Variable_Density_Compressive_Image_Sampling_EUSIP2009
V ARIABLE DENSITY COMPRESSED IMAGE SAMPLING Zhongmin Wang?,Gonzalo R.Arce?and Jose L.Paredes
?Dep.of Electrical and Computer Engineering, University of Delaware,Newark,DE,19716USA. e-mail:{zhongmin,arce}@https://www.wendangku.net/doc/f513362310.html, Electrical Engineering Department, Universidad de Los Andes,Venezuela e-mail:paredesj@ula.ve
ABSTRACT
Compressed sensing(CS)provides an ef?cient way to acquire and reconstruct natural images from a reduced number of linear pro-jection measurements at sub-Nyquist sampling rates.A key to the success of CS is the design of the measurement ensemble.This pa-per addresses the design of a novel variable density sampling strat-egy,where the“a priori”information about the statistical distribu-tions that natural images exhibit in the wavelet domain is exploited. Compared to the current sampling schemes for compressed image sampling,the proposed variable density sampling has the following advantages:1)The number of necessary measurements for image reconstruction is reduced;2)The proposed sampling approach can be applied to several transform domains leading to simple imple-mentations.In particular,the proposed method is applied to the compressed sampling in the2D ordered discrete Hadamard trans-form(DHT)domain for spatial domain imaging.Furthermore,to evaluate the incoherence of different sampling schemes,a new met-ric that incorporates the“a priori”information is also introduced. Extensive simulations show the effectiveness of the proposed sam-pling methods.
1.INTRODUCTION
By exploiting the sparsity of natural images,compressed sensing has shown that it is feasible to acquire and reconstruct natural im-ages from a limited number of linear projection measurements at sub-Nyquist sampling rates[1,2].A key component to the success of CS is the design of the measurement ensemble,which is based on the evaluation of the incoherence between the measurement en-semble and the sparsity basis.Incoherence property requires that the components of the measurement ensemble should have a dense representation in the sparsity basis.Due to the large scale of nat-ural images and their high resolution,it is also required that the generation of the measurement ensemble be both computationally ef?cient and memory ef?cient.Furthermore,the sampling scheme should enable fast algorithms to perform image reconstruction.
Measurement matrices where each entry is an independent and identically distributed(i.i.d.)random variable obeying Gaussian or Bernoulli distribution have been proposed for compressed image sampling[1,2].Recently,it has been shown that the performance of CS sampling can be improved if the random measurement matrices are suitably optimized[3].Due to their unstructured nature,how-ever,large memory space and demanding computation resources are needed,making them prohibitively expensive for a practical imple-mentation.A more desirable way to obtain linear measurements is by incoherent sampling in a transform domain equipped with fast transform algorithms[4].Measurement ensembles in the transform domain that enable fast computations include partial Fourier ensem-ble,scrambled Fourier ensemble(SFE),scrambled block Hadamard ensemble(SBHE)and Noiselets[1,4,5].Although these sampling approaches have shown to obtain good performance in CS applica-tions,their formulation do not exploit any“a priori”information of natural images that could lead to improved CS performance[6,7].
In this paper,we propose a new method for compressed image sampling by exploiting the“a priori”information about the statisti-cal distributions that natural images exhibit in the wavelet domain.
A novel family of variable density sampling patterns are designed for compressed image sampling in the frequency transform domains which includes Discreet Fourier Transform(DFT),Discrete Cosine Transform(DCT)and ordered DHT.Equipped with fast transform algorithms,the sampling process is simple,fast and easily https://www.wendangku.net/doc/f513362310.html,pared with other sampling strategies,the CS perfor-mance obtained with the proposed sampling method is signi?cantly improved.
To design the proposed measurement matrices,a simpli?ed model to describe the statistical distribution of image wavelet co-ef?cients is?rst established.According to the subband incoherent sampling method proposed in[4],the Fourier coef?cients in the subband where signi?cant energy of the wavelet exits should be sampled randomly to minimize the coherence between the sparse wavelets and the measurement Fourier atoms.Thus,based on sub-band incoherent sampling,we derive a variable density sampling function p(m,n)in the frequency transform domains according to the statistical wavelet model.Here p(m,n)indicates the probabil-ity that the(m,n)th coef?cient is sampled.The sampling pattern in the frequency transform domain is generated randomly from the proposed density sampling function,where samples are generated according to the underlying probability function.
Compared with other sampling patterns,such as radial sam-pling pattern or variable density spiral that also exploit some“a priori”information about the image of interest[1,6],the proposed sampling patterns are not heuristically constructed,but are based on reliable statistical models of wavelet coef?cients and thus the pro-posed sampling patterns are analytically justi?https://www.wendangku.net/doc/f513362310.html,pared with the method proposed in[8],which samples the signals directly in the wavelet domain,the proposed sampling method is more general and does not rely on any speci?c wavelet basis.
Image reconstruction from a reduced set of random samples is inevitably exacerbated by the interference resulting from undersam-pling,which depends on the incoherence of the sampling patterns. To compare different sampling patterns,an effective metric to quan-titatively evaluate the incoherence of the sampling patterns is desir-able.The point spread function(PSF)and transformed point spread function(TPSF)has been proposed in[7]to evaluate the sampling pattern incoherence,but they do not consider the“a priori”informa-tion of the underlying signals and thus are not suitable for this work. Therefore,a new incoherence metric is proposed in this paper which incorporates the underlying wavelet statistical models.Simulation results show that there is strong consistence between the low value of the proposed metric and the good quality of the reconstructed image.
https://www.wendangku.net/doc/f513362310.html,PRESSED SENSING IN A TRANSFORM DOMAIN
A signal x∈R N is S sparse on some basisΨ=[ψ
1
,ψ
2
,...,ψ
N
]if x can be represented by a linear combination ofΨwith S N.Thus,the signal can be expressed as:x=Ψθ,whereθis an N×1vector with only S non-zero https://www.wendangku.net/doc/f513362310.html,pressive signals, such as natural images,can be well approximated by this sparse signal model.
For the application at hand,we consider compressed image sampling in a transform domainΦ.To obtain the sparse signal in-formation,we acquire a small set of transform coef?cients of x in Φ.The measurements are given by:y=Φ?x,whereΦ?is a M×N
matrix with M N and y=[y1,y2,...,y M]T represents the M mea-
surements.Each row ofΦ?,denoted byφ
i ,is taken from a subset
??{1,...,N}of atoms ofΦ.IfΨandΦare incoherent with each
other andφ
i is randomly chosen,then given M=CS log N N,
where C is an oversampling factor,x can be recovered from M measurements with high probability[1,2].The incoherence prop-erty holds for many pairs of bases(Ψ,Φ).If noise is present,the measurements can be modeled as:y n=Φ?Ψθ+n,where n is zero-mean additive white Gaussian noise.The reconstructed sig-nal can be obtained by using Basis Pursuit Denoising algorithm (BPDN)which solves the following optimization problem[9]:
min
θ
Φ?Ψθ?y n 22+λ θ 1,(1)
where θ 1=∑i|θi|,y n=Φ?Ψθ+n,andλ>0depends on the noise level.Note that BPDN works equally well for the approxi-mate reconstruction of compressible signals[9].If there exist fast algorithms associated with bothΦandΨ,then a fast reconstruc-tion algorithm can be implemented for signal reconstruction[10].
A second image reconstruction algorithm widely used is the mini-mization of total variation(min-TV)with quadratic constraints.Let x(i,j)be a N×N image,then min-TV with quadratic constraints aims to solve the following problem[11]:
min
x
x TV s.t. Φ?x?y n ≤ε,(2)
where x TV=∑i,j
[x(i+1,j)?x(i,j)]2+[x(i,j+1)?x(i,j)]2is
the total variation andε>0is a constant depending on the noise level.Min-TV with quadratic constraints usually provides better quality of image reconstruction at the cost of more computation than BPDN.
3.V ARIABLE DENSITY SAMPLING IN THE FOURIER
DOMAIN
Wavelets have well de?ned spectral characteristics in the Fourier domain[12].A coarse scale wavelet has its spectrum localized in the low frequency band whereas a?ne scale wavelet has its spectrum widely spread out in the high frequency band.In[4], Fourier sampling of wavelet subbands is proposed to acquire sig-nals sparse in the wavelet domain.To minimize the coherence between the sparse wavelets and the selected Fourier atoms,the Fourier atoms are chosen in the region where signi?cant energy in the wavelet spectrum exists.Let k=1,2,...,K denote the scale of a1D wavelets where k=1is the?nest scale and k=K is the coarsest scale.Without loss of generality,it is assumed that the 1D wavelet has length N=2K.Let?k,l denote the1D wavelet at a scale k with a shift l∈[0,N2?k?1],then the discrete Fourier transform(DFT)spectrum of?k,l is approximately over the band B k=[?N2?k,?N2?k?1]∪[N2?k?1,N2?k].To reconstruct?k,l from its DFT samples,we need to select the DFT atoms randomly within the band B k[4].The probability that a DFT atom within B k will be sampled depends on the size of B k and on the num-ber of signi?cant wavelets over B k.The sampling strategy can be naturally extended to2D image sampling.
Assuming a natural image of size N×N,we establish a statisti-cal model to describe the distribution of the image in the2D wavelet domain.For mathematical tractability,we assume that the statisti-cal distribution of2D wavelet coef?cients at each subband can be modeled as Gaussian distributions with variance at each scale de-creasing exponentially from coarse scales to?ne scales[13,14]. Let w B i,j,k denotes the(i,j)th wavelet coef?cient in the k th scale of the subband B∈LH,HL,HH.It is assumed that w B i,j,k obey i.i.
d.zero-mean Gaussian distribution.That is:w B i,j,k～N(0,σ2k). The varianceσ2k decreases exponentially from coarse scales to?ne scales[14]:
σ2k=2?a(K?k),(3)where a>0.It was shown in[14]that a can range from2.25to3.1 based on the inference from empirical studies.The coarsest scaling coef?cient of the natural image belongs to a uniform distribution w0～U(0,1).The model used in this work is simpli?ed where correlations between the wavelet coef?cients are not accounted for. Although more accurate statistical models for wavelet coef?cients exist,this model is suf?ciently accurate for sampling purposes and allows us to pursue further analysis.[14,15].
Assume that the most signi?cant wavelet coef?cients are those whose magnitudes are larger than a given thresholdμ>0.Since the number of wavelet coef?cients at scale k λk≈3(4K?k) 2 σk √ 2π ∞ μ e ?x2 2σ2k dx=3(4K?k) 1?erf μ √ 2σk , where erf(x) 2√ π x e?t2dt.Furthermore,let s=max{|m|,|n|} where?N/2≤m,n ?p k(m,n)∝ λk ?k ≈1?erf( μ √ 2σk ).(4) We are interested in the asymptotic behavior of?p k(m,n)when k→1.Note that in that case,σk 1and thusμ √ 2σk 1.Then,using the fact that erf(x)≈1?e?x2 x √ π ,for x 1,?p k(m,n)can be approximated as: ?p k(m,n)∝ σk μ e ?μ2 2σ2k≈ (2s)?a2 μ e?(μ 2 21?a s a)(5) In Eq.(5),the term e?(μ 2 21?a s a)determines the asymptotic behavior of ?p k(m,n).Thus,it is clear that the decreasing rate of?p k(m,n)alone with the atom coordinate s depends on the the term e?(μ 2 21?a s a).For the random selection of DFT atoms,it is convenient to construct a sampling density function in the DFT domain and generate a sam-pling pattern according to the sampling density function. To conform to the decaying behavior of the sampling probabil-ity with increasing coordinates while keeping a simple form of the sampling function,we propose a new family of sampling density functions containing only exponential terms.Assuming the size of the underlying image is M×N,the probability that the(m,n)th co-ef?cient is sampled reduces to: p F (m,n)=e ? √ (m M)2+(n N)2 a F σ2F,(6) where?M/2≤m F depends on the characteristics of the underlying image and is directly related to a in(3).From Eq.(5),it is clear that a F should be slightly larger than a to capture the decaying behavior of the sampling probabil-ity.If an estimation of a is not available,then setting a F =3.5is recommended since this value leads to robust and satisfactory CS performance.σ F is a positive parameter which is tuned to obtain desired number of samples. The sampling patterns generated from the sampling density function are binary where1at(m,n)indicates a sampling point and 0means no measurement on that point is made.With a probability given by p F (m,n),1will be generated at(m,n);otherwise,0will be generated at(m,n).The local density of the samples approximates the value of the sampling density function and it can be shown that the number of samples generated is monotonically increasing with σ F .Thus,it is easy to tune the parametersσ F iteratively to get the desired number of samples. Figure1:Ordered DHT and DCT spectra of Daubechies-4wavelets(a)at scale4;(b)at scale3;(c)at scale2;(d)at scale1. 4.V ARIABLE DENSITY SAMPLING IN THE ORDERED DHT DOMAIN Sampling in the ordered DHT domain is suitable for image sam- pling with hardware capable of representing binary measurement matrices.An appealing feature of ordered DHT is that the2D basis images are binary and can be easily implemented by digital micro-mirror devices(DMD)or liquid crystal spatial light modula- tors(LC-SLM)[16].Furthermore,ordered DHT is equipped with fast transform algorithms.In ordered DHT,the atoms are ordered by the number of sign changes(zero crossing)between consecu- tive entries in a Hadamard atom[17].Thus,ordered DHT can be regarded as a generalized class of DFT and more speci?cally,a bi- narized version of DCT sharing thus many properties of DFT and DCT[18]. To design the incoherent sampling pattern in the ordered DHT domain,we exploit the fact that ordered DHT can be thought of as a binary approximation of the DCT.To this end,we?rst describe the spectrum characteristics of the1D wavelets in the DCT domain. Let v k,l(m),0≤m≤N?1be the the DCT of a wavelet?k,l(n), n=0,1,...,N?1.De?ne the averaged DCT spectrum of a wavelets at scale k as: V(m)k=N2?k?1 ∑ l=0 abs v k,l(m) .(7) It can be shown that the DCT spectrum of wavelets at scale k has ap-proximately the same shape as their Fourier spectra[19].The DCT spectrum band B k of wavelets at scale k is B k≈[N2?k,N2?k+1] [19].Much like the incoherent sampling in the Fourier domain,to reconstruct?k,l from its DCT samples,we need to select the DCT atoms randomly within the band B k.Now consider the incoherent sampling in the ordered DHT domain.We can de?ne the averaged ordered DHT spectrum of wavelets at scale k as: H(m)k=N2?k?1 ∑ l=0 abs(h k,l(m)),(8) where h k,l(m),0≤m≤N?1is the ordered DHT of?k,l(n).Since ordered DHT is the binary approximation of the DCT,H(m)k can be approximated as:H(m)k≈V(m)k.To further illustrate this point, Fig.1shows H(m)k for k=1to4where N=256and Daubechies-4wavelets are used.The corresponding DCT spectrum at k scale is also shown.It can be seen that the averaged ordered DHT spectrum and the averaged DCT spectrum are similar,which indicates that the principles of sampling in the DCT domain should be equally ap-plied to the sampling in the ordered DHT domain.Such conclusion also hold for2D DCT and2D ordered DHT.Following a similar procedure to that described in Sec.3,the variable density sampling function in2D ordered DHT domain is designed as follows: p H (m,n)=e ? √ (m M)2+(n N)2 a H σ2H,(9) where0≤m≤M?1,0≤n≤N?1.a H depends on the parameter a in the image statistical model(3)andσ H >0depends on the number of samples.The sampling patterns are then obtained as realizations of the statistical model p H (m,n). 5.EV ALUATION OF INCOHERENCE SAMPLING Undersampling in the transform domain inevitably brings aliasing interference in signal reconstruction.However,if the sampling is incoherent,the aliasing interference has noise-like effect and can can be removed by compressed sensing without degrading the im- age quality[7].To evaluate the incoherent sampling patterns for natural images we consider the interference in the wavelet domain resulted from undersampling in the measurement transform domain Φ.To this end,we extend the TPSF analysis in[7]and incorporate the statistical model of signal distribution in the wavelet domain. Let w i,j,(i,j)∈[0,N?1]×[0,N?1],be the simpli?ed notation of the(i,j)th2D wavelet coef?cient of a natural image that obeys the proposed model in Sec.3.Incoherence is measured by the inter- ference in the wavelet domain caused by the undersampling inΦ. Without loss of generality,letΦu be the undersampled ordered DHT operator andΦ+u be the back-projection ordered DHT operator.Let Ψbe the sparse wavelet transform andΨi,j be the(i,j)th wavelet atom.For an image of size N×N,we de?ne a weighted TPSF as:T i,j w=|w i,j|ΨΦ+uΦuΨi,j.T i,j w is the interference in the wavelet domain caused by the undersampling of|w i,j|Ψi,j in ordered DHT. T i,j w is the distribution of the energy of|w i,j|Ψi,j to other wavelets through back-projeciton.A nonzero value of T i,j w at(m,n)=(i,j) means that the wavelet component at(m,n)suffers from the inter- ference caused by the undersampling of the wavelet component at (i,j).Note that if test image is available,w i,j can also be drawn from the test image directly.The incoherence of the sampling pat- tern can be evaluated by the following metricξde?ned as: ξ= 1 N2×N2 ∑ (i,j) ∑ (m,n) |?T i,j w(m,n)|2,(10) where?T i,j w(m,n)is de?ned as:?T i,j w(m,n)=T i,j w(m,n)for(m,n)= (i,j)and?T i,j w(m,n)=0for(m,n)=(i,j).ξcan effectively mea- sure the strength of the aliasing interference.The smaller theξ,the smaller the incoherent interference.As will be shown in the simula- tions,among several sampling patterns,ξcan be used to select the sampling pattern that yields the lowest incoherence interference. 6.SIMULATIONS Due to the space constraints,only compressed image sampling in the2D ordered DHT domain is illustrated.The proposed sampling method is applied to acquisition and reconstruction of the natural image“Boat”shown in Fig.2(a)with size256×256.The test im- age is assumed sparse in the Daubechies-8wavelet domain and the pixel values are scaled within interval[0,1].It is also assumed that each measurement is corrupted by additive white Gaussian noise with varianceσ2=1e?4.The BPDN algorithm and min-TV with quadratic constraints algorithm are used for image reconstruction. (a)(b) (c)(d) (e)(f) (g)(h) Figure2:(a)The original“boat” age.(c)Proposed variable density reconstructed image from the proposed sampling pattern.(f)Part of the reconstructed sampling pattern.(g)Logarithmic spiral sampling pattern.(h)Part of the reconstructed image from the Logarithmic spiral sampling pattern. Figure2(c)depicts a realization of the proposed variable den-sity sampling pattern in the ordered DHT domain that contains 20000sampling points generated from Eq.(9)with a H =3.5and σ H =0.501.To illustrate more details of the reconstructed im- age,we compare part of the original image and part of the recon-structed image corresponding to the region within the white frame in Fig.2(a).The framed area of the original image and the framed area of the reconstructed image are shown in Fig.2(b)and Fig.2(d), respectively.It is clear that with an undersampling ratio of30.5%, the test image is reconstructed with only small distortion. 10000 15000 20000 25000 5000 10000 15000 20000 25000 28 28.5 29 29.5 30 30.5 31 31.5 P S N R ( d B ) Figure3:Reconstruction the proposed ber of measurements For comparison from the samples rithmic spiral spectively. patterns are of the corresponding and Fig.2(h), aliasing artifacts, formation design of the density sampling achieves much better performance than logarith-mic spiral patterns and radial patterns.The performance gain is 2～4dB with5000samples and0.5～0.8dB with25000sam-ples.To verify that the proposed sampling pattern,which exploits the a priori information of natural images,achieves better perfor-mance than methods that do not exploit the a priori information, the simulation results using the Noiselet ensemble,SFE and SBHE under different number of measurements are also presented in Ta-ble1.To acquire the image information,samples of the Noiselets, SFE and SBHE are taken randomly.The proposed sampling method achieves the best performance in all simulations.Such comparison clearly shows the performance gain achieved by exploiting the a priori information. The incoherence of the proposed sampling patterns can also be evaluated by using the proposed incoherence metric proposed in this paper.We test three sampling patterns in the ordered DHT domain using the image“Boat”and take5000measurements.The inco-herence metrics are0.0034,0.0062and0.0078for the proposed variable density sampling pattern,radial sampling pattern and log-arithmic spiral sampling pattern,respectively.The lowest metric value is yielded by the proposed variable density sampling pattern leading to the least aliasing interference,which,in turn,is in accor-dance with the best reconstruction performance. Finally,we show that the reconstruction performance is not very sensitive to the parameters a F or a H in the sampling functions.Here, we test the reconstruction performance as the parameter a H changes. Figure3shows the reconstruction of images“Boat”and“Lena”with20000measurements.Images“Boat”and“Lena”have differ-ent curves and textures,thus have different statistical model param-eters.The sampling patterns are constructed from sampling density functions with a H ranging from2to5.It is shown in Fig.3that the sampling patterns lead to similar performance for both images when a H ranges from2.5to4.5.For both images,the difference of performance measured by PSNR is within0.3dB with different a H .Thus,the image reconstruction is robust to variations over a H . Interestingly,note also that the reconstruction performance tends to become worse as a H increases.Since larger a H means that more low frequency samples are taken,the simulation shows that sam- Figure4:Reconstruction results for image“Boat’using min-TV algorithm with quadratic constraints. Figure4shows the simulation results of image recovery us-ing min-TV.For illustrative purposes,only the following sampling schemes are considered:the proposed variable density sampling, logarithmic spiral sampling and SBHE[5].The simulation results show that the proposed variable density sampling again achieves the best performance,which is more obvious when the sampling ratio is low.It can be concluded that the advantages of using the proposed variable density sampling does not depend on a speci?c reconstruc-tion algorithm. 7.CONCLUSIONS In this paper,a family of variable density sampling patterns are pro-posed for compressed sensing of natural images in the Fourier do-main,DCT and ordered DHT domain,which are based on the sta-tistical model of natural images in the sparse wavelet https://www.wendangku.net/doc/f513362310.html,-pared with other sampling scheme,our proposed method is simple, fast and can be extended to a wide range of applications.Further-more,the“a priori”information needed is pretty general and no data training for parameter estimation is needed.The proposed met-ric,along with extensive simulation study,show that the proposed sampling pattern leads to the least coherent interference. 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