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Multichannel Sampling and Reconstruction in Multidimensional Shift-Invariant Spaces

Riya Ghosh, Satish Mulleti, Chandra Sekhar Seelamantula

arXiv:2610.36142v1eess.SP

TL;DR

Recovering signals from measurements in multidimensional principal shift-invariant spaces is the central problem. The paper develops a polyphase framework for periodic sampling and an interpolation-projection iteration for irregular derivative samples. It reports stable recovery under stated conditions and geometric convergence for the irregular-sampling method.

  • Problem

    The paper seeks to recover continuous signals in principal shift-invariant spaces from multichannel measurements, including periodic and irregular derivative samples.

    Source: §I · p. 2, C. A multichannel sampling problem · p. 3

  • Method

    A finite-dimensional polyphase formulation handles periodic measurements, while irregular derivative sampling uses local bivariate Hermite interpolation followed by orthogonal projection.

    Source: §I · pp. 1–2

  • Results

    Uniform singular-value bounds characterize stable recovery in the polyphase framework, and a sufficient contraction condition yields geometric convergence for the irregular-sampling iteration.

    Source: §I · p. 2, §V · p. 12

  • Takeaways & Limitations

    The approaches apply to separable and genuinely non-separable generators and are demonstrated under exact measurements, noise, and sampling-location perturbations.

    Source: §V · p. 12

Abstract

from arXiv · show

We develop a multichannel sampling and reconstruction framework for multidimensional principal shift-invariant spaces based on a finite-dimensional polyphase representation. Periodic nonuniform shifts and derivative measurements are modeled as analysis channels, leading to a frequency-domain linear system relating the measurements to the polyphase components of the underlying coefficient sequence. Stable recovery is characterized through uniform singular-value bounds on the associated polyphase matrix, and explicit synthesis kernels are obtained from its Moore Penrose left inverse. The formulation applies to both separable and genuinely non-separable generators and includes several classical one dimensional sampling formulas as special cases. For irregular derivative measurements, where the periodic polyphase structure is no longer directly available, we develop an iterative reconstruction method based on local bivariate Hermite interpolation followed by orthogonal projection onto the shift-invariant space. A sufficient contraction condition is derived to guarantee geometric convergence of the iteration. Numerical experiments demonstrate the reconstruction performance for separable and non-separable generators and illustrate robustness to additive measurement noise and sampling-location perturbations.

I. INTRODUCTION

The introduction places the work among extensions of sampling theory to shift-invariant spaces, derivative measurements, and multichannel acquisition. It presents a multidimensional polyphase framework and a separate iterative approach for irregular derivative sampling.

I. INTRODUCTION

  • Prior work addressed nonuniform sampling in shift-invariant spaces,, derivative sampling,, and multivariate generalized sampling,.The introduction also situates the framework among multichannel and periodic nonuniform sampling methods –.Source: §I · p. 1
  • The paper develops a multidimensional coefficient-domain polyphase formulation for periodic nonuniform mixed-derivative sampling in principal shift-invariant spaces.Its sampling matrix explicitly depends on the generator, derivative orders, and sampling shifts.Source: §I · p. 1
  • The formulation requires no tensor-product factorization and applies to separable and genuinely non-separable generators.Numerical examples include a sinc-Gaussian generator and a radial generator constructed from the Gauss hypergeometric function.Source: §I · p. 2
  • For irregular derivative samples on two-dimensional Cartesian grids, local bivariate Hermite approximation and orthogonal projection yield an iteration with a sufficient geometric-convergence condition.The condition relates sampling gaps and interpolation orders to generator spectral properties, while the error estimate accounts for mixed derivatives and coordinate coupling.Source: §I · p. 2
  • Uniform singular-value bounds characterize stable recovery, with synthesis kernels obtained from the polyphase matrix’s Moore–Penrose left inverse.Examples include classical sampling formulas and multidimensional cases with separable and non-separable generators.Source: §I · p. 2

A. Notation and basic spaces

  • The preliminaries establish notation for vectors, Fourier transforms, sequence spaces, and polyphase components.The polyphase representation has P components, with P defined as the product of the dimension-wise sampling intervals.Source: A. Notation and basic spaces · p. 2

B. Shift-invariant spaces (SISs)

The paper models signals in principal shift-invariant spaces using stable generators and unique coefficient representations. Its multichannel sampling problem combines derivative channels with periodic nonuniform measurements to recover the signal from samples.

B. Shift-invariant spaces (SISs)

  • A principal shift-invariant space is generated by integer translates of a basis function, and the paper assumes those translates form a Riesz basis.This assumption gives each signal a unique coefficient representation.Source: B. Shift-invariant spaces (SISs) · pp. 2–3
  • The generator is assumed sufficiently smooth for the sampled derivatives, making derivative evaluations bounded linear functionals on the space.The required regularity depends on the highest total derivative order used in the sampling model.Source: B. Shift-invariant spaces (SISs) · p. 3

C. A multichannel sampling problem

  • Each channel measures a selected derivative at periodic nonuniform locations, while channels use coarser sampling lattices and jointly provide information for recovery.The goal is to recover the coefficient sequence and thereby the continuous signal.Source: C. A multichannel sampling problem · p. 3
  • In d dimensions, the model permits arbitrary selected derivative orders and combines derivative channels with multiple uniform sampling patterns.A necessary recovery condition is KM ≥ P; stable reconstruction is established through uniform lower and upper singular-value bounds on the polyphase matrix.Source: C. A multichannel sampling problem · p. 3

III. MULTIDIMENSIONAL RECONSTRUCTION VIA POLYPHASE ANALYSIS

The section represents multichannel measurements as a finite-dimensional polyphase system and characterizes stable recovery through uniform singular-value bounds. Under those bounds, synthesis kernels come from the matrix’s Moore–Penrose left inverse; examples connect the framework to classical one-dimensional formulas and non-separable generators.

III. MULTIDIMENSIONAL RECONSTRUCTION VIA POLYPHASE ANALYSIS

  • The finite-dimensional polyphase representation relates multichannel measurements to the underlying coefficient components and tests whether those measurements uniquely and stably determine the signal.The associated matrix is frequency-dependent and has KM rows and P columns.Source: §III · p. 4
  • Uniform lower and upper singular-value bounds on the polyphase matrix guarantee unique and stable reconstruction, with synthesis kernels obtained from its Moore–Penrose pseudoinverse.The stated necessary condition is KM ≥ P.Source: §III · p. 4
  • The reconstruction examples recover classical one-dimensional sampling formulas and extend the formulation to non-separable multidimensional generators.The reconstruction pipeline includes multichannel sampling, polyphase recovery, coefficient recombination, and signal synthesis.Source: §III · pp. 4–5

A. Illustrative Examples

The examples illustrate that the polyphase formulation recovers classical one-dimensional derivative-sampling and periodic nonuniform formulas. They validate explicit reconstruction by computing relative recovery error.

A. Illustrative Examples

  • Each numerical example validates the explicit reconstruction formula by computing the relative recovery error.Source: A. Illustrative Examples · p. 5
  • The sinc example with P = 2 and M = 2 yields the classical derivative-sampling formula in for σ = π.Source: A. Illustrative Examples · p. 5
  • The periodic nonuniform reconstruction matches the formula reported in when −τ1 = τ2 = τ.The example uses two distinct shifts and inverts the corresponding 2 × 2 polyphase matrices.Source: A. Illustrative Examples · p. 5

B. Multidimensional Separable and Non-Separable Generators

The examples apply the polyphase reconstruction framework to separable and genuinely non-separable generators, with numerical reconstructions illustrating its stability and accuracy. Their nonzero errors arise from numerical implementation, while the reconstruction identities are exact under the theorem’s hypotheses.

B. Multidimensional Separable and Non-Separable Generators

  • The joint multidimensional polyphase formulation recovers tensor-product structure for separable generators and also applies to non-separable generators without factorization.A later example explicitly considers a strictly non-separable generator.Source: B. Multidimensional Separable and… · pp. 6–7
  • 7.202266 × 10−7 relative error is reported for the Figure 2 reconstruction obtained using the Moore–Penrose pseudoinverse.The figure compares the original function, its reconstruction, and absolute error.Source: B. Multidimensional Separable and… · p. 6, Figure 2 · p. 7
  • 0.0157 relative error is reported for the Figure 3 reconstruction after the non-separable example’s polyphase matrix is found numerically bounded away from singularity.The aliasing sum was truncated to |n1|, |n2| ≤10 for numerical evaluation.Source: B. Multidimensional Separable and… · p. 7, Figure 3 · p. 7
  • 8.268 × 10−7 relative error is reported for the Figure 4 reconstruction, with minω σmin(Φ(ω)) ≈ 0.164 supporting numerical stability.The synthesis kernels are determined by the Moore–Penrose pseudoinverse Φ†(ω).Source: B. Multidimensional Separable and… · p. 8, Figure 4 · p. 8
  • The nonzero reconstruction errors in Examples III.3–III.6 arise from truncation, discretization, numerical approximation, and floating-point arithmetic; identities are exact under Theorem III.1’s hypotheses.Source: B. Multidimensional Separable and… · p. 8

IV. ITERATIVE RECONSTRUCTION FROM IRREGULAR DERIVATIVE SAMPLES

For irregular derivative locations, the paper replaces the periodic sampling formulation with local bivariate Hermite interpolation followed by projection onto the shift-invariant space. Under a derived density condition, the resulting error operator is contractive and convergence is geometric.

IV. ITERATIVE RECONSTRUCTION FROM IRREGULAR DERIVATIVE SAMPLES

  • Irregular derivative samples are handled by constructing a local bivariate Hermite interpolant from the measurements and projecting it onto V2(ϕ).Source: §IV · p. 8
  • The Hermite interpolation and projection composition is an approximation operator whose error is contractive under the derived density condition.Source: §IV · p. 8
  • The contraction condition yields geometric convergence of the iterative reconstruction.Source: §IV · p. 8

A. Local Hermite Interpolation

Local tensor-product Hermite interpolation constructs a bivariate polynomial from values and partial derivatives at grid nodes. Its approximation error is bounded using coordinate-wise contributions and mixed derivatives, motivating a Bernstein-type inequality for functions in V2(ϕ).

A. Local Hermite Interpolation

  • The local Hermite interpolant uses m + 1 x-nodes and derivatives through order r, and n + 1 y-nodes and derivatives through order s.The corresponding polynomial degrees are Rx = m(r + 1) + r and Ry = n(s + 1) + s.Source: A. Local Hermite Interpolation · p. 8
  • The tensor-product operator H = IXIY coincides with the local Hermite interpolant and incorporates mixed derivative data.Source: A. Local Hermite Interpolation · p. 8
  • The Hermite approximation bound separates errors in the two coordinates and includes a mixed-derivative contribution.Its constants are independent of the interval indices and depend on interpolation orders and normalized sampling-set geometry.Source: A. Local Hermite Interpolation · p. 8
  • Because the interpolation estimate involves higher-order derivatives of f, the section next derives a Bernstein-type inequality for functions in V2(ϕ).Source: A. Local Hermite Interpolation · p. 8

B. Bernstein-Type Inequality

A Bernstein-type inequality is established for functions in V2(ϕ), under generator and derivative regularity conditions. The proof uses Plancherel’s theorem and periodization.

B. Bernstein-Type Inequality

C. Approximation operator

The approximation operator combines local bivariate Hermite interpolation with orthogonal projection onto the shift-invariant space. When its error bound satisfies q < 1, the resulting iterative reconstruction converges geometrically.

C. Approximation operator

  • The interpolant is projected back onto V2(ϕ) because the Hermite interpolant Hf does not generally lie in the signal space.The projection is expressed in the Fourier domain for a stable generator.Source: C. Approximation operator · p. 9
  • If q < 1, the approximation error is contractive and the iteration reconstructs every signal in V2(ϕ) from irregular derivative samples.The error bound gives ∥eν∥ ≤ q^(ν+1)∥f∥, implying convergence in L2.Source: C. Approximation operator · p. 9
  • The convergence theorem assumes a stable generator, bounded required derivative evaluations, and separated sampling sets with finite block gaps.Under the theorem’s stated conditions, every signal in V2(ϕ) is recoverable from the irregular derivative samples.Source: C. Approximation operator · p. 9
  • The iteration starts from f0 = Af, with initial error e0 = (I − A)f.The iteration updates through the stated scheme; the error analysis uses the operator norm bound ∥I − A∥ ≤ q.Source: C. Approximation operator · p. 9

D. Numerical Experiments

The numerical examples report convergence-bound values below one for two sampling configurations, then examine reconstruction under measurement noise and sampling-location perturbations. The noise experiments show rapid initial error decay followed by an SNR-dependent floor.

D. Numerical Experiments

  • The first sinc-generator example satisfies the sufficient convergence condition with q ≈ 1.04 × 10^-3 < 1.It uses r = s = 1, m = n = 2, with sampling gaps δ2X = 0.70 and δ2Y = 0.80.Source: D. Numerical Experiments · p. 10

Simulation 2:

  • The Gaussian-generator example satisfies the sufficient convergence condition with q ≈ 2.44 × 10^-3 < 1.This configuration uses r = 1, s = 2, m = 1, n = 2, with δ1X = 0.27 and δ2Y = 1.18.Source: Simulation 2 · p. 10

E. Robustness to measurement noise and sampling jitter

F. Reconstruction of an image-generated signal with a non-separable generator

The image experiment applies the iterative method to a 17 × 17 Cameraman-derived coefficient array with a non-separable generator. The paper reports convergence behavior under exact, noisy, and jittered measurements and summarizes image-reconstruction performance across those conditions.

F. Reconstruction of an image-generated signal with a non-separable generator

  • The image experiment uses the six derivative measurements f, fy, fyy, fx, fxy, and fxyy, with each iteration locally interpolating residuals and projecting the correction onto V2(ϕ).The 289 pixel intensities of a resized 17 × 17 Cameraman image form the coefficient sequence.Source: F. Reconstruction of an… · p. 11
  • Figure 10 compares convergence under exact measurements, 40 dB additive noise, and 2% sampling-location perturbation; Table III summarizes reconstruction performance across these conditions.Figure 10’s noisy and jittered curves average 50 realizations.Source: F. Reconstruction of an… · p. 11, Figure 10 · p. 12, Table III · p. 12

V. CONCLUSION

  • The framework covers separable and genuinely non-separable generators, while irregular derivative sampling uses a Hermite-interpolation/projection iteration with a sufficient contraction condition for geometric convergence.The conclusion also reports numerical demonstrations under exact measurements, additive noise, and sampling-location perturbations.Source: §V · p. 12
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