Source-linked AI summary
Optimal transport of image sources for interpolation of room impulse responses with moving sources
Jesper Brunnström, Filip Elvander, Isabel Haasler
TL;DR
Moving physical sources also move their image sources, complicating RIR interpolation from endpoint measurements, especially when visibility changes leave unmatched sources. The paper derives geometry-informed costs and a dummy cost for POT; simulations show the approach is more accurate than linear interpolation, with maximum-likelihood and source-informed costs particularly effective.
Problem
Method
The method uses a statistical model of image-source motion to derive POT costs and a dummy-point cost for matching and interpolating endpoint image-source sets.
Source: §1 · p. 1, §4.1 · p. 2, §5 · p. 3Results
Simulations show the proposed POT approach is more accurate than linear interpolation, with maximum-likelihood and source-informed costs particularly effective.
Source: §7 · p. 4Takeaways & Limitations
Source-informed costs are an overall pragmatic choice, providing performance similar to maximum likelihood while being simpler to use.
Source: §6.4 · p. 4The NMSE metric is ill-posed at infinite bandwidth, where any non-zero position error yields the maximum NMSE for that image source.
Source: §6.1 · p. 4
Abstract
from arXiv · showhide
In geometrical acoustics, room impulse responses (RIRs) can be represented by a set of image sources in free space. For a fixed source position, the image sources allow for computing RIRs at arbitrary receiver positions. However, for a moving physical source, the image sources also move, making interpolation more difficult. In this paper we develop an interpolation method for image source positions of a moving source, given image source positions at the start and end of the trajectory. The method exploits the fact that each image source moves the same distance as the physical source. A statistical model is developed to derive cost functions and an appropriate dummy cost used in the proposed partial optimal transport (POT) approach. Through simulated experiments, POT using the proposed cost functions is shown to be effective compared to the alternatives.
1. INTRODUCTION
RIR interpolation matters across audio applications, and image sources offer a way to represent acoustic paths. For moving sources, the image-source sets change, motivating a method that matches shared sources and discards unmatched ones.
1. INTRODUCTION
- RIRs are important in audio applications, and representing them with free-space sources makes interpolation to other receiver positions straightforward.The image-source method represents each source as corresponding to a sequence of specular reflections.Source: §1 · p. 1
- Because image-source positions depend on physical-source position, moving sources require an interpolation method.The image-source positions are independent of receiver position, but not of source position.Source: §1 · p. 1
- Visibility changes can make endpoint image-source sets differ, so the assignment must match shared sources and discard poor matches using partial optimal transport.The paper assumes room-wall geometry is unavailable and obtains image-source positions from collected sound data.Source: §1 · p. 1
- The method models the shared-distance motion of image sources to derive POT costs and a principled way to determine which points to discard.It addresses a moving physical source, unlike the moving-receiver POT approach, and explicitly models image sources unlike the OT approach in.Source: §1 · p. 1
2. BACKGROUND
The image-source method represents an RIR using reflected-source contributions. Image-source locations follow reflections through room faces, although visibility determines which sources are available at a receiver.
2.1. Image-source method
- An RIR is modeled as a sum of image-source contributions, each a scaled and delayed impulse; in practice, the impulse is bandlimited.The ideal infinite-bandwidth model is a sum of time-domain Green’s functions [24, Ch.7].Source: §2.1 · p. 1
2.2. Position of image sources
- First-order image sources are formed by mirroring the physical source in each room face, and higher-order sources by reflecting lower-order sources in sequence.The reflection sequence defines an affine transformation whose linear component is orthogonal and whose offset depends on room geometry.Source: §2.2 · pp. 1–2
- Only a subset of geometrically placed image sources may be visible to a receiver, so sources can appear or disappear as source or receiver positions change.Visibility limits which image sources are simulated or measurable in practice.Source: §2.2 · p. 2
3. PROBLEM STATEMENT
The task is to estimate intermediate image-source measures from noisy endpoint estimates without knowing room geometry. The endpoint sets may be unordered and differ because of visibility changes or missed detections.
3. PROBLEM STATEMENT
- Image-source weights are one for visible sources and zero otherwise, reflecting potential occlusion as the source moves.The endpoint image-source positions are modeled as noisy estimates.Source: §3 · p. 2
- Estimate the image-source measure along the source trajectory from noisy endpoint estimates, without explicit room geometry, so RIRs can be interpolated.The physical source follows a linear path between its known endpoint positions.Source: §3 · p. 2
- Each measured endpoint set may contain a different number of unordered image sources because of visibility changes or missed detections.The endpoint estimates are denoted ν0 and ν1 and correspond to known physical-source positions.Source: §3 · p. 2
- The unknown correspondence is an injection that identifies measurements of the same image source, with unmatched sources representing detections present at only one endpoint.When all sources are detected at both endpoints, the correspondence reduces to a permutation.Source: §3 · p. 2
4. ALGORITHM
The algorithm uses partial optimal transport to match image sources while allowing unmatched points to go to dummy points. It interpolates matched positions by displacement and fades unmatched points, becoming exact under ideal assignment and noise-free conditions.
4.1. Partial optimal transport
- POT uses a ground-cost matrix and dummy-point transport plans to estimate a partial one-to-one matching between endpoint image-source sets.The dummy-point transport cost is ξ, and an optimal plan exists with binary matching entries.Source: §4.1 · p. 2
4.2. Interpolation
- Because each image source moves the same distance as the physical source, the method interpolates matched image-source positions by displacement along linear trajectories.The displacement interpolation is applied after solving the POT problem.Source: §4.2 · p. 2
- The interpolation fades out points assigned to dummy points, and is exact when assignment is recovered, no mass goes to dummy points, and endpoint positions are noise-free.Matched sources are transported between endpoints; interpolated weights may take any value in [0, 1].Source: §4.2 · p. 2
5. STATISTICAL MODEL
The statistical model represents matched image-source residuals using a known displacement length but unknown direction, then uses that model to derive assignment costs and a principled rejection threshold.
5. STATISTICAL MODEL
- The residual model treats the displacement as a known length γ with unknown direction and models measurement error as Gaussian noise.The section assumes a permutation, so all image sources are matched across endpoints.Source: §5 · pp. 2–3
5.1. Assignment maximum likelihood estimator
- Assuming independent residuals, the maximum-likelihood assignment is obtained by maximizing the likelihood of the measured image-source positions.Independence is motivated by the lack of room-geometry knowledge; knowledge of geometry or reflection order would imply dependence.Source: §5.1 · p. 3
5.2. Ground cost
- The maximum-likelihood assignment is equivalent to balanced optimal transport with a statistically motivated ground cost.The paper also considers Euclidean and source-informed costs; the maximum-likelihood cost is asymptotically equivalent to Euclidean when σ ≫ γ, and to source-informed when γ ≫ σ.Source: §5.2 · p. 3
5.3. Choosing dummy cost parameter ξ
- The dummy cost ξ is chosen so that the probability a matched pair exceeds cost 2ξ equals the selected rejection probability α.The model’s distribution for the distance r makes this probability solvable numerically for each cost function.Source: §5.3 · p. 3
6. EXPERIMENTAL VALIDATION
Across simulated statistical and image-source-method data, the proposed POT methods generally outperform linear interpolation, with source-informed and maximum-likelihood costs especially effective. The experiments define assignment and RIR error metrics and compare methods on both data types.
6.1. Error metrics
- Assignment error measures how well the correspondence is estimated, while NMSE evaluates RIR error aggregated along the trajectory.The NMSE metric is ill-posed at infinite bandwidth: any nonzero position error produces its maximum value.Source: §6.1 · pp. 3–4
6.4. Results
- Source-informed and maximum-likelihood costs nearly attain oracle performance on statistical data, while the gap is smaller on ISM data; all POT methods beat linear interpolation except at very high noise.For high-noise ISM data, Euclidean performs better; source-informed is the pragmatic choice overall because it performs similarly to maximum likelihood with a simpler cost.Source: Figure 2 · p. 4, §6.4 · p. 4
6.2. Methods
- The comparisons include oracle, linear interpolation, and no-transport baselines alongside the proposed maximum-likelihood, source-informed, and Euclidean POT methods.Oracle uses the ground-truth assignment; no transport provides an assignment-error baseline.Source: §6.2 · p. 4
6.3. Data
- The experiments use both statistical-model data and ISM data generated for random rooms, with 16 receivers in each data type.The statistical data varies visibility along the trajectory; the ISM data uses image sources generated up to order 3.Source: §6.3 · p. 4
7. CONCLUSION
Simulated experiments show that the proposed POT approach is more accurate than linear interpolation, with maximum-likelihood and source-informed costs particularly effective.
7. CONCLUSION
- The proposed POT approach is more accurate than linear interpolation in simulated experiments.The experiments use statistical data and data from the image-source method for random rooms.Source: §7 · p. 4
- Maximum-likelihood and source-informed costs are particularly effective in the simulated experiments.Source: §7 · p. 4
- The method interpolates image sources using POT and derives appropriate costs through a statistical model informed by a maximum-likelihood assignment estimator.Source: §7 · p. 4