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arxiv: 2606.09264 · v1 · pith:CVXHZTGInew · submitted 2026-06-08 · 📡 eess.SP

Block-Term Decomposition Approach to Blind Multi-trial Functional Ultrasound Unmixing

Sofia-Eirini Kotti , Eleftherios Kofidis , Borb\'ala Hunyadi This is my paper

Pith reviewed 2026-06-27 15:44 UTC · model grok-4.3

classification 📡 eess.SP
keywords functional ultrasoundblind source separationtensor decompositionhemodynamic response functionneurovascular couplingmulti-trial imaging
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The pith

A block-term tensor decomposition model enables blind unmixing of multi-trial functional ultrasound data into spatial maps, activations, and responses.

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

The paper introduces a tensor-based model to separate latent neuronal sources from functional ultrasound signals, which only capture blood flow changes indirectly. Each source is represented by a low-rank spatial map, a piecewise-constant activation signal, and a trial-specific hemodynamic response function of plausible shape. An optimization method using alternating projected gradient descent computes the factors from multi-trial measurements. Simulations demonstrate reliable recovery of maps and activations even with noise, though HRF estimation proves more difficult. This matters for interpreting high-resolution brain imaging without assuming known response shapes.

Core claim

We propose a data-driven convolutive block-term tensor decomposition-based model for multi-trial fUS measurements, where each source has a spatiotemporal representation comprising a low-rank spatial map and a piecewise-constant neuronal activation signal convolved with a trial- and source-dependent hemodynamic response function (HRF) with a physiologically plausible shape. We propose a constrained optimization framework for the model computation, which consists of alternating projected gradient descent iterations. Simulation results demonstrate accurate recovery of spatial maps and reliable estimation of activation temporal profiles across various noise levels.

What carries the argument

The convolutive block-term tensor decomposition that models each source's activity as the convolution of a low-rank spatial map with a piecewise-constant neuronal signal and a source- and trial-dependent HRF.

Load-bearing premise

Each source's neuronal activation is piecewise-constant and the trial- and source-dependent HRF has a physiologically plausible shape that can be estimated jointly with the other factors.

What would settle it

If simulations with known ground-truth activations that are not piecewise-constant yield spatial maps that fail to align with expected regions even at low noise levels.

Figures

Figures reproduced from arXiv: 2606.09264 by Borb\'ala Hunyadi, Eleftherios Kofidis, Sofia-Eirini Kotti.

Figure 1
Figure 1. Figure 1: Proposed model for multi-trial fUS measurements. are denoted by h (k) r ∈ R Nh×1 , and are assumed to have common length Nh < Nt, without loss of generality. If we stack all Y (k) into a four-dimensional tensor and subsequently reshape it to a three-dimensional tensor by combining the spatial dimensions into one, we arrive at tensor X ∈ R Ns×Nt×K, with Ns = NzNx, whose frontal slices are the trial-specific… view at source ↗
Figure 2
Figure 2. Figure 2: Boxplots of the relative error for the estimated quantities per BTD component during training at different SNR levels, defined as ∥x − xˆ∥2/∥x∥2 for quantity x and its estimate xˆ. Each boxplot in (a) and (b) includes 25 points (datasets), and, in (c), 25 × 4 (datasets × number of trials). λ differs per dataset and SNR. Colored boxes represent the IQR, containing the middle 50% of the data between the 25th… view at source ↗
Figure 3
Figure 3. Figure 3: As in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: As in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Estimation results for a single dataset. [13] N. D. Sidiropoulos et al., “Tensor decomposition for signal processing and machine learning,” IEEE Trans. Signal Process., vol. 65, no. 13, pp. 3551–3582, Jul. 2017. [14] B. Hunyadi et al., “Block term decomposition for modelling epileptic seizures,” EURASIP J. Adv. Signal Process., vol. 2014, no. 1, 2014. [15] C. Chatzichristos et al., “Blind fMRI source unmix… view at source ↗
read the original abstract

Functional ultrasound (fUS) has emerged as a powerful neuroimaging modality due to its high resolution in both space and time, low cost and potential portability. Nevertheless, fUS signals provide only indirect observations of neuronal activity through the neurovascular coupling, and hence require the blind separation of latent neuronal sources while also deconvolving their hemodynamic responses. In this work, we propose a data-driven convolutive block-term tensor decomposition-based model for multi-trial fUS measurements, where each source has a spatiotemporal representation comprising a low-rank spatial map and a piecewise-constant neuronal activation signal convolved with a trial- and source-dependent hemodynamic response function (HRF) with a physiologically plausible shape. We propose a constrained optimization framework for the model computation, which consists of alternating projected gradient descent iterations. Simulation results are reported that demonstrate accurate recovery of spatial maps and reliable estimation of activation temporal profiles across various noise levels, while confirming that HRF estimation remains the most challenging part of the problem.

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 / 2 minor

Summary. The paper proposes a convolutive block-term tensor decomposition model for blind unmixing of multi-trial functional ultrasound (fUS) data. Each neuronal source is represented by a low-rank spatial map and a piecewise-constant activation signal convolved with a trial- and source-dependent HRF of physiologically plausible shape. The authors introduce a constrained optimization framework solved via alternating projected gradient descent and report simulation results claiming accurate recovery of spatial maps and temporal profiles across noise levels, while identifying HRF estimation as the most challenging component.

Significance. If validated with quantitative metrics and real data, the approach could advance blind source separation in fUS by providing a data-driven tensor framework that jointly handles spatial unmixing, activation estimation, and HRF deconvolution. The explicit modeling of trial-dependent HRFs and the use of block-term structure are strengths, as is the constrained alternating optimization. However, the current reliance on simulations without reported error metrics, ablation studies, or baseline comparisons limits the assessed impact.

major comments (2)
  1. [Abstract / Simulation results] Abstract and simulation results section: the claims of 'accurate recovery of spatial maps' and 'reliable estimation of activation temporal profiles' are presented without any quantitative metrics (e.g., correlation, NMSE, or Dice scores), specific noise models, number of trials, or comparisons to existing methods, which is load-bearing for evaluating whether the simulations support the central claims.
  2. [Constrained optimization framework] Model definition and optimization section: no convergence guarantees, initialization procedure, or sensitivity analysis for the alternating projected gradient descent are provided, which is critical because the reported recoveries depend on successful optimization of the joint factors including the HRF.
minor comments (2)
  1. [Abstract] The abstract should be expanded to include at least one quantitative result or table reference from the simulations to substantiate the recovery claims.
  2. [Model formulation] Notation for the block-term decomposition and the convolutive model should be introduced with explicit tensor dimensions and rank parameters for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and will revise the manuscript to incorporate quantitative evaluations and additional optimization details where feasible.

read point-by-point responses

  1. Referee: [Abstract / Simulation results] Abstract and simulation results section: the claims of 'accurate recovery of spatial maps' and 'reliable estimation of activation temporal profiles' are presented without any quantitative metrics (e.g., correlation, NMSE, or Dice scores), specific noise models, number of trials, or comparisons to existing methods, which is load-bearing for evaluating whether the simulations support the central claims.

    Authors: We agree that quantitative metrics are necessary to substantiate the claims. In the revised manuscript, we will add Pearson correlation coefficients and NMSE values for the recovered spatial maps and activation profiles, specify the additive white Gaussian noise model and number of trials (e.g., 10), and include comparisons against baselines such as ICA and NMF to demonstrate relative performance across noise levels. revision: yes

  2. Referee: [Constrained optimization framework] Model definition and optimization section: no convergence guarantees, initialization procedure, or sensitivity analysis for the alternating projected gradient descent are provided, which is critical because the reported recoveries depend on successful optimization of the joint factors including the HRF.

    Authors: We will expand the optimization section to describe the initialization procedure (random factors with a canonical HRF template) and include a sensitivity analysis reporting success rates over multiple random initializations. Convergence guarantees are not provided because the joint optimization is non-convex; we will instead report empirical convergence via objective value plots and discuss practical reliability in the revised text. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper proposes a data-driven convolutive block-term tensor decomposition model for multi-trial fUS measurements, with each source represented by a low-rank spatial map, piecewise-constant activation convolved with a trial- and source-dependent HRF, and a constrained optimization framework solved via alternating projected gradient descent. Simulation results demonstrate recovery of spatial maps and temporal profiles. No equations, fitting procedures, or self-citations are shown that would reduce the reported recoveries or estimates to quantities defined by the same fitted parameters by construction. The model is explicitly data-driven with stated assumptions, and the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; the model rests on domain assumptions about signal structure and HRF shape that are stated but not derived.

axioms (1)
  • domain assumption Each source has a piecewise-constant neuronal activation signal convolved with a trial- and source-dependent HRF of physiologically plausible shape.
    Explicitly stated in the abstract as part of the generative model.

pith-pipeline@v0.9.1-grok · 5705 in / 1173 out tokens · 20248 ms · 2026-06-27T15:44:15.191083+00:00 · methodology

discussion (0)

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Reference graph

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