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arxiv: 2509.11932 · v2 · submitted 2025-09-15 · 📡 eess.IV

The Filter Echo: A General Tool for Filter Visualisation

Daniel Gaa , Joachim Weickert , Iva Farag , \"Ozg\"un \c{C}i\c{c}ek This is my paper

Pith reviewed 2026-05-18 16:19 UTC · model grok-4.3

classification 📡 eess.IV
keywords filter echodiffusion echoimage inpaintingosmosisoptic flowfilter visualizationimage processingcompression
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The pith

The filter echo generalizes diffusion echoes to visualize filters used in inpainting, osmosis, and variational optic flow computation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper presents the filter echo as a direct generalization of diffusion echoes, which previously visualized only nonlinear diffusion filters. The new concept supports inspection of space-adaptive impulse responses across additional applications, including image inpainting, osmosis, and optic flow estimation. A supporting framework is provided to generate and examine these echoes for different filter types. The work also introduces a compression technique that lowers the storage cost of each echo by a factor between 20 and 100. These steps remove the two main barriers that had kept echo-based visualization from wider use.

Core claim

The filter echo is introduced as a generalisation of the diffusion echo, enabling the visualization of space-adaptive impulse responses for filters beyond adaptive smoothing, including image inpainting, osmosis, and variational optic flow computation, supported by a visualization framework and a compression method that reduces storage by a factor of 20 to 100.

What carries the argument

The filter echo, defined as a space-adaptive impulse response that extends the diffusion echo to arbitrary filters for visualization purposes.

Load-bearing premise

The assumption that the interpretability and useful visualization properties of diffusion echoes transfer directly to non-diffusion filters such as inpainting and optic flow without substantial new validation.

What would settle it

An experiment showing that filter echoes computed for an inpainting method fail to display the expected filling behavior or lose clear interpretability relative to their diffusion counterparts.

read the original abstract

To select suitable filters for a task or to improve existing filters, a deep understanding of their inner workings is vital. Diffusion echoes, which are space-adaptive impulse responses, are useful to visualise the effect of nonlinear diffusion filters. However, they have received little attention in the literature. There may be two reasons for this: Firstly, the concept was introduced specifically for diffusion filters, which might appear too limited. Secondly, diffusion echoes have large storage requirements, which restricts their practicality. This work addresses both problems. We introduce the filter echo as a generalisation of the diffusion echo and use it for applications beyond adaptive smoothing, such as image inpainting, osmosis, and variational optic flow computation. We provide a framework to visualise and inspect echoes from various filters with different applications. Furthermore, we propose a compression approach for filter echoes, which reduces storage requirements by a factor of 20 to 100.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper introduces the filter echo as a generalisation of the diffusion echo, defined as a space-adaptive impulse response, to visualise the behaviour of a range of image filters. It extends the concept to applications including image inpainting, osmosis, and variational optic flow computation, supplies a framework for inspecting these echoes, and proposes a compression scheme claimed to reduce storage requirements by a factor of 20 to 100.

Significance. If the generalisation holds and the visualisations prove insightful for non-diffusion operators, the work could provide a practical diagnostic tool for understanding and refining filters in image processing. The compression approach directly addresses a noted practical barrier. No machine-checked proofs, reproducible code, or falsifiable predictions are described.

major comments (2)
  1. [§3] §3 (Filter Echo Framework): the assertion that visualisation interpretability transfers from diffusion echoes to operators based on energy minimisation (variational optic flow) or filling constraints (inpainting) lacks an explicit re-derivation of the impulse-response properties for these operator classes; the PDE structure exploited in the diffusion case is absent, so the claimed insightfulness is not guaranteed by the original construction.
  2. [§5] §5 (Applications and Experiments): no quantitative validation, error metrics, or baseline comparisons are supplied to demonstrate that the computed echoes isolate adaptive behaviour comparably to the diffusion case or that the compression preserves visual fidelity; this leaves the central claim that the framework is useful for the listed applications unsupported beyond plausibility.
minor comments (1)
  1. [Abstract] Abstract: the factor-of-20-to-100 storage reduction is stated without reference to the specific compression algorithm or the filters and image sizes used in the measurement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and the recommendation of major revision. We respond to each major comment below and indicate the changes we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Filter Echo Framework): the assertion that visualisation interpretability transfers from diffusion echoes to operators based on energy minimisation (variational optic flow) or filling constraints (inpainting) lacks an explicit re-derivation of the impulse-response properties for these operator classes; the PDE structure exploited in the diffusion case is absent, so the claimed insightfulness is not guaranteed by the original construction.

    Authors: The filter echo is introduced as a direct generalisation: it is the output obtained by applying the given filter operator to an input image containing a single impulse. This definition is operator-agnostic and does not presuppose a PDE formulation; it simply records how any filter modifies the impulse. Consequently, the same visual interpretation—local adaptation and support of the response—applies by construction to variational optic flow, inpainting, and osmosis. We nevertheless agree that an explicit paragraph linking the general definition back to the original diffusion-echo properties would improve clarity. We will insert this justification in §3 of the revised manuscript. revision: yes

  2. Referee: [§5] §5 (Applications and Experiments): no quantitative validation, error metrics, or baseline comparisons are supplied to demonstrate that the computed echoes isolate adaptive behaviour comparably to the diffusion case or that the compression preserves visual fidelity; this leaves the central claim that the framework is useful for the listed applications unsupported beyond plausibility.

    Authors: The experiments in §5 are deliberately illustrative, showing that the same visualisation pipeline works across qualitatively different operators. We accept that quantitative support would make the utility claim more robust. In the revision we will add (i) an L2-error table comparing compressed and uncompressed echoes for the compression factor range 20–100 and (ii) a simple quantitative measure (e.g., deviation from isotropy) for at least one non-diffusion application to confirm that the echo isolates the expected adaptive behaviour. These additions will be placed in §5. revision: yes

Circularity Check

0 steps flagged

No significant circularity; filter echo is a conceptual generalization with independent framework and compression method

full rationale

The paper defines the filter echo as a direct generalization of the diffusion echo concept and applies it to new tasks (inpainting, osmosis, variational optic flow) while adding a visualization framework and a compression technique that reduces storage by 20-100x. No equations, derivations, or fitted parameters are presented that reduce the central claims to quantities defined by construction from the inputs or from self-citations. The contribution is self-contained as an independent extension of an existing visualization idea, with the compression approach providing separate practical value. Self-citations to prior diffusion work, if present, are not load-bearing for the new generalization or compression claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the echo visualization concept extends usefully beyond diffusion filters; no free parameters or invented physical entities are mentioned.

axioms (1)
  • domain assumption The visualization utility of diffusion echoes generalizes to other classes of filters such as inpainting and variational optic flow.
    This premise is required for the claim that the filter echo is a practical tool for the listed applications.

pith-pipeline@v0.9.0 · 5699 in / 1250 out tokens · 60714 ms · 2026-05-18T16:19:06.440025+00:00 · methodology

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