sampling matrix, that we call Radon-like CS matrix. Whilst the. Radon transform evaluates projections of a given image along different directions, the Radon-like ...
Proceedings of the 5th International Symposium on Communications, Control and Signal Processing, ISCCSP 2012, Rome, Italy, 2-4 May 2012
Computationally-Efficient Compressive Sampling of Pulse Stream Images Using Radon-like Measurements Stefania Colonnese, Roberto Cusani, Stefano Rinauro, Gaetano Scarano Dip. DIET, Universitá “La Sapienza” di Roma via Eudossiana 18 00184 Roma, Italy (stefania.colonnese, roberto.cusani, stefano.rinauro, gaetano.scarano)@uniroma1.it Abstract— This paper introduces a compressive sampling (CS) scheme for pulse stream images. We adopt a particular sparse sampling matrix, that we call Radon-like CS matrix. Whilst the Radon transform evaluates projections of a given image along different directions, the Radon-like CS matrix evaluates randomly weighted projections of the image along a few directions. We demonstrate that the such CS measurements are invertible and assess the reconstruction accuracy of CS with the Radonlike sampling matrix by numerical trials. The Radon-like CS performs almost as well as state of the art techniques, with a reduced number of operations. As an application example, we show that, when implemented in a resource limited framework such as a Smart Sensors Grid, the sampling matrix significantly reduces inter-node signaling and then the associated energy consumption.
I. I NTRODUCTION Compressive Sensing (CS) is an emerging signal processing research field concerning non-adaptive sampling and compression of sparse signals. The objective of CS is to find a transformed domain where the signal is non-sparse. Since this condition allows downsampling the transformed components while maintaining roughly the same signal energy as dictated by the downsampling ratio, it also assures the conditions required for signal reconstructions [1]. The unifying sampling and compression vision underlying CS has enabled the development of several procedures for sparse signal representation; the sampling rate is related to signal sparseness rather than to signal bandwidth. An overview of CS is found in [2] and references therein. The problem of sampling signals that are not band-limited but have a finite number of degrees of freedom per unit of time is dealt in [3], where the authors present local and global sampling schemes. In [5], the authors show that, if the sparse signal to be sampled is built by convolution of two sparse signals, i.e. the signal is a so called pulse stream, the number of measurements to be adopted in CS can be further reduced without affecting the reconstruction accuracy. A potential application of CS is found in wireless sensor networks, where CS appears a useful technique for sensor data aggregation. Nonetheless, throughput improvement depends on the adopted CS structure [4]. In this paper, we deal with CS of images obeying the pulse stream generation model. In classical CS [2], each sample represents a randomly weighted sum of all the image values. Here, we seek for a random sampling matrix such that each sample is a weighted sum of a subset of the image pixels. Towards this aim, we observe that all the Radon projections of a pulse stream image capture the contribution
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of an isolated image spike, i.e. the sparse signal has a nonsparse representation in the Radon domain. Thereby, randomly weighted Radon projections are expected to assure the signal energy preservation which underlies the CS approach. In this paper, we introduce the so called Radon-like CS random sampling matrix and we establish that it statistically satisfies the Restricted Isometry Property (RIP) required for signal reconstruction from CS measurements. Numerical simulations assess the CS scheme performance by comparison with stateof-the-art sampling schemes. Finally, as a possible application of the herein introduced CS scheme, we show how, when implemented in a smart sensor grid, the envisaged CS structure reduces the number of inter-node signaling required for evaluation of the compressed sample data set. The remainder of the paper is organized as follows. In Sect.II we recall the pulse stream signal model [5]; in Sect.III we introduce the novel, Radon-like random sampling matrix for which we verify the RIP. In Sect.IV we present a numerical example of the CS scheme. In Sect.V we discuss the application of the sampling structure in a smart sensor grid, and Sect.VII concludes the paper. II. P ULSE STREAM SIGNAL MODEL Sampling and recovery of pulse streams is formerly introduced in [5]. A pulse stream is a signal z ∈ RN obeying the following generation model: z = x∗h
where x ∈ R and h ∈ RN are L-sparse and F -sparse signals, i.e. they have only L and F nonzero components. For such signals, the authors in [5] show that the signal can be restored after sampling with a M × N i.i.d. Gaussian matrix Φ for M above a suitable value M depending on L, F , i.e. M = M (L, F ). In [5] the authors provide two algorithms for signal reconstruction given a set of measurements: N
y = Φz being Φ a M × N i.i.d. Gaussian random sampling matrix satisfying M > M (L, F ). Here, we consider image CS, and we focus on images obeying the said generation model. Let us then consider a bi-dimensional sequence z[n1 , n2 ] of finite size N1 × N2 generated by circular convolution of a finite bi-dimensional sequence x[n1 , n2 ] and a F -sparse FIR filter h[n1 , n2 ]. The sequence x[n1 , n2 ] is built by isolated bi-dimensional discrete ! (l) (l) pulses, i.e. x[n1 , n2 ] = L−1 l=0 ξl δ[n1 − n1 , n2 − n2 ]. Let
us denote by x and h the image x[n1 , n2 ] and the filter h[n1 , n2 ] represented in lexicographic form. For those images, the generation model is written as z = Hx = X h, being H, X the doubly circulant matrices representing the circular convolution between the image and the filter as a vector product. With these positions, in the followings we determine a set of Radon-like random projections of the image z[n1 , n2 ] for CS purposes. III. CS USING R ADON - LIKE RANDOM PROJECTIONS To introduce the Radon-like CS measurements, let us consider the random projection of the image z[n1 , n2 ] along the vertical direction, as illustrated in Fig.1 ! p0 [m] = ϕ(0) (1) m [i]z[i, m], m = 0, . . . , N1 − 1 i
Fig. 1.
Original image z[n1 , n2 ] with its vertical projection p0 [m]
(0)
where ϕm [i], i = 0, N2 − 1, m = 0, . . . , N2 − 1 are i.i.d. zero 2 mean Gaussian random variables of equal variance σϕ . The expression in (1) can be generalized to represent a projection along a selected direction ϑp . Let us denote by z (ϑp ) [n1 , n2 ] a ϑp -radiant clockwise rotated version of the image z[n1 , n2 ], defined on an extended support of K(p) columns, with K(p) ≤ N1 + N2 − 1. Then, evaluating the following columnwise accumulation ! (ϑp) p) pϑp [m] = ϕ(ϑ [i, m]. (2) m [i]z i
we obtain a random projection along the direction ϑp . The expression in (2) allows to obtain different random projections pϑp [m], m = 0, . . . , K(p), along selected directions ϑp , p = 0, . . . P − 1. We refer to the projections in (2) as Radon-like random projections, since they can be considered as Radon projections of a Gaussian modulated field. When P Radon-like random projections are considered, each bidimensional discrete pulse in the image domain appears, randomly weighted, in each and every Radon-like random projection. This observation forms the basis for using such Radon-like random projections for CS of a pulse stream image. To elaborate, let us consider the measurement vector y built by collecting P Radon-like random projections. Such measurements are obtained by CS as: y = ΦR z
(3)
using the following random sampling (ϑ ) ΦR 0 .. ΦR = . (ϑ
ΦR P −1
)
matrix:
(π/2)
For the sake of concreteness, we report the matrices ΦR (0) and ΦR , corresponding to the horizontal and the vertical projection, in (4) and (5) respectively. While the sampling matrices usually adopted in CS are full, the sampling matrix ΦR is sparse; as we will discuss in Sect.V, such a peculiar form significantly simplifies the CS measurements computation when this latter is realized in a resource limited framework. To formally state that such a reduced sampling structure is feasible for accurate image reconstruction, we need to demonstrate the RIP for the sampling matrix ΦR . Specifically,
let Ez , Ey be the quadratic norm of the vectors z, and y; then, it should be shown that (1 − δ) Ez ≤ Ey ≤ (1 + δ) Ez
(6)
with high probability. In the light of the work in [6], we observe that the RIP is verified when the measurements energy Ey is highly concentrated, in probability, around the value Ez . Towards this aim, let us introduce the following property. Property: Let us assume the nonzero entries in ΦR to be i.i.d. zero mean normally distributed variables, with equal 2 variance σϕ = 1/P . Then, the measurements energy Ey satisfies the following concentration inequality: o(P ) (7) %2 Dim.: The demonstration of the above property, that we just outline here for the sake of compactness, is conducted resorting to the Chebishev inequality1. For the case under( con) cern, we have evaluated the expected values E {Ey } , E Ey2 . Omitting the derivation details, we report the results in (8) 2 and (9) (top of next page).2 Recalling that σϕ = 1/P , we observe that the variance Var {Ey } decays as 1/P . Hence, by evaluation of Var {Ey }, Eq.(7) follows. The concentration inequality guarantees that the image z can be recovered from the measurements y with high probability, as far as the number P of considered projections increases. Although the variance of Ey diminishes slowly with the number of measurements because of the sparse nature of the sampling matrix ΦR , the herein introduced sampling structure performs as well Gaussian sampling matrix, while it can greatly simplify the CS measurements collection. Pr{|Ey − Ez | ≤ %} ≤
IV. N UMERICAL S IMULATIONS Here, we present numerical simulation results obtained in CS using the Radon-like random projection in (3). The pulse stream images restoration Algorithm 2 in [5] has been adopted, assuming oracle knowledge for the pulse stream support. The 1 The Chebishev inequality states that, for a random variable x of mean m x and variance σx2 , the following inequality holds: Pr{|x−mx| ≤ "} ≤ σx2 /"2 2 The constant K i1,j1,i2,j2 in (8) and (9) exploits the property of Gaussian multi-variates g1 , g2 , g3 , g4 : E {g1 g2 g3 g4 } = E {g1 g2 } E {g3 g4 } + E {g1 g3 } E {g2 g4 } + E {g1 g4 } E {g2 g3 }. Then, Ki1,j1,i2,j2 takes on values in {0, 1, 2, 3} depending on the indexes i1 , i2 , j1 , j2 .
(π/2)
ΦR
=
(0)
ΦR
(π/2)
ϕ0 0 0 0
(0)
−1 K(p)−1 P! ! !
p=0
m=0
(π/2)
[1] · · · 0 ··· 0 ... 0 ···
ϕ0 [0] 0 ··· (0) 0 ϕ1 [0] · · · .. .. . . 0 0 0 ···
=
E {Ey } = E
(π/2)
[0] ϕ0
ϕ0
(0)
0 0
0 ··· ϕ0 [N1 − 1] (0) 0 ϕ1 [N1 − 1] · · · ······ .. .. . 0 . 0 (0) ϕN2 −1 [0] 0 0 ···
(p) ψm [i]z (ϑp ) [i, m]
i
0 0 [N2 − 1] 0 0 0 · · ·· · · 0 0 0 (π/2) (π/2) ϕN1 −1 [0] ϕN1 −1 [1] 0
.2
=
P −1 K(p)−1 ! ! ! p=0
m=0
i
0 0 0 (π/2) ϕN1 −1 [N2 − 1] 0 0 0 (0) ϕN2 −1 [N1 − 1]
··· ··· ··· ···
(4)
(5)
3 2 (p) 2 2 E ψm [i] z (ϑp ) [i, m]2 = P · σϕ · Ez(ϑp ) = Ez
(8) P −1 K(p)−1 ! ! !!!! !! 4 Var {Ey } = σϕ Ki1,j1,i2,j2z (ϑp ) [i1 , m]z (ϑp ) [i2 , m]z (ϑp ) [j1 , m]z (ϑp ) [j2 , m]− z (ϑp ) [i1 , m]2 z (ϑp ) [i2 , m]2 p=0
m=0
i1
j1
i2
j2
i1
i2
(9)
64 × 64 image z[n1 , n2 ] is built by 7 pulses, around which a 5 × 5 pulse response is centered. We have considered both the cases of fixed and random amplitude pulses; since the results obtained are similar, we show here pulses whose amplitudes are set in the range (0.5 − 0.85]. We consider noise-free CS, using P = 3 directions with ϑ0 = 0, ϑ1 = π/2, ϑ2 = π/4, corresponding to the vertical, horizontal, and diagonal projections 3 and to a number of measurements M = 255. For comparison sake, we report also the reconstruction accuracy achieved when a full M × N sampling matrix with Gaussian entries is adopted for CS. In Fig.2, we report the original image, the image decoded using the Radon-like sampling matrix ΦR and the full sampling matrix Φ. The Radon-like measurements involve a smaller assortment of the original image values the images restored using the full sampling matrices ΦR and Φ achieve the same visual quality. In Fig.3, we report the M SE = (z − zˆ)T (z − ˆ z) per iteration number observed during 10 runs corresponding to different pulse locations. In all the considered cases, the corresponding SNR largely overcomes 70dB. Although from a visual point of view the final reconstruction accuracy is just the same, the algorithm convergence is typically slower using Radon-like measurements than using a full Gaussian sampling matrix, due to the Radon-like matrix sparsity, which reduces the effectiveness of each CS sample. Thereby, the Radon-like trades sampling complexity with restoration algorithm convergence rate. This trade-off is much in the spirit of CS, where complex restoration algorithms are adopted in favor of a simplified sampling rate. In the next Section, we show an example of applications of the Radonlike CS scheme in a sensor network scenario. 3 In generating the Radon-like sampling matrix, outcomes under a predefined threshold have been discarded, in order to prevent an uncontrolled increase of the sampling matrix sparsity. Implications in terms of matrix coherence are currently under investigation.
Fig. 3.
MSE versus iteration number.
V. R ADON - LIKE CS IN SMART SENSOR GRIDS
Fig. 4. Radon-like CS of a pulse stream image by means of an SSG: an example of the data aggregation path realizing the computation of a CS sample.
Some remarks about the application of our CS scheme are in order. In case of compressing while sampling, adoption of the ΦR sampling matrix can yield significant saving of system resources. For concreteness sake, we analyze here the case of a Smart Sensor Grid (SSG) in which distributed sensor nodes
Fig. 2. Examples of CS results (from left to right): original pulse stream image; a particular zoomed in from the original image, from the image decoded using Alg.2 in [5] from Radon-like Gaussian random projections, and from the image decoded using Alg.2 in [5] from Gaussian random projections.
interact among themselves or with network infrastructure for data acquisition, processing and transmission purposes. Let us consider a grid realizing CS of a physical quantity extracted from a bi-dimensional field (temperature field, etc.), such that the acquired image size corresponds to the number of grid sensor nodes, as illustrated in Fig.4. In this framework, classical CS requires each of the output measurements |y|m to be computed on the basis of the data generated by each and every sensor in the network. Thereby, uncompressed data shall be individually collected to a sink, or to a network infrastructure for computation purposes. Conversely, according to our Radon-like sampling, each CS measurement |y|m is computed as a randomly weighted sum of the field values acquired on a fixed direction , which corresponds to a selected node path (see Fig.1). The path identifies a set of sensor nodes that compute the sample |y|m in a distributed fashion. the i-th node of the path 4Specifically, i−1 receives a partial sum k=0 ϕk ζk from the (i−1)-th path node, adds its own data ζi , scaled by a random Gaussian weight ϕi , and transmits the so obtained partial sum to the next node in the path. Thereby, only N1 data transmission are required for computation of the sample |y|m (see Fig.1). Besides, given P directions elected for CS random projections computation, each node transmits its own sample value P times only. We have evaluated the values assumed by NT as a function of the number of grid sensors nodes for the afore described acquisition geometry. We have assumed that data are collected towards the edge grid nodes, which finally transmit them to a wired network infrastructure for storing and/or further processing. In Radon-like CS, each measurement |y|m is computed after a number of transmission nm equal to the corresponding path length, that is nm = n for the horizontal and vertical projections samples and nm ∈ {1, . . . n} for the diagonal projections samples. For full CS sampling matrix, data from the generic node has been assumed to be collected towards the next vertical edge node.The overall number of transmission NT required by the Radon-like CS is coarsely evaluated as 63 × n2 using Radon-like sampling matrix and 1 5n as + 1 × n2 using a full sampling matrix. Table I 2 2 illustrates the result for different grid sizes. The reduction of the number of transmission resulting from the sparsity of the Radon-like CS matrix is evident. Thereby, our CS sampling scheme efficiently trade-offs the reconstruction algorithm convergence speed with reduced inter-node signaling, and consequently reduced transmission power.
SSG sensors nodes 32 × 32 64 × 64 128 × 128 176 × 176
NT using ΦR 3072 12288 49152 195075
NT using Φ 8704 67584 532480 4177856
TABLE I N UMBER OF TRANSMISSIONS NT REQUIRED TO PROVIDE SSG DATA AT THE GRID BORDERS .
Therefore, in this energy-constrained framework, the slower convergence of the restoration algorithm is widely compensated by the gain in terms of employed transmission energy. VI. ACKNOWLEDGEMENTS The authors would like to thank Giorgia Ruggiero for providing simulations. VII. C ONCLUSION AND FURTHER WORKS In this paper, we have presented a sparse random sampling matrix for CS of pulse stream images, using a sparse sampling matrix, for which we have demonstrated the RIP. The reconstruction accuracy of the CS process has been assessed by numerical simulations. The Radon-like CS has been demonstrated to perform at least as well as state of the art techniques, with a significantly simpler sampling structure. The implications of such simplifications have been illustrated with reference to a smart sensor grid, where adoption of our Radon-like sampling significantly reduces the inter-node signaling, and the grid energy consumption. R EFERENCES [1] J. Romberg,”Imaging via compressive sampling”, IEEE Sig. Proc. Mag., vol., March.2008. [2] Baraniuk, R.G.; Candes, E.; Elad, M.; Yi Ma; , "Applications of Sparse Representation and Compressive Sensing," Proc. of the IEEE, vol.98, no.6, June 2010 [3] P. Shukla, P.L. Dragotti, ”Sampling Schemes for Multidimensional Signals With Finite Rate of Innovation” IEEE Trans. on Sig. Proc., vol. 55, no. 7, July. 2007. [4] J. Luo, L. Xiang, C. Rosenberg, ”Does Compressed Sensing Improve the Throughput of Wireless Sensor Networks?”, IEEE Int. Conf. on Comm., (ICC 2010), 23-27 May 2010. [5] C. Hegde, R.G. Baraniuk, “Sampling and Recovery of Pulse Streams” IEEE Trans. on Sig. Proc., vol. 59, no. 4, Apr. 2011. [6] R.G. Baraniuk, M. Davenport, R. DeVore, M. Wakin, “A simple proof of the restricted isometry property for random matrices” Constructive Approximation , vol. 28, no.3, December 2008.
