Resolution enhancement of ultrasonic signals by up-sampled sparse deconvolution Ewen Carcreff†,‡ , Sébastien Bourguignon† , Jérôme Idier† , Laurent Simon‡ † LUNAM Université, École Centrale de Nantes / IRCCyN UMR CNRS 6597, 1 rue de la Noë, 44321 Nantes, France ‡ LUNAM Université, Université du Maine / LAUM UMR CNRS 6613, avenue Olivier Messiaen, 72085 Le Mans, France
[email protected] A BSTRACT
R ESULTS
MISO SYSTEM WITH SPARSE INPUTS
This paper deals with the estimation of the arrival times of overlapping ultrasonic echoes. We focus on approaches based on discrete sparse deconvolution. Such methods are limited by the time resolution imposed by the model discretization, which is usually considered at the data sampling rate. In order to get closer to the continuous-time model, we propose to increase the time precision by introducing an up-sampling factor in the discrete model. The problem is then recast as a Multiple Input Single Output (MISO) deconvolution problem. Then, we propose to revisit standard sparse deconvolution algorithms for MISO systems. Specific and efficient algorithmic implementation is derived in such setting. Algorithms are evaluated on synthetic data, showing improvements in robustness toward discretization errors and competitive computational time compared to the standard approaches.
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• Deconvolution of synthetic data (following model): 8 2 X (nTS − ti ) sin (2πf0 (nTS − ti )) + en yn = exp − 2 2σ i=1 • ti randomly and continuously distributed on the time axis • f0 = 5 MHz, fS = 1/TS = 25 MHz, σ = 0.18 µs, SNR = 10 dB
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with x(t) sparse signal and e(t) additive noise
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• Constraint: y(t) is sampled at the sampling period TS such that yn = y(nTS )
• Specific optimized implementations for MISO deconvolution: • MP and OMP selection step: - H · −→ K products (Hk )T · where Hk are Toeplitz matrices - (Hk )T · equivalent to a cross-correlation computation (2 FFTs + 1 IFFT) - pre-computation of the FFTs of the K waveforms hk
• Discretization of (h ∗ x)(t) at TS −→ Standard discrete convolution model: h
- OMP solution update: inversion of performed by Cholesky factorization [2]
hm xn−m + en ⇐⇒ y = Hx + e
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(?: subset of active indices)
• Discretization of (h ∗ x)(t) at TS /K with up-sampling factor K: h
hp xnK−p + en ⇐⇒ y = H x + e
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with up-sampled sequences hp = h(pTS /K) and xp = x(pTS /K) (P=KM) −→ One can show that this is equivalent to a sum of K discrete convolutions: ! 1 h2 h3 K M −1 K h X X X yn = hkm xkn−m +en ⇐⇒ y = Hk xk +e i.e. a Multiple Input Single Output (MISO) system
A CKNOWLEDGEMENTS This work has been partially supported by the French Region Pays de la Loire as part of the scientific program "Non-Destructive Testing and Evaluation-Pays de la Loire" (ECND-PdL).
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(a): Waveforms hk , k = 1 . . . 6, (b): data and true spikes, results with (c) K=1 and (d) K=6
◦ true spikes estimated spikes
• Monte-Carlo simulations from 2000 data sets composed of 15 spikes: Spike error computed from a spike distance value [4]: 1. amplitudes are binarized to ±1 to give the same importance to all detections 2. the spike trains are convolved with a double-side exponential kernel e−|t|/τ , with τ = TS , producing slight spike spreading 3. the `2 -norm between the two convolved spike trains is computed 2
• OLS and SBR: T - extensive accesses to elements of H H −→ elements of (Hk )T H` - pre-computation of the K(K + 1)/2 cross-correlations between hk and h` - core computational cost remains roughly constant as K increases
MP OMP OLS SBR L1
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R EFERENCES [1] T. Blumensath and M. E. Davies, “On the difference between Orthogonal Matching Pursuit and Orthogonal Least Squares,” Tech. Rep., University of Edinburgh, March 2007. [2] C. Soussen, J. Idier, D. Brie, and J. Duan, “From Bernoulli Gaussian Deconvolution to Sparse Signal Restoration,” IEEE Transactions on Signal Processing, vol. 59, no. 10, pp. 4572–4584, October 2011.
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• Computation times averaged from the Monte-Carlo simulations:
• `1 -HC: - addition tests (similar to OMP): two products H · −→ 2(K + 1) FFTs T - removal tests: inversion of H? H? by Cholesky factorization
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with sequences hn = h(nTS ) and xn = x(nTS )
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• MP and OMP solution updates: same as standard algorithms T H? H?
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D ISCRETE UP - SAMPLED CONVOLUTION
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• Method: Sparse representation of the data using the continuous model: Z +∞ y(t) = (h ∗ x)(t) + e(t) = h(τ ) x(t − τ )dτ + e(t)
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2 4. Single Best Replacement (SBR) [2]: minimization of y − Hx + µkxk0
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P 5. `1 -norm penalization (`1 -HC): minimization of y − Hx + µ ` |x` | performed by homotopy continuation [3]
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Spike errors
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3. Orthogonal Least Squares (OLS) [1]
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• Goal: estimation of the arrival times ti and amplitudes ai
yn =
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• Revisited algorithms to estimate the sparse inputs xk :
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• Ultrasonic data model:
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C ONCLUSIONS
[3] D. M. Malioutov, M. Cetin, and A. S. Willsky, “Homotopy continuation for sparse signal representation,” in IEEE International Conference on Acoustics, Speech, and Signal Processing, Philadelphia, USA, March 2005, vol. 5, pp. 733–736.
• Up-sampled model of discrete convolution: better performances of deconvolution for synthetic ultrasonic data
[4] M. C. W. Van Rossum, “A novel spike distance,” Neural Computation, vol. 13, no. 4, pp. 751–763, April 2001.
• Optimized implementation of sparse approximation algorithms for MISO data: controlled computation time
