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Reviving Etter method for autoregressive inpainting: Generalization, evaluation, implementation

Ondřej Mokrý, Matěj Hrdlička, Pavel Rajmic

arXiv:2609.18401v1eess.ASeess.SP

TL;DR

Audio inpainting restores missing waveform segments, and this paper revisits Etter’s forward-and-backward autoregressive interpolation method. It extends the formulation to gaps shorter than the AR order and to causal packet-loss concealment. On musical excerpts with gaps up to 80 ms, Etter is competitive with most comparison methods, while gap-wise Janssen remains superior in reconstruction quality.

  • Problem

    Etter’s full autoregressive interpolation method was rarely applied, with no public implementation or prior raw-audio application identified by the authors.

    Source: §I · p. 1

  • Method

    The paper revisits Etter’s joint forward- and backward-AR interpolation and extends it to AR orders exceeding gap length and to left-context-only packet-loss concealment.

    Source: A. Estimating missing samples using… · p. 1, §III · p. 3, §V · pp. 5–6

  • Results

    On musical inpainting gaps up to 80 ms, Etter is competitive with other AR methods, but gap-wise Janssen remains superior in reconstruction quality.

    Source: §V · p. 6

  • Takeaways & Limitations

    Etter ranks among the stronger comparison approaches and is the top-performing method according to PEAQ.

    Source: §IV · p. 5

    The reported experiment concerns musical signals with gaps up to 80 ms.

    Source: §V · p. 6

Abstract

from arXiv · show

Audio inpainting aims to restore missing segments in an audio waveform, as encountered in dropouts and packet losses. This paper revisits the autoregressive (AR) interpolation method proposed by Etter, which combines forward and backward AR prediction through a structured linear system, yet lacks a widely used full implementation for audio signals. We provide an open implementation and propose two practical extensions: A formulation that allows high-order AR models even when the gap length is shorter than the model order, and a causal variant tailored to packet loss concealment. The method is evaluated on musical excerpts with gaps up to 80 ms, using SNR and perceptually motivated objective difference grades, complemented by a listening test. The results show that Etter inpainting matches the performance of most AR- and sparsity-based baseline methods, with the exception of the iterative, gap-wise Janssen approach.

I. INTRODUCTION

Audio inpainting fills missing waveform segments, which can create audible artifacts even when interruptions are short. This paper revisits Etter’s autoregressive interpolation method, whose full application to raw audio has been rare and lacked a public implementation.

I. INTRODUCTION

  • Short gaps in audio can cause clearly audible artifacts, motivating methods that restore missing waveform segments.The gaps arise in settings such as field-recording dropouts and packet losses.Source: §I · p. 1
  • Audio inpainting methods include spectrum-sparsity,, social-sparsity,, dictionary-learning, low-rank, and autoregressive approaches such as Janssen –.Source: §I · p. 1
  • Autoregressive methods remain attractive when low latency and modest computational cost matter, including in packet loss concealment.Source: §I · p. 1
  • Etter’s method was widely cited but rarely used in its full, non-simplified form,, and lacked a publicly available implementation or prior raw-audio application.The authors identify these gaps as limiting reproducible evaluation and wider adoption.Source: §I · p. 1

II. OVERVIEW OF ETTER INTERPOLATION METHOD

This section summarizes the principal steps in the derivation of Etter’s method.

II. OVERVIEW OF ETTER INTERPOLATION METHOD

  • The overview recaps the principal steps of Etter’s method derivation [11, Sec. IV.].Source: §II · p. 1

A. Estimating missing samples using left-sided and right-sided AR coefficients

Etter’s interpolation estimates the missing-sample vector jointly from forward and backward AR predictions based on the known signal on either side of the gap.

A. Estimating missing samples using left-sided and right-sided AR coefficients

  • The missing segment is represented by a single estimated vector, with separate AR models fitted to the known left and right signal portions.Source: A. Estimating missing samples using… · p. 1
  • Left-sided AR coefficients define forward prediction, while right-sided coefficients define backward prediction across the gap.Source: A. Estimating missing samples using… · p. 2
  • For the standard matrix construction, the gap length M is assumed to exceed the AR order K, and filter and known-data matrices separate unknown gap samples from known context.The forward filter is lower-triangular Toeplitz; the backward filter is upper-triangular.Source: A. Estimating missing samples using… · p. 2
  • A unified gap estimate equates the forward and backward predictions and minimizes their total residual energy through a linear system.The symmetric matrix D and vector y incorporate the model structure and boundary conditions from both known segments.Source: A. Estimating missing samples using… · p. 2

B. Estimating AR model

The paper describes two standard strategies for estimating the left- and right-sided AR coefficients: LPC and Burg.

B. Estimating AR model

  • Etter does not specify how to estimate its left- and right-sided AR coefficients, so the paper presents LPC and Burg as standard options.Source: B. Estimating AR model · p. 2
  • LPC estimates AR coefficients by minimizing residual energy, a convex problem efficiently solved with Levinson–Durbin [15],.Source: B. Estimating AR model · p. 2
  • Burg assumes shared AR parameters for the signal and its time-reversed counterpart, ensuring stability of the all-pole filter [17], [18, Sec. 12.3.3].Source: B. Estimating AR model · p. 2

C. Weighted extrapolation

Weighted extrapolation avoids the matrix inversion used to estimate the missing samples by cross-fading independent forward and backward predictors.

C. Weighted extrapolation

  • Weighted extrapolation cross-fades the independent forward and backward predictors using suitable weights, avoiding the matrix inversion required by the full method.The predictors are denoted ˆxL and ˆxR.Source: C. Weighted extrapolation · p. 2
  • The construction neglects both excitation vectors over the gap, setting ˆuL and ˆuR to zero.Source: C. Weighted extrapolation · p. 3
  • The paper calls this approach extrapolation-based inpainting, or simply extrapolation.Source: C. Weighted extrapolation · p. 3

III. PROPOSED MODIFICATIONS

The proposed modifications target practical applicability by extending Etter's derivation to shorter gaps and adapting it to packet-loss concealment constraints.

III. PROPOSED MODIFICATIONS

  • The modifications aim to improve the practical applicability of the original Etter procedure.Source: §III · p. 3
  • The generalized derivation covers gaps whose length M is no greater than the AR model order K.Source: §III · p. 3
  • A further adjustment addresses constraints typical of packet-loss concealment.Source: §III · p. 3

A. Generalization for M ≤K

For gaps no longer than the AR order, the method truncates its structured matrices to the gap length and retains the original linear system for estimating missing samples.

A. Generalization for M ≤K

  • When M ≤ K, the filter and data matrices retain their construction by truncating the Toeplitz structures to the gap length.Source: A. Generalization for M ≤K · p. 3
  • The forward prediction filter has a lower-triangular matrix structure, whereas the backward prediction filter is represented by an upper-triangular matrix, as shown by the displayed coefficient arrangements.Source: A. Generalization for M ≤K · p. 3
  • Known left- and right-context samples are embedded in data matrices, while unknown samples enter the backward prediction through a structured matrix.Source: A. Generalization for M ≤K · p. 3
  • Despite the modified matrices, interpolated samples are computed by solving the same linear system (13) as in the original formulation.Source: A. Generalization for M ≤K · p. 3

B. PLC case

For low-latency packet-loss concealment, the estimate can rely only on left-sided context and reduces to forward extrapolation; a short crossfade can smooth the return to available audio.

B. PLC case

  • With only left-sided context available, the missing segment is estimated directly from the left-sided AR parameters.Source: B. PLC case · p. 3
  • This causal estimate is equivalent to sequential forward extrapolation with the excitation term omitted.Source: B. PLC case · p. 3
  • A short crossfade can smooth the transition to the first available sample after the gap and help avoid click-like artifacts.The paper gives crossfades as one practical option.Source: B. PLC case · p. 3

IV. EXPERIMENTS AND RESULTS

Etter performs strongly against reference inpainting methods, though the listening-test ranking differs in part from objective comparisons. Results also compare estimator choices, extrapolation, and computational cost.

IV. EXPERIMENTS AND RESULTS

  • Objective evaluation estimated AR coefficients using 4096 samples on each side of a gap; validation used a MUSHRA-like listening test on three signals and three gap lengths, omitting SPAIN.The test involved 19 assessors, prescreened as a knowledgeable public.Source: §IV · pp. 4–5
  • Burg estimation consistently yields higher reconstruction quality than LPC, with a Wilcoxon test rejecting equal medians in its favor in all cases.The estimator comparison is shown in Fig. 1.Source: §IV · p. 4, Figure 1 · p. 4
  • PEMO-Q favors Etter over extrapolation except at p = 3072, while other objective metrics suggest extrapolation may be preferable at higher model orders.At lower model orders, the other metrics are inconclusive.Source: §IV · p. 4, Figure 4 · p. 4
  • Model order and gap length affect elapsed time; for long gaps, Burg’s extra runtime over LPC becomes negligible except for extrapolation.For Etter, solving the linear system is the main cost, and its size and structure do not depend on the AR estimator.Source: §IV · pp. 4–5, Figure 5 · p. 5
  • Etter ranks among the stronger reference methods and is the top-performing method according to PEAQ.Fig. 6 presents the baseline comparison; its PEMO-Q ODG and PEAQ ODG axes are cropped.Source: Figure 6 · p. 5, §IV · p. 5
  • Listening-test scores mostly follow PEMO-Q’s ranking, but favor A-SPAIN-MOD more strongly, placing it alongside Etter and extrapolation.Gap-wise Janssen remains superior in the listening-test ranking.Source: §IV · p. 5

V. CONCLUSION

The paper extends Etter inpainting for high-order AR models and packet-loss concealment, and finds it competitive with other AR methods while gap-wise Janssen remains superior in reconstruction quality.

V. CONCLUSION

  • The proposed extensions allow AR model order to exceed gap length and adapt Etter to packet-loss concealment constraints of causality and tight latency.The causal modification uses only left-sided context and AR parameters, yielding sample-wise signal extrapolation.Source: §V · pp. 5–6
  • Fig. 7 shows the distribution of scores in the MUSHRA listening test.Its color coding corresponds to Fig. 6.Source: Figure 7 · p. 6
  • On musical gaps up to 80 ms, Etter is competitive with other AR methods, but gap-wise Janssen remains superior in reconstruction quality.Extrapolation-based inpainting scores similarly to Etter and is preferable in computation time.Source: §V · p. 6
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