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1638 J. Opt. Soc. Am. A/Vol. 27, No. 7 /July 2010 Z. Xu and E. Y. Lam

Image reconstruction using spectroscopic andhyperspectral information for compressive

terahertz imaging

Zhimin Xu and Edmund Y. Lam*

Imaging Systems Laboratory, Department of Electrical and Electronic Engineering, The University of Hong Kong,Pokfulam Road, Hong Kong, China

*Corresponding author: elam@eee.hku.hk

Received December 10, 2009; revised May 19, 2010; accepted May 25, 2010;posted May 25, 2010 (Doc. ID 121306); published June 16, 2010

Terahertz (THz) time-domain imaging is an emerging modality and has attracted a lot of interest. However,existing THz imaging systems often require a long scan time and sophisticated system design. Recently, a newdesign incorporating compressed sensing (CS) leads to a lower detector cost and shorter scan time, in exchangefor computation in an image reconstruction step. In this paper, we develop two reconstruction algorithms thatcan estimate the underlying scene as accurately as possible. First is a single-band CS reconstruction method,where we show that by making use of prior information about the phase and the correlation between the spa-tial distributions of the amplitude and phase, the reconstruction quality can be significantly improved overpreviously published methods. Second, we develop a method that uses the multi-frequency nature of the THzpulse. Through effective use of the spatial sparsity, spectroscopic phase information, and correlations acrossthe hyperspectral bands, our method can further enhance the recovered image quality. This is demonstrated bycomputation on a set of experimental THz data captured in a single-pixel THz system. © 2010 Optical Societyof America

OCIS codes: 110.6795, 100.3010, 100.3020, 100.3190, 110.1758.

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. INTRODUCTIONn recent years, advances in terahertz (THz) science havettracted increasing attention in time-domain THz imag-ng for a very diverse array of applications. These variousotential applications range from biology and medical sci-nces [1] and non-destructive evaluation [2] to qualityontrol processes [3] and homeland security [4]. In con-rast to the conventional optical modality that is based onelative intensity measurements only, one important ad-antage of THz time-domain imaging systems is that theransient electric field itself can be measured, from whiche can determine the amplitude and phase of each spec-

ral component that makes up the entire THz pulse.hus, THz imaging systems provide opportunities forpectroscopic studies, which are not possible with manyompeting modalities. Unfortunately, most existing THzmaging systems suffer from slow acquisition because ofheir raster-scanning nature [2,5,6]. For example, usingne of the fastest raster-scanning time-domain THz imag-ng system to date [6], we still need 6 min to scan a00 mm2 area at 0.25 mm resolution (equivalently, a00�400 pixel image). Although recent efforts by usingore sophisticated imaging technologies, such as inter-

erometric and tomographic approaches, have shown pre-iminary successes, raster-scanning is still a latent factorhat limits the acquisition rate of such systems. The highomplexity and hardware requirements are also majorimitations to their practical applications.

To meet the requirements of practical, time-crucial ap-lications, Chan et al. [7,8] propose two fast time-domainHz imaging schemes by applying the compressed sens-

1084-7529/10/071638-9/$15.00 © 2

ng (CS) theory [9,10]. In [7], the imaging process is accel-rated by randomly sampling only a subset of the Fouriermplitude measurements. However, signal acquisition inhis system still requires the THz receiver to raster scanhe focal plane. To replace this mechanical raster-canning, they then present a proof-of-concept single-ixel THz imaging setup in [8]. The system schematic isllustrated in Fig. 1. Instead of collecting the pixels/oxels, a single detection element is used to sample theoncentrated THz beams, which are spatially modulatedith a set of random patterns. According to the CS theory,uch fewer samples than the total number of image pix-

ls are needed to fully reconstruct an image, which thusmplement fast compressed imaging. In exchange forigh-speed signal acquisition, the challenge of imple-enting such a system comes in developing efficient spar-

ifying representations and the corresponding algorithmsor signal recovery from the linear measurements. The CSheory suggests the conditions that need to be fulfilled tollow nearly perfect reconstruction with a much smallerumber of measurements. However, in reality, these con-itions are hard to satisfy fully. In addition, the signal ob-ained from time-domain THz imaging systems is theransient electric field. That is, the received signal is com-lex [8,11]. Several CS reconstruction algorithms haveeen proven effective in dealing with real-valued signals,ut for complex-valued signals, challenges still exist. Theuthors in [12,13] extend the considerations to the com-lex domain. However, aside from the �1 norm regularizern the complex data, they did not exploit the other a pri-ri information.

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Z. Xu and E. Y. Lam Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. A 1639

In this paper, we present a reconstruction algorithm tostimate the underlying complex signals based on theingle-pixel THz imaging system described in [8]. Wehow that by using the prior knowledge of the phase androperties contained in the system, our algorithm can sig-ificantly improve the reconstruction quality. In addition,e note that time-domain THz modalities, such as THz

ime-domain spectroscopy (THz-TDS) and THz imagingystems, also offer an important piece of information thats not commonly found in other ones: the THz pulse con-ains information at multiple frequencies, and thereforeas hyperspectral information [14,15]. We thus consider

urther enhancing the reconstruction image quality byaking use of such hyperspectral information. Prelimi-

ary work along this direction has been reported in [16].This paper is organized as follows. Following a brief re-

iew in Subsection 2.A concerning the CS theory and theeconstruction techniques commonly used, we propose aparse reconstruction method for signal recovery at a spe-ific frequency in Subsection 2.B. The work presentedere is an extension of our previous work in [17], andore technical details are provided. Then, in Subsection

.C we introduce our partition-based multiscale hyper-pectral image estimation approach for the complex-alued data. In Subsection 2.D, we combine the single-and sparse reconstruction method with multiscaleyperspectral image estimation and devise a hyperspec-ral image reconstruction algorithm for the compressiveHz imaging system. For illustration purposes, we dem-nstrate in Section 3 the output of our algorithm on theractical THz data. Last, based on the experimental re-ults, we draw our conclusions in Section 4.

. METHODS. Compressed Sensing Backgroundompressed sensing is concerned with encoding a sparseignal using a relatively small number of linear measure-ents and ensures accurate reconstruction with a veryigh probability when a sufficient condition called the re-tricted isometry property (RIP) is satisfied [9,10]. Math-matically, given an M�N measurement matrix � with

�N and a vector b that denotes the linear measure-ents of an N-dimensional sparse signal x, say b=�x,

he optimal estimate of the underlying sparse signal cane solved by

ig. 1. (Color online) Schematic diagram of a single-pixel,ulsed THz imaging system based on that in [8].

minimize �x�0 subject to �x = b, �1�

ith �x�0 the number of nonzero entries in x, also calledhe �0 norm. That is, we can estimate the underlyingparse/compressible signals (e.g., THz signals) by search-ng the sparsest solution satisfying �x=b. But this isomputationally intractable and highly sensitive to noise18]. Consequently, an easier-to-solve linear program issed for searching the sparsest solution of �x=b; that is,

minimize �x�1 subject to �x = b, �2�

here �x�1=�i=1N �xi� denotes the �1 norm of a vector x. In

ther words, one minimizes the �1 norm of the signal in-tead of the sparsity itself.

To solve such an inverse problem, several methods andode packages have been developed, such as the SPGL1lgorithm [19,20], gradient projection for sparse recon-truction [21], �1-magic [22], etc. Since the signal x con-idered in the conventional CS theory is real, most of thelgorithms focus only on processing real-valued signals.lthough some researchers have tried to extend theirethods to the complex domain [19], the reconstruction

uality in practical applications is often not satisfactory.or instance, one such attempt can be observed from Fig.in [8]. If we consider the optimization problem in Eq. (2)

arefully, we may note that for the complex case (i.e., xCN), the term �x�1 actually imposes constraints only on

he amplitude of x. So aside from sparsity of the ampli-ude intensity, any other a priori information (e.g., thehase) has not been exploited. The pulsed THz imagingystems are well-known for providing spectroscopic phasenformation, and this leads us to further considerationsn the use of the phase. On the other hand, in the time-omain setups, such as the THz imaging systems in [7,8],ome underlying relations have not been used for furthermprovement in the image reconstruction. This includeshe similarities between spatial intensity distributions ofhe amplitude and phase and the correlation across theyperspectral bands. In the following sections, we demon-trate the capabilities of this prior knowledge in the re-overed image quality improvement.

. Single-Band Compressed Sensing Reconstructionhe single-pixel pulsed THz imaging system in Fig. 1

mplements a CS process. The basic principle behind suchCS imaging system can be described as, at a particular

requency fk,

b�fk� = �x�fk�, �3�

here b�fk��CM is a column vector of measurements and�fk� represents the underlying N�N complex-valued sig-al ordered in an N2�1 vector, spatially modulated by aet of random patterns that form the measurement ma-rix ��RM�N2

. As mentioned above, the CS theoryrovides us with the mathematical basis for the accurateecovery of the original signals with only a few measure-ents (i.e., M�N2). For simplicity, one could directly

dapt the optimization program in the form of Eq. (2) tour case, just recognizing that the true and observed datare both complex. Since seeking the sparsest exact solu-ion may be useless because of the additive noise, an ap-

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1640 J. Opt. Soc. Am. A/Vol. 27, No. 7 /July 2010 Z. Xu and E. Y. Lam

roximate reconstruction is often preferred. It thenakes sense to replace Eq. (2) by

minimize �x�fk��1 subject to ��x�fk� − b�fk��2 � �,

�4�

ith � the tolerance to be defined. Unfortunately, such aimple extension to the complex domain can hardly pro-uce a satisfactory estimation of the original scene. An ex-mple of this can be found in Fig. 4 in [8], and also in ourxperimental results in Section 3.

In our recent work [17], we propose an effective methodhat can significantly improve the reconstruction qualityn combination with the phase information carried in theeceived THz pulses. The basic consideration relies on theact that the phase should be smooth and not vary rapidly23,24]. Hence, to interpret the smoothness constraint onhe phase, we apply the similarities between the ith pixelnd the spatial neighborhood �i of a certain size centeredt pixel i. The size of �i controls the degree of spatialmoothness. Meanwhile, this kind of smoothness con-traint of the phase can also alleviate phase wrapping er-ors and noise to some degree. The size of the neighbor-ood area should be chosen according to the specificituation.

Let ��xi� and ���i� be the phase value of the ith pixelnd the mean phase value in �i, respectively. The smooth-ess constraint on the phase intensities is then defined as

���x� − ��x��2 = ��i=1

N2

���xi� − ���i��21/2

� , �5�

ith

��xi� = − j logxi

�xi�, if �xi� T

0, otherwise� � �− �,��. �6�

ere the parameter is used to control the similaritiesetween the ith pixel and its nearest neighbors, and T re-ers to a given threshold for separating the regions con-aining signal and noise only. Note that the reconstructedhase is arbitrary and carries no physical information athe location where the signal is very weak (i.e., the am-litude intensity is less than a given value T). Such arti-acts will seriously affect the appearance of the restoredesults. We will show its effectiveness in detail in the ex-erimental section.In addition, many existing time-domain THz imaging

ystems, especially the one in Fig. 1, are based on aransmission-type spectroscopy. With no loss of generality,e assume that the object is piecewise homogeneous andas uniform thickness. Thus, we get one additional piecef prior knowledge about the original signals, which ishat the smooth regions in the spatial distribution of am-litude should be the same as those in the spatial distri-ution of phase. Mathematically, let x1=A1 exp�j�1 and2=A2 exp�j�2 be the complex intensities of two differentixels. If these two pixels are in the same homogeneousegion, then A1=A2 and �1=�2. Accordingly, we can con-lude that the first-order difference of the complex inten-ities in a homogeneous region is zero.

According to this assumption, if we define the totalariation (TV) of a two-dimensional complex datum basedn the �0 norm as

�i

��ihx�0 + ��i

vx�0,

ith �ih and �i

v as linear operators corresponding to, re-pectively, horizontal and vertical first-order differencest pixel i, then minimizing the TV will be a more appro-riate choice for sparse complex image reconstruction inHz systems. Such a definition is not only good for signalparsifying, but also emphasizes the correlation betweenhe spatial distributions of the amplitude and phase. Thats, for the solution that produces the minimum value ofhat �0 norm TV, any two different pixels in the same ho-ogenous region will not only have the same amplitude

ntensities, but also the same phase. The other optimiza-ion processes, such as minimizing the �1 norm of the sig-al itself or the coefficients in a certain transform do-ain, however, cannot efficiently utilize such spatial

orrelation. Meanwhile, although some transform opera-ion (e.g., wavelet transform) may sparsify the signal, thepatial correlation between the amplitude and phase wille destroyed in that transform domain.For tractable programming, we replace �0 norm with �1

orm, i.e.,

�x�TV � �i

��ihx� + ��i

vx�. �7�

uch a form, which is defined as the complex-valued TV,lso appears in [25], as holography is another common op-ical system that involves capturing complex signals26,27]. Then minimizing Eq. (7) will give us the same so-ution as the one by minimizing the �0 norm TV. At thisoint, we would like to mention that in some sources [28],nother form of the �1 norm TV on complex data has beensed as follows:

�x�TV � �i

���ihx�2 + ��i

vx�2�1/2. �8�

owever, the distinction between these two regularizershould be kept in mind, since, at least in our optimizationroblems, the definition in Eq. (7) leads to better resultsith much sharper edges, as illustrated in Section 3. Herend below, the term �x�TV refers to our first definition inq. (7).Considering the case at a particular frequency fk, the

parse reconstruction algorithm for the CS THz imagingystem can be interpreted as an optimization given by

minimize �x�fk��TV

subject to ��x�fk� − b�fk��2 � �,

���x�fk�� − ��x�fk���2 � ,

�9�

r, equivalently, by the following criterion

x�fk� = arg minx�fk�

12 ��x�fk� − b�fk��2

2 + �x�fk��TV + ����x�fk��

− ��x�fk���2. �10�

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Z. Xu and E. Y. Lam Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. A 1641

uch a solution of Eq. (10) can be found by convex pro-ramming methods, e.g., the nonlinear conjugate gradi-nt method combined with backtracking line search.

. Complex Multiscale Hyperspectral Imagestimationime-domain THz systems provide the capability to cap-ure multiple spectral information in the far-infraredange. Recently, researchers have begun to pay more andore attention to THz hyperspectral imaging, such as se-

urity screening [14,15]. However, degradation (i.e., noise,lurring, etc.) of the observations acquired from hyper-pectral imaging systems is always inevitable, and thusecovery of hyperspectral imagery from degraded obser-ations becomes a vital task. A direct and naïve thought iso treat each spectral channel as an independent signalnd then process them separately. The main shortcomingf such an approach is that it fails to account for the in-ercorrelated relationships present in the original hyper-pectral signal. Researchers thus consider more sophisti-ated techniques to alleviate the degradation by takingot only the individual channels, but also the correlationcross them, into account. One major approach isavelet-based hyperspectral image estimation, becauseavelet-based techniques can provide good spatial adap-

ivity by virtue of the well-localized property of waveletsn both time and frequency.

In [29,30], Atkinson et al. present a near-optimal hy-erspectral image estimation scheme by addressing theifferent nature of spatial and spectral dimensions, whichhus avoids the requirement of second-order signal statis-ics. In particular, based on some assumptions of the in-erchannel correlation, the hyperspectral signal can bepproximately decorrelated in the spatial domain by us-ng a 2-D discrete wavelet transform and the discreteourier transform (DFT) in the spectral domain. How-ver, their work is essentially derived from the analysis ofhe hyperspectral Wiener filter, which is based on the as-umption that the system is corrupted by additive whiteaussian noise. Besides, to choose a suitable wavelet ba-

is to guarantee an optimal converge rate, some a priorinformation about the underlying signals has to be pro-ided, e.g., the degrees of smoothness.

Another class of hyperspectral image estimation ap-roaches, which can be categorized as partition-basedethods, automatically adapts to the signal functionithout any user input or a priori notion. This enjoys thedvantages of wavelet-based techniques, such as fast con-ergence rates. The work by Willett and Nowak [31]hows a way for Poisson intensity estimation of a single-hannel signal by using a penalized likelihood method onecursive dyadic partitions. Then Krishnamurthy andillett [32] use such a partition-based multiscale Poisson

ntensity estimator in hyperspectral imaging reconstruc-ion. While these methods are concerned with real-valuedignal estimation from photon-limited observations cor-upted by Poisson noise, in this paper we develop aartition-based multiscale estimation scheme foromplex-valued hyperspectral signals gathered fromime-domain THz imaging systems.

It is reasonable to think of a hyperspectral signal ac-uired from a time-domain THz imaging system as a

hree-dimensional dataset, where the first two dimen-ions correspond to the spatial locations and the third di-ension indicates the index of each spectral band. Let x

nd y respectively denote the true complex hyperspectralignal and the distorted complex hyperspectral observa-ion on a grid of size N�N�K, where K represents theumber of spectral bands. Our goal is to estimate x fromas accurately as possible. Note that in this subsection,

he hyperspectral data x and y are of size N�N�K.In accordance with [32], a key feature of the hyperspec-

ral images is that the boundaries and singularities areocated at the same spatial positions across all the spec-ral bands, no matter how bad the contrast or perceptibil-ty is at some band. This is also true in our THz imagingcenario. In each spectral band, the underlying amplitudend phase intensities contain the same spatial bound-ries in their respective domains. Therefore, in what fol-ows we process the amplitude and phase separately. Spe-ifically, we decompose the underlying signal x into themplitude part xA and phase part x�, and equally defineA and y� for the observation y. Since the signals con-erned here are basically spatially piecewise constant, toetter adapt to the spatial distributions and preserve theiscontinuities, we choose the multiscale approach by re-pectively performing recursive dyadic partitioning (RDP)n the amplitude and phase domains. The RDP process onn image produces a tree representation by recursivelyecomposing any part of an existing partition into dyadicquares, replacing a square by four similar squares of halfhe size [33]. Thus, the spatial RDPs can be representedn terms of quad-trees. Figure 2(a) gives a sample of anncomplete RDP and its tree. Since the partition definedy the RDP is not unique, we use penalized likelihood es-imation to select the optimal partition P that provideshe best fit to the observations from the space of possibleartitions �P. Each of the terminal squares of this data-daptive RDP P corresponds to a spatially homogeneousegion. An example of such partitioning on a two-imensional image is shown in Fig. 2(b).In particular, at the beginning, complete spatial RDPs

n amplitude and phase domains are obtained by recur-ively partitioning their respective spaces into cells withyadic side length until reaching the pixel-level reso-ution. Then starting from this finest level, we searchackward to find the fittest partition over every level ofhe quad-tree representation by merging adjoining

ig. 2. (Color online) (a) A Example of an incomplete RDP andts quad-tree. (b) Sample partition of a two-dimensional image.

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1642 J. Opt. Soc. Am. A/Vol. 27, No. 7 /July 2010 Z. Xu and E. Y. Lam

quares based on the similarities of the intensities in theeighboring cells. We let PA and P� denote the respectivepatial RDP representations of amplitude and phase do-ains. Thus in each partition cell ca�PA and c��P�,

ased on the respective likelihood of observing yA�ca� and

��c�� given the amplitude estimate xA

�ca� and phase esti-ate x�

�c��, p�yA�ca� �xA

�ca�� and p�y��c�� �x�

�c���, we separatelyompute the following penalized likelihood (in a log-ikelihood sense):

xA�ca� = arg min

xA�ca�

LA�ca� = arg min

xA�ca�

�− log p�yA�ca��xA

�ca�� ,

�11a�

x�

�c�� = arg minx�

�c��L�

�c�� = arg minx�

�c���− log p�y�

�c���x�

�c��� .

�11b�

Apparently, the conditional probability functions�yA

�ca� �xA�ca�� and p�y�

�c�� �x��c��� are directly related to the

oise distributions in the particular THz system. Whileome work has been performed on modeling the distor-ions in the amplitude and phase domains [34], it is stilln open problem in deciding an accurate noise estimation;n this paper, we choose the Gaussian distributions with

ean zero and variance A2 and �

2) for the likelihood es-imation, i.e.,

p�yA�ca��xA

�ca�� = ��i,j��ca

�k=0

M−1 1

�2�A2

�exp�−�yA�i,j,k� − xA�i,j,k��2

2A2 � ,

�12a�

p�y�

�c���x�

�c��� = ��i,j��c�

�k=0

M−1 1

�2��2

�exp�−�y��i,j,k� − x��i,j,k��2

2�2 � .

�12b�

Certainly, p�yA�ca� �xA

�ca�� and p�y��c�� �x�

�c��� can be modifiedor other kinds of distribution in the cases where the noisean be accurately estimated. Anyhow, such an algorithmives the users high design freedom. Finally, the optimalDP representations in the amplitude and phase domainsre respectively calculated by selecting the ones with theinimal total penalized likelihood, and then the respec-

ive optimal estimators for the amplitude and phase in-ensities are the ones fit best to the observations over ev-ry cell in that optimal RDP, i.e.,

PA = arg minPA

� �ca�PA

LA�ca� + �A�PA�� , �13a�

xA = � �ca�PA

xA�ca�� , �13b�

nd

P� = arg minP�

� �c��P�

L�

�c�� + ���P��� , �14a�

x� = � �c��P�

x�

�c��� . �14b�

ote that in the above equations, the superscripts ca and� restrict the areas of interest, and the variables withhese superscripts are zero outside the corresponding ar-as. The terms �A�PA� and ���P�� are penalties for con-rolling the spatial smoothness in the amplitude andhase domains, respectively. The values of these two pen-lties are respectively proportional to the number of cellsn the partitions PA and P�. More technical details abouthe penalties can be found in [31]. Except for other hyper-pectral image estimation methods, such as the ones men-ioned in [29,30], the key distinction is the use of the cor-elated relationships in the hyperspectral images. In ourethod, such relationships are exploited by separately

orcing the spatial RDPs of the amplitude and phase di-ensions at each spectral band to be the same.

. Compressed Sensing Hyperspectral Imageeconstruction

ncorporating the single-band sparse reconstruction ap-roach presented in Subsection 2.B with the complexultiscale hyperspectral image estimation technique in

ubsection 2.C, we devise a hyperspectral image recon-truction algorithm for the CS THz imaging system. Thelgorithm flow chart is shown in Fig. 3. It consists of twolternating steps:

Step 1: Let x�t� be the input of the tth iteration. Obtainhe solution y�t� by computing Eq. (10) over each observedpectral band.

Step 2: Considering y�t� as the observations in this es-imation process, perform our complex multiscale hyper-pectral estimation method, then obtain x�t+1�. Specifi-ally, deal with the amplitude and phase parts separatelynd get the respective optimal estimators xA

�t+1� and x��t+1�

ccording to Eqs. (11)–(14). Combine the amplitude andhase and then obtain x�t+1�.

These two steps are executed repeatedly, and the algo-ithm terminates when

�x�t+1� − x�t��1

�x�t��1

s small enough.

. RESULTS AND DISCUSSION. Single-Band CS Reconstruction Resultse expand on the preliminary results first presented in

17] with further experiments and analyses. The experi-ental data are acquired with the help of Wai Lam (Wil-

iam) Chan and Daniel M. Mittleman of Rice University,ho are the designers of the single-pixel THz imaging

ystem [8]. The test object is a rectangular hole in anpaque screen filled with two transparent plastic plates ofifferent thickness, as shown in Fig. 4. Figure 5 shows theS reconstruction results by using different approaches

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Z. Xu and E. Y. Lam Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. A 1643

nly from the linear measurements acquired at a specificrequency �0.1 THz�. In these experiments, 600 measure-ents are used to estimate the original complex signal of

ize 32�32.We first perform the reconstruction by directly mini-izing the �1 norm of the underlying signal itself. The re-

ig. 3. Hyperspectral image reconstruction algorithm flowhart.

Fig. 4. (Color online) Rectangular object mask [17].

ults are shown in Figs. 5(a) and 5(b). Figures 5(c) and(d) and Figs. 5(e) and 5(f) are the respective results bysing two different wavelet functions (i.e., Daubechies-4avelet and Daubechies-8 wavelet) for sparsification.hese results are obtained by using the Matlab code pack-ge of SPGL1 algorithm [20]. The SPGL1 algorithm in19] is an often used method which is designed to solvehe conventional sparse reconstruction problem expressedn Eq. (4). Moreover, the SPGL1 algorithm is one of a fewpproaches which can deal with the problems in the com-lex domain. Although the improvement is recognizablen comparison with the direct �1 norm minimization

ethod, the visual quality of these recoveries is still un-atisfactory. For example, one still cannot identify thedges of each plastic plate, and the artifacts distributed inhe background of both amplitude and phase images areuite obvious.

ig. 5. (Color online) CS reconstruction results on the single-and THz data at the frequency of 0.1 THz. Each image is of size2�32. The left column shows the amplitude images, and thehase images are in the right column. (a), (b) Results obtained byirectly minimizing the �1 norm of the underlying signal itself.c), (d) Results reconstructed by minimizing the �1 norm of theaubechies-4 wavelet coefficients. (e), (f) Results obtained by us-

ng Daubechies-8 wavelet transform for sparsification. (g), (h) Re-ults obtained by applying our single-band CS reconstructionethod with Eq. (7) as the complex TV norm. (i), (j) Results ob-

ained by using our single-band approach with the TV norm inhe form of Eq. (8).

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1644 J. Opt. Soc. Am. A/Vol. 27, No. 7 /July 2010 Z. Xu and E. Y. Lam

Next, we show the recovered amplitude and phase im-ges by performing our single-band CS reconstructionethods. Note that in Subsection 2.B, we introduce two

ifferent forms of the complex TV norm, i.e., Eqs. (7) and8). Figures 5(g) and 5(h) correspond to using Eq. (7), andhe results with the TV norm of Eq. (8) are shown in Figs.(i) and 5(j). From the results, we can easily see that ouringle-band algorithm with either of these two forms ofV norm outperforms the conventional method. Yet theistinction between these two different forms also needso be dealt with. The recovery with the form of Eq. (7)ields images consisting of smoother regions with muchharper edges. Hence, in the following experiments, wehoose it as our complex TV regularizer. In addition, inhe phase smoothing regularizer, the smoothness degrees controlled by the size of the neighborhood area aroundach pixel. Since too large a window will cause over-moothing in the estimation, accordingly we use a win-ow of 3�3 pixels.

ig. 6. (Color online) Hyperspectral reconstruction results withhe practical THz data of size 600�16. (a), (b) Amplitude andhase obtained by minimizing the �1 norm of the Daubechies-8avelet coefficients only with the measurements at 0.1 THz. (c),

d) Amplitude and phase reconstructed by using our single-bandeconstruction method only with the measurements at 0.1 THz.e), (f) Amplitude and phase obtained by performing our proposedyperspectral algorithm with data across all 16 spectral bands,isplayed at 0.1 THz.

. CS Hyperspectral Multiscale Reconstruction Resultsn the experiments discussed below, we demonstrate theffectiveness of our proposed CS hyperspectral image re-onstruction method on a set of practical hyperspectralHz measurements. The experimental platform is theame as in the previous experiment. Since each measure-ent obtained from the time-domain THz system is actu-

lly a whole pulse signal containing frequency informa-ion across the THz frequency range, we sample theeasurements at 16 spectral bands uniformly distributed

ver the frequency range between 0.1 THz and 0.2 THz.e first conduct the experiments with 600 measurements

t each spectral band. Therefore, our goal is to find theptimal estimation of the underlying hyperspectral dataf size 32�32�16 with only 600�16 linear measure-ents.Figures 6(a) and 6(b) show the amplitude and phase

mages obtained by applying the conventional CS recon-truction method with the information only at 0.1 THz.

ig. 7. (Color online) Hyperspectral reconstruction results withhe practical THz data of size 400�16. (a),(b) Amplitude andhase obtained by minimizing the �1 norm of the Daubechies-8avelet coefficients only with the measurements at 0.1 THz. (c),

d) Amplitude and phase reconstructed by using our single-bandeconstruction method only with the measurements at 0.1 THz.e), (f) Amplitude and phase obtained by performing our proposedyperspectral algorithm with data across all 16 spectral bands,isplayed at 0.1 THz.

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Z. Xu and E. Y. Lam Vol. 27, No. 7 /July 2010 /J. Opt. Soc. Am. A 1645

he amplitude and phase reconstructed by using ouringle-band reconstruction algorithm in Subsection 2.B athe same frequency are shown in Figs. 6(c) and 6(d). Ingreement with our previous discussion, the latter per-orms much better than the conventional method. How-ver, these methods fail to preserve some fine edges, ande still cannot easily estimate the object edges accurately.Now let us compare our proposed hyperspectral algo-

ithm, shown in Figs. 6(e) and 6(f), with two reconstruc-ion approaches that are based only on the considerationf single-band information. Visually, the reconstructionuality is significantly improved in both the amplitudend phase dimensions, i.e., clearer contours with less no-iceable artifacts and smoother in the homogeneous re-ions. More importantly, the amplitude and phase recov-ries corresponding to our proposed hyperspectral methodre more coincident with the correlation between the am-litude and phase (see Subsection 2.B) and also closer tohe reality that the two plastic plates are homogeneous.n addition, since each pixel corresponds to a 1 mm area,e get the size of the outer rectangular by counting pixels

15 mm�14 mm�, which is equal to the actual measure-ent.Furthermore, we also experiment with decreasing the

umber of the measurements to 400�16 (see Fig. 7). Wean see that the results reconstructed from only the infor-ation at a single spectral band are still worse in quality,

ompared with our hyperspectral recovery. Our hyper-pectral algorithm produces almost the same results,ven though fewer measurements are used for reconstruc-ion.

. CONCLUSIONhis paper studies methods for accurate image recovery

n compressive THz imaging. First, we present a single-and CS reconstruction method. Unlike the conventionalpproach, which emphasizes the sparsity of only the am-litude, our single-band algorithm results in significantmprovement by adding control on the phase and involv-ng the correlation between the spatial distributions ofmplitude and phase into the reconstruction process. Sec-nd, the hyperspectral nature of the THz pulse inspiress to devise a CS hyperspectral image reconstruction al-orithm. In addition to the features of the single-bandne, our hyperspectral method effectively employs corre-ations across the hyperspectral bands and shows gooderformance in preserving edges and alleviating artifactsn both amplitude and phase domains. Experimental re-ults support these claims. Although the discussion inhis paper is mainly based on the single-pixel THz imag-ng system in [8], many of contributions and discoveriesan be readily utilized in other THz systems, such asime-domain THz spectroscopies.

CKNOWLEDGMENTShis work was supported in part by the Research Grantsouncil of the Hong Kong Special Administrative Region,hina, under Projects HKU 713906 and 713408. We ac-

nowledge the help of Wai Lam (William) Chan andaniel M. Mittleman of Rice University in providing thexperimental data.

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