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Compressive sensing of images based ondiscrete periodic Radon transform
G.-W. Ou, D.P.-K. Lun and B.W.-K. Ling
ELECT
A new compressive sensing (CS) scheme using the structured randommatrix and the discrete periodic Radon transform (DPRT) is proposed.The new scheme first pre-randomises the sensing image and the DPRTis applied to the randomised samples to generate the so-called DPRTprojections. They are then randomly selected to obtain the finalsensing measurements. As the DPRT is friendly to hardware/opticsimplementation, it improves the operability and lowers the cost forreal-time CS applications. Compared with other similar transformssuch as the Walsh–Hadamard transform, the proposed DPRT schemegives much better reconstructed images as shown in the simulationresults.
Introduction: Compressive sensing (CS) is currently one of the hottesttopics in the signal processing area. It has attracted wide attention andresearch [1–5]. CS emphasises that signals can be reconstructed usingfar fewer samples or measurements compared with traditional samplingmethods. To be specific, the CS framework can be described by the fol-lowing formulation [3]:
y = Fx = FCa (1)
where x is the original N-point signal that can be sparsely represented byits transform coefficients α. We denote α as K-sparse which means onlyK =N coefficients are non-zero (or significant). In (1), Ψ is the inversesparse transform operator for signal x. Φ is the sensing operator thatgives y with M samples from the N-point signal x, where M≪ N. TheCS framework allows us to reconstruct the signal x from the compressedy by solving an optimisation problem. To guarantee the feasibility of CS,there are two basic rules: sparsity and incoherence [4]. By sparsity, itrequires that signals should have sparse representation in some specialdomains. As for images, they show good sparsity in Fourier andwavelet domains. By incoherence, it requires that the sparse transformoperator Ψ should have low coherence with the sensing operator Φ. Itis proved that random matrices are largely incoherent with any fixedbasis Ψ [5].
However, the implementation of random matrices in practical com-pressive image sensing applications requires huge memory resourcesand high computational complexity because of their completely unstruc-tured nature. Thus it is suggested in [6] to replace the random matricesby the so-called structured random matrix (SRM) as follows:
F = DFR (2)
where the operators R, F and D represent the three steps of the SRMsensing model. R is an operator that randomises the original signal x.F is an orthonormal transform operator that is applied to the randomisedsignal. It can be the discrete Fourier transform (DFT), discrete cosinetransform (DCT), Walsh–Hadamard transform (WHT) or other ortho-normal transforms that have fast implementation methods. D is thesensing operator that extracts the M-point measurements from theN-point transformed signal. Although the use of the SRM sensingmodel can significantly reduce the memory requirement and compu-tational complexity in the reconstruction process, it imposes the require-ment of implementing an orthonormal transform in the data acquisitionprocess. Since the implementation of most orthonormal transformsrequires certain computation and memory resources, it can significantlyincrease the hardware cost and/or processing time in data acquisition,which defeats the purpose of using CS. For this reason, it is suggestedin [6] that the WHT can be used because it requires only additionsand subtractions for its implementation. However, we note in our simu-lations that the quality of the reconstructed images will be significantlydegraded as compared with using the DFT or DCT, particularly whenthe compression rate is high.
To address the above difficulty, we suggest in this Letter using thediscrete periodic Radon transform (DPRT). Similar to the WHT, theDPRT requires only additions for its implementation. However, basedon the discrete Fourier slice theorem [7], it can be shown that arandom sampling of an image in the Fourier domain can be achievedby randomly selecting the DPRT projections. The result is similar tousing the SRM with the DFT. Our simulation results show that the pro-posed SRM-DPRT CS scheme can give a performance significantly
RONICS LETTERS 10th April 2014 Vol. 50
better than using the WHT, while it can be implemented using simplehardware/optics.
Proposed SRM CS scheme using the DPRT: Denote the set of realnumbers as R, R× R as R2, the set of integers as Z and Z× Z asZ2. Denote a subset of integers {0, 1, 2, 3, . . . , n− 1} , Z as Zn,Zn × Zn as Z
2n. Denote l
2 Z2n
( )to be a set of doubly summable 2D func-
tions over Z2. The DPRT of an image f (x, y) [ l2 Z2p
( )in the case that p
is prime which is defined as follows [7] (it is also called finite Radontransform [8]):
cm(d) =∑p−1
x=0
f x, kd + mxlp( )
b0(d) =∑p−1
y=0
f (d, y)
(3)
where m [ Zp, d [ Zp and ⟨a⟩P refer to the modulo of an integer a by apositive integer P. As shown in (3), the DPRT of an image is obtainedfrom a set of ‘discrete line’ summations of the image pixels. Similar tothe traditional Radon transform, these summations can be considered asthe projections of the image taken from different angles m, but in a per-iodic fashion. As shown in (3), all DPRT projections of an image can beobtained solely by additions, which can be implemented by simple hard-ware/optics.
C 2
C 3
u
v
Fig. 1 Sampling points of Fourier transform of cm, where m = 2 and 3, on theoriginal discrete image spectrum; the image is assumed to have the size of7 × 7
One of the important properties of the DPRT is the discrete Fourierslice theorem [7]. Given that the spectrum of an image
f (x, y) [ l2 Z2p
( )is defined as
F(u, v) =∑p−1
x=0
∑p−1
y=0
f (x, y)e−j2p(ux+vy)/p (4)
where (u, v) [ Z2p, the spectrum of an image is related to the spectrum
of its DPRT projections as follows:
F k− mvlp, v( ) = ∑p−1
d=0
cm(d)e−j2p(vd)/p
F(u, 0) =∑p−1
d=0
b0(d)e−j2p(ud)/p
(5)
The above relationship is known as the discrete Fourier slice theorem,which states that the Fourier transform of a DPRT projection in fact isjust one slice of the original discrete 2D spectrum, although in a periodicfashion as shown in Fig. 1. Such a periodic nature implies that if we ran-domly select some of the DPRT projections, the Fourier transform ofthese projections will give the Fourier coefficients of the originalimage at random spatial frequencies. Fig. 1 illustrates the above obser-vation. In the Figure, the Fourier transforms of cm where m = 2, 3 areshown on the original discrete image spectrum, where the image is
No. 8 pp. 591–593
assumed to have the size of 7 × 7. As both slices are periodic, they coverthe original discrete spectrum at some irregular spatial frequencies.
Thus, the proposed SRM-DPRT CS strategy contains three steps:
Step 1: Apply the operator R of the SRM sensing model to randomisethe original image, i.e. xr = Rx.Step 2: Apply the DPRT to the randomised image, which is the operatorF of the SRM sensing model, i.e. xf = Fxr = {cm, b0} = DPRT(xr).Step 3: Randomly select M DPRT projections, i.e. y =Dxf. In effect, theoperator D randomly selects M projections from the set of cm and b0.
In fact, since the DPRT only involves additions, steps 2 and 3 can becombined such that only the selected DPRT projections will be gener-ated. It can further reduce the data acquisition time because only MDPRT projections, rather than P + 1, need to be computed. When theM DPRT projections are obtained, the original image is reconstructedthrough an optimisation process as follows:
Step 1: Compute the DFT of each of theM DPRT projections. Make useof (5) to map the results to a 2D spectrum. Then compute the inverse 2DDFT of the resulting spectrum to give the initial estimation of the orig-inal image.Step 2: With the initial estimation obtained in Step 1, reconstruct theoriginal image x̂ by solving the following minimisation problemusing, for instance, the fast composite splitting algorithm (FCSA) [9]:
x̂ = argminx̃
1
2||F x̃− y||22 + g||C′x̃||1 + b||x̃||TV
{ }(6)
where ||.||1, ||.||2 and ||.||TV are the 1-norm, 2-norm and total variation(TV) norm, respectively. γ and β are the parameters to control the con-tribution of the 1-norm and TV-norm in the optimisation. They areselected empirically. Φ is the SRM with the transform F replaced bythe DPRT. Ψ′ is the inverse DFT. Note that when solving the optimis-ation problem in (6), both the DPRT and inverse DPRT (IDPRT) will beused iteratively. Fast algorithms for the DPRT and IDPRT [10] can beapplied to speed up the operation.
Simulation results: A series of simulations was carried out to verify theproposed SRM-DPRT CS scheme. In the simulations, natural images ofsize 256 × 256 were used. The compression rate is first set at 15%. Theproposed scheme is compared with the original SRM CS scheme usingthe FFT and WHT. Fig. 2 shows the reconstruction results of the testingimages, house, pepper and girl. Table 1 shows the peak signal to noiseratio achieved by different schemes. Fig. 3 further shows the perform-ance of different schemes at various compression rates (the testingimage house is used). It can be seen from the simulation results thatthe proposed SRM-DPRT CS scheme performs much better than theSRM-WHT scheme. The artefact in the reconstructed images usingthe SRM-WHT is largely because of the discontinuity of the WHTkernel. Although the performance is similar to using the FFT, the pro-posed scheme requires only additions, which is more friendly to hard-ware/optics implementation.
original image SRM-WHT SRM-FFT proposed
Fig. 2 Reconstruction performance at 15% compression rate
ELECTRONICS LETTER
Table 1: PSNR comparison of different SRM CS schemes
S
10thHouse
April 2
Pepper
014
Girl
SRM-WHT
25.6603 24.6939 28.8895SRM-FFT
29.0386 25.8338 32.1306Proposed
29.5038 26.0287 32.2613sample rate0.10
20
22
24
26
28
30
32
34
36
38
0.15 0.20 0.25
proposedSRM-WHTSRM-FFT
0.30 0.35 0.40 0.500.45
PS
NR
Fig. 3 Performance of different SRM CS schemes at various compressionrates
Conclusion: In this Letter, we propose a new CS strategy based on theSRM and DPRT. Instead of applying a complicated orthonormal trans-form to the randomised image, the proposed method can get the sensingmeasurements using only the addition operator, which is friendly tohardware/optics implementation. Compared with the SRM-WHTscheme which also uses only the addition (or subtraction) operator,the proposed SRM-DPRT scheme gives significantly better recon-structed images.
Acknowledgment: This work is fully supported by the Hong KongPolytechnic University under research grant G-YJ85.
© The Institution of Engineering and Technology 20144 March 2014doi: 10.1049/el.2014.0770One or more of the Figures in this Letter are available in colour online.
G.-W. Ou and D.P.-K. Lun (Department of Electronic and InformationEngineering, Centre for Signal Processing, The Hong Kong PolytechnicUniversity, Hung Hom, Kowloon, Hong Kong)
E-mail: [email protected]
B.W.-K. Ling (Faculty of Information Engineering, GuangdongUniversity of Technology, Guangzhou 510006, People’s Republic ofChina)
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