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Extended Target Imaging in Non-Ideal Distributed ISAC Systems: A Hierarchical Off-Grid Approach

Xin Tong, Yu Ge, Alireza Pourafzal, Dario Tagliaferri, Henk Wymeersch

arXiv:2610.33500v1eess.SP

TL;DR

Extended-target imaging in distributed ISAC must handle continuous anisotropic surfaces, occlusion, and synchronization errors. This paper uses coarse incoherent initialization followed by visibility-constrained coherent off-grid reconstruction, and reports stable convergence where OG-AMP does not converge effectively because it leaves spatial-orientation mismatches uncompensated.

  • Problem

    Continuous target geometry, anisotropic scattering, and changing visibility make reconstruction difficult, while synchronization errors further couple with geometry and scattering.

    Source: §I · p. 1

  • Method

    A hierarchical framework initializes geometry and synchronization incoherently, then performs coherent position-orientation off-grid reconstruction with BiOG-AMP updates to visibility and dictionary matrices.

    Source: §I · p. 1, A. Pixel Model for Extended Target · p. 6, B. First-Order Taylor Approximation… · p. 6

  • Results

    BiOG-AMP converges stably with major gains within a few iterations, while OG-AMP fails to converge effectively due to uncompensated spatial-orientation mismatches.

    Source: C. Robustness and Convergence Evaluation · p. 12

  • Takeaways & Limitations

    The framework is reported to provide accurate reconstruction and robustness against synchronization errors and occlusions in D-ISAC extended-target imaging.

    Source: §VIII · p. 12

Abstract

from arXiv · show

In future integrated sensing and communication (ISAC) systems, distributed multiple-input multiple-output (MIMO) architectures offer high-resolution imaging capabilities. However, reconstructing real-world extended targets is physically challenging due to continuous anisotropic surface scattering and complex occlusions, especially in the presence of system synchronization errors. To overcome these limitations, this paper proposes a hierarchical two-stage computational imaging framework for extended target reconstruction. In Phase I, an incoherent coarse imaging process is performed to serve as a reliable initializer for subsequent high-resolution coherent image generation. By performing line-of-sight (LOS) interference suppression and leveraging network-wide topological constraints, timing and phase offsets are calibrated. This enables image construction by back-projection (BP) and the coarse identification of target contours using the probabilistic Hough transform (PHT). In Phase II, a joint target position and orientation off-grid compensation model is established, capturing Lambertian diffuse and specular reflections on continuous facets via a first-order Taylor expansion. To tackle severe occlusion and spatial grid mismatches, we propose the bilinear off-grid approximate message passing (BiOG-AMP) algorithm. BiOG-AMP iteratively updates surface visibility and dictionary matrices, enforcing physical consistency across distributed transceiver views. Theoretical analysis derives compressed sensing error bounds, Fisher information limits, and linearization error scaling. Simulation results demonstrate that the proposed framework effectively eliminates imaging artifacts and exhibits superior robustness against synchronization offsets and grid mismatches compared to state-of-the-art baselines.

I. Introduction I

Distributed MIMO can use multi-view LOS and NLOS measurements for high-resolution imaging, but continuous target geometry, anisotropic scattering, occlusion, and synchronization errors complicate reconstruction. The paper proposes a two-stage framework that initializes geometry and synchronization before coherent off-grid reconstruction with visibility constraints.

I. Introduction I

  • Distributed MIMO captures multi-perspective scattering, while LOS and NLOS paths together provide spatial measurements for reconstructing continuous target contours.The paper identifies high-resolution RF imaging as useful for fine-grained environment sensing.Source: §I · p. 1
  • Independent local oscillators introduce timing and phase offsets that couple with target geometry, scattering, and visibility, making direct coherent optimization unstable.Phase ambiguities can trap algorithms in physically distorted local solutions.Source: §I · p. 1

A. Related Works

  • Existing methods do not jointly model continuous position-orientation mismatch, anisotropic scattering, and dynamic visibility under distributed synchronization errors.Prior approaches often assume isotropic point targets or recover discrete point clusters before connecting them into surfaces.Source: A. Related Works · p. 2

B. Contributions

  • The framework combines robust incoherent initialization with coherent reconstruction to address coupled geometry-dependent dictionaries and visibility patterns.Phase I uses LOS suppression, BP, and PHT for coarse layout and boundary initialization; Phase II refines position and orientation off-grid while updating visibility.Source: B. Contributions · pp. 2–3

A. Environment Setting and Synchronization Model

  • The D-ISAC model aggregates CSI across links and subcarriers, with direct LOS links and target-reflected NLOS paths; clutter outside the ROI is treated as noise.LOS links support OTA synchronization, while low wave penetration yields surface scattering and orientation-dependent occlusion.Source: A. Environment Setting and… · p. 3

B. Computational Imaging Model for Extended Targets

  • Each pixel is modeled by reflectivity and surface orientation, making effective reflectivity direction-dependent across links.The hybrid reflection model combines quasi-isotropic diffuse scattering with directional specular energy; inter-pixel occlusion is handled through visibility indicators.Source: B. Computational Imaging Model for… · p. 3

A. LOS Suppression and Coarse Synchronization

Phase I detects and suppresses dominant LOS components, estimates their timing and phase offsets, then uses network topology to calibrate offsets across links, including links without direct LOS observations.

A. LOS Suppression and Coarse Synchronization

  • The dominant LOS component is suppressed to expose weaker target-related NLOS reflections before imaging.Frequency-domain CSI is transformed to the delay domain to obtain an impulse response for LOS detection.Source: A. LOS Suppression and Coarse… · p. 4
  • The method detects a dominant LOS path near its predicted delay, then estimates coarse timing and phase from the frequency-dependent CSI phase.The predicted delay uses known node positions; the search window accommodates discrete sampling offsets, and regression estimates phase slope and intercept.Source: A. LOS Suppression and Coarse… · p. 4
  • LOS timing and phase estimates are refined across the network by recovering antenna-level offsets from valid LOS links and reconstructing offsets for every link.The least-squares system uses the incidence matrix; the LOS-detected Tx-Rx graph must be connected for the stated inverse operations.Source: A. LOS Suppression and Coarse… · p. 5
  • For NLOS-only links, phase offsets are inferred from the recovered antenna-level phase vector rather than directly observed.The link phase is reconstructed as the relative difference between its Tx and Rx node-level offsets, with phase wrapping handled in the inversion.Source: A. LOS Suppression and Coarse… · p. 5

B. Incoherent Imaging via Space-Time Matched Filter

The incoherent Phase I image uses calibrated NLOS power profiles and back-projection to estimate coarse geometry. Probabilistic Hough processing then extracts and merges line segments into target boundaries and orientation priors for Phase II.

B. Incoherent Imaging via Space-Time Matched Filter

  • Links with detected LOS paths have the dominant geometric reference subtracted, while LOS-occluded links use their synchronized signal as calibrated NLOS output.The resulting target-related NLOS CSI is the input to the incoherent imaging stage.Source: B. Incoherent Imaging via Space-Time… · p. 5
  • Calibrated NLOS signals produce power delay profiles that feed back-projection to form a coarse target-layout image.For each candidate location, BP accumulates energy over links and subcarriers using the theoretical bistatic propagation delay.Source: B. Incoherent Imaging via Space-Time… · p. 5

C. Target Extraction

  • Thresholding the BP image and applying PHT identifies principal line segments as coarse extended-target boundaries.PHT samples image points to detect discrete segments; nearby fragments are merged by spatial and angular proximity, then PCA estimates target extent and orientation.Source: C. Target Extraction · pp. 5–6
  • Each extracted target is represented by segment endpoints and a surface normal that initialize Phase II pixel positions and normals.The endpoint and orientation estimates are derived from the merged segment clusters.Source: C. Target Extraction · p. 6

IV. Phase II-A: Coherent Precision Imaging Model

Phase II coherently refines reflectivity, off-grid position and orientation, and link-pixel visibility within the identified ROI. Its oriented-facet model and first-order approximation represent continuous surfaces while accommodating deviations from the reference grid.

IV. Phase II-A: Coherent Precision Imaging Model

  • Phase II jointly refines reflectivity, off-grid offsets, and link-pixel occlusion status through coherent imaging within the identified ROI.The initialization comes from Phase I.Source: §IV · p. 6

A. Pixel Model for Extended Target

  • Each pixel is modeled as a continuous linear facet whose propagation kernel integrates scattering across the facet rather than evaluating a single point.The facet is parameterized by its grid-center position and orientation.Source: A. Pixel Model for Extended Target · p. 6

B. First-Order Taylor Approximation and Linearization

  • First-order Taylor expansion approximates the unknown kernel using position and orientation offsets from the Phase-I reference grid and surface normal.The resulting per-pixel state includes reflectivity and auxiliary position and orientation offset variables.Source: B. First-Order Taylor Approximation… · pp. 6–7

C. Problem Formulation

The reconstruction is formulated as joint estimation of target state and link-pixel visibility under geometric constraints. The model combines a likelihood, sparsity and state priors, and visibility determined by target geometry.

C. Problem Formulation

  • The formulation jointly estimates the state and visibility by maximizing the posterior probability given the NLOS observations and a geometry-dependent dictionary.Visibility is constrained by the geometric visibility function for each link and pixel.Source: C. Problem Formulation · p. 7
  • Visibility accounts for self-occlusion from surface orientation and mutual occlusion from other targets, with facing constraints forcing some pixels to be invisible.The shadowing term represents obstruction along the transmitter-to-pixel or pixel-to-receiver paths.Source: C. Problem Formulation · p. 7, Figure 5 · p. 7
  • The factorization combines an observation likelihood, a jointly sparse prior over each pixel's state components, a geometric visibility factor, and a sparse activation prior.The pixel-state components are activated together through a shared activation variable.Source: C. Problem Formulation · pp. 7–8

V. Phase II-B: BiOG-AMP Algorithm

BiOG-AMP jointly estimates target state and occlusion while tying visibility to target geometry. It uses iterative message passing and alternates state updates with geometric-visibility updates to stabilize convergence.

V. Phase II-B: BiOG-AMP Algorithm

  • BiOG-AMP couples occlusion status to extended-target geometry rather than treating the visibility mask as an unconstrained random variable.The stated aim is stable convergence and physical consistency.Source: §V · p. 8
  • The state-vector messages follow sum-product and AMP approximations, using Gaussian approximations for aggregated factor-node inputs.The updates for reflectivity and the position and orientation offset variables share the same mathematical structure.Source: A. Approximate Message Passing · p. 8
  • The algorithm alternates state-vector message passing with geometric-visibility updates, treating visibility as fixed during each state-message stage.The alternating scheme is reported to reduce cyclic-graph oscillations and stabilize convergence.Source: A. Approximate Message Passing · p. 8

B. Estimation and Iterative Execution

The iterative procedure estimates active pixels and their states, then updates the dictionary and occlusion matrix using the current geometry. Its reported complexity scales linearly with the numbers of links, pixels, and subcarriers.

B. Estimation and Iterative Execution

  • The procedure estimates active pixels and their state components, then feeds the refined estimates back to update the dictionary and occlusion matrices.The reported outputs include reflectivity and position and orientation offsets.Source: 2) Estimation for the State Vector · p. 9, 4) Iterative Execution · p. 9
  • BiOG-AMP dynamically updates the dictionary and visibility from the current estimated geometry, rather than keeping either fixed during reconstruction.Updated dictionaries are aligned with offset estimates, while visibility masking removes blocked paths from subsequent message passing.Source: 3) Occlusion and Dictionary Matrix… · p. 9
  • O(MNK) operations per iteration give the overall imaging complexity, compared with O((4N)^3) for MMSE matrix inversion.The dominant cost is matrix-vector multiplication; four state components are used per pixel.Source: 4) Iterative Execution · p. 9

VI. Theoretical Performance Analysis

The theoretical analysis establishes conditions for stable sparse recovery with incomplete, visibility-limited measurements. It also shows that error depends on target orientation, not simply the total number of links.

A. Ideal Imaging Performance

  • Extended-target occlusion leaves each pixel illuminated by only a portion of the links, producing incomplete observations even in the ideal baseline.The baseline assumes perfect synchronization and grid alignment.Source: A. Ideal Imaging Performance · pp. 9–10
  • Stable sparse recovery is guaranteed when the effective number of unoccluded, well-aligned measurements scales as K times the number of links times visibility, at least C s log(N/s).Here s is the sparsity level, and C is governed by the structured dictionary's restricted isometry property.Source: A. Ideal Imaging Performance · p. 10
  • Reconstruction error may increase despite a large link count when target orientation causes severe self-occlusion or unfavorable specular reflection angles.Thus accuracy is not a simple linear function of the total number of links.Source: A. Ideal Imaging Performance · p. 10

B. Impact of Extended Geometry on Sensing Performance

The analysis relates extended-target geometry to reflectivity Fisher information and characterizes how position and orientation mismatch contribute to linearization error.

B. Impact of Extended Geometry on Sensing Performance

  • Specular alignment sharply increases local Fisher information and reduces the reflectivity Cramér–Rao bound, while complete occlusion creates a sensing blind zone.Diffuse scattering contributes a positive term from every visible link, making the local Fisher information increase monotonically with link count.Source: B. Impact of Extended Geometry on… · p. 10
  • Visibility acts as a binary mask on each link's contribution to local Fisher information; removing visible links reduces that information.With all links occluded, the local Fisher information is zero.Source: B. Impact of Extended Geometry on… · p. 10

C. Model Approximation Error

  • First-order Taylor expansions accommodate continuous target positions and orientations on discrete grids, with spatial mismatch producing a second-order remainder.The passage describes the variance of this remainder under uniform pixel displacement.Source: C. Model Approximation Error · p. 10
  • Orientation error retains the λ^-4 pixel scaling of spatial grid mismatch and adds a quantization penalty proportional to the fourth moment of orientation offset.For a continuous linear facet, orientation offset induces a position displacement along the facet.Source: C. Model Approximation Error · p. 10

VII. Numerical Results

The simulations compare the proposed framework with GAMP and OG-AMP across imaging stages and robustness settings. BiOG-AMP achieves millimeter-level precision at high SNR, remains robust to synchronization and pixel-size variation, and converges stably.

A. Simulation Setting and Metrics

  • The simulations use 30 transmitters and 30 receivers around a 20 m × 20 m region, with performance curves averaged over 1000 Monte Carlo runs.Phase I uses a 100 × 100 pixel grid, while Phase II uses 0.1 m spacing around identified targets.Source: A. Simulation Setting and Metrics · pp. 10–11

B. Performance Evaluation

  • Across a representative T-shaped target, Phase I extracts coarse geometry under severe synchronization errors, while complete BiOG-AMP restores the profile after baseline degradation from off-grid and orientation offsets.Neglecting occlusion produces physically inconsistent reconstruction with spurious artifacts.Source: B. Performance Evaluation · p. 11, Figure 6 · p. 11
  • Phase I maintains low chamfer distance across SNR regimes, closely approaching ideal performance and providing a geometric prior for Phase II.LOS suppression and graph-based mapping of link timing offsets to node delays support this performance.Source: B. Performance Evaluation · p. 11, Figure 7 · p. 12
  • BiOG-AMP outperforms GAMP and OG-AMP in Phase II, achieving millimeter-level precision at high SNR.GAMP plateaus near 0.35 m from grid quantization mismatch, while OG-AMP degrades under unmodeled orientation offsets and anisotropic scattering.Source: Figure 8 · p. 12, B. Performance Evaluation · p. 12
  • BiOG-AMP steadily reduces Phase II RMSE as transmitter count increases, leveraging spatial diversity while GAMP and OG-AMP suffer from model mismatch.The comparison fixes the receiver count at 30.Source: B. Performance Evaluation · p. 12, Figure 9 · p. 12

C. Robustness and Convergence Evaluation

Appendix A Derivation of Reflection Gradients

The appendices derive reflection gradients and approximate message-passing quantities used in the reconstruction procedure.

Appendix A Derivation of Reflection Gradients

  • Diffuse and specular reflection gradients are set to zero outside their respective stated directional conditions.The conditions use incident, receiver, and perfect specular reflection vectors.Source: Appendix A · p. 12

Appendix B Approximation of Message µ(t)

  • The message from the observation factor to a reflectivity variable is derived by marginalizing the joint likelihood over the other involved variables.The appendix approximates the aggregate interference term as complex Gaussian under a large-scatterer assumption.Source: Appendix B · p. 13

Appendix C Approximation of Message µ(t)

  • The message from a reflectivity variable to its prior factor is formed as the product of incoming observation-factor messages.The derivation substitutes the Gaussian approximations from Appendix B.Source: Appendix C · p. 13

Appendix D Derivation of Z(t)

  • The derivation of Z(t) considers separate cases for the active and inactive states of c_n.The active state uses a Gaussian prior, while the inactive state uses a point mass at zero.Source: Appendix D · p. 13

Appendix E Approximation of Message µ(t+1)

Appendix E forms a message by combining the two sparsity-variable cases with normalized weights. Appendix F combines a Bernoulli-Gaussian prior with a Gaussian cavity message to derive activation probabilities and moments.

Appendix E Approximation of Message µ(t+1)

  • The message to x_n sums over sparsity states and integrates over offset variables, combining active and inactive cases as a normalized weighted sum.For c_n = 1 the term is Gaussian; for c_n = 0 it includes a point mass at zero.Source: Appendix E · p. 13

Appendix F Approximation of Message µ(t+1)

  • The message to z_{n,i} combines the prior with observation-node messages other than the current one, forming a Gaussian cavity message.The cavity mean and variance are used in the subsequent update.Source: Appendix F · p. 13
  • Combining the Bernoulli-Gaussian prior with the cavity Gaussian yields a Bernoulli-Gaussian message whose activation probability and first two central moments provide the updated quantities.The text identifies the normalized activation probability and derives the mean and variance from the message's central moments.Source: Appendix F · p. 14

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