppression of an unwanted residual signal.