IJBIInternational Journal of Biomedical Imaging1687-41961687-4188Hindawi Publishing Corporation12587110.1155/2009/125871125871Research ArticleA General Total Variation Minimization Theorem for Compressed Sensing Based Interior TomographyHanWeimin1YuHengyong2WangGe2WeiGuowei1Department of MathematicsUniversity of IowaIowa CityIA 52242USAuiowa.edu2Biomedical Imaging DivisionVT-WFU School of Biomedical Engineering and SciencesVirginia TechBlacksburg, VA 24061USAvt.edu200904112009200907092009011120092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Recently, in the compressed sensing framework we found that a two-dimensional interior region-of-interest (ROI) can be exactly reconstructed via the total variation minimization if the ROI is piecewise constant (Yu and Wang, 2009). Here we present a general theorem charactering a minimization property for a piecewise constant function defined on a domain in any dimension. Our major mathematical tool to prove this result is functional analysis without involving the Dirac delta function, which was heuristically used by Yu and Wang (2009).

1. Introduction

While in general an interior region-of-interest (ROI) cannot be uniquely reconstructed from projection data only associated with lines through the ROI [1, 2], in the compressed sensing framework, we recently found that a two-dimensional interior ROI can be exactly reconstructed via the total variation minimization if the function on the ROI is piecewise constant [3, 4]. The major idea behind our analysis is that the total variations of a piecewise constant function and a smooth artifact function are separable. The main mathematical tool is the expression of the two-dimensional gradient in terms of the Dirac delta function. In our analysis , the Delta function was instrumental but applied heuristically without mathematical rigor. In this note, we will prove rigorously a more general theorem, as an extension of the total variation minimization property presented in , to characterize the total variation minimization property for a piecewise constant function defined on a domain in any dimension. Such a theorem may serve as a theoretical basis for further development of interior tomography algorithms.

2. Theoretical Result

For piecewise constant or piecewise smooth functions, it is natural to use the space of functions of bounded variation  to capture the discontinuities. For an integer d>0, let Ωd be a d-dimensional open bounded set and denote its boundary by Ω. Then the space of functions of bounded variation is BV(Ω)={υL1(Ω)υBV(Ω)<}. It is a Banach space with the norm υBV(Ω)=υL1(Ω)+Ω|Dυ|, where υL1(Ω) is the integral of |υ(x)| over Ω, and Ω|Dυ|=sup{Ωυdivϕdx:ϕC01(Ω)d,|ϕ|1inΩ} is the total variation of the function υBV(Ω). Here C01(Ω) is the space of continuously differentiable functions that vanish on Ω, and for ϕ=(ϕ1,,ϕd)TC01(Ω)d, divϕ=i=1dϕi/xi. The Sobolev space W1,1(Ω)={υL1(Ω):|υ|L1(Ω)} is a subspace ofBV(Ω) and Ω|Dυ|=Ω|υ|dx,υW1,1(Ω).

We that assume Ω has a piecewise C1 boundary, and it is decomposed into a union of a finite number of subsets with disjoint interiors Ω̅=m=1MΩm¯ such that each subset Ωm has a piecewise C1 boundary. The unit outward normal vector on Ωmis denoted by υm. Denote Γij=Ωi¯Ωj¯, which may be empty for some pairs of i and j between 1 and M. We write meas(Γij) for the (d-1)-dimensional measure of Γij; it is the area of Γij for d=3, and the length of Γijfor d=2. The symbol i<j will refer to a summation for those i and j with a nonempty Γij in the range 1i<jM. The main result of this note is the following.

Theorem 1.

Let f be a piecewise constant function corresponding to the decomposition (6): f(x)=cm for xΩm, 1mM. Then we have Ω|D(f+g)|=Ω|Df|+Ω|g|dx,gW1,1(Ω),Ω|Df|=i<j|ci-cj|meas(Γij). Consequently, we have the minimization property Ω|Df|Ω|D(f+g)|,gW1,1(Ω).

Proof.

After an integration by parts and some rearrangement, we have, for any ϕC01(Ω)d, Ω(f+g)divϕdx=m=1McmΩmϕ(x)·υm(x)ds-Ωg(x)·ϕ(x)dx=i<j(ci-cj)Γijϕ(x)·υi(x)ds-Ωg(x)·ϕ(x)dx. By the definition (3), Ω|D(f+g)|=sup{i<j(ci-cj)Γijϕ(x)·υi(x)ds-Ωg(x)·ϕ(x)dx:ϕC01(Ω)d,|ϕ|1inΩ{i<j(ci-cj)Γijϕ(x)}. Taking g(x)=0 in (11), the formula (8) follows (cf. the argument in the next paragraph for a more general situation). Moreover, from (11) again, Ω|D(f+g)|Ω|Df|+Ω|g|dx.

For the opposite inequality, we first consider the case where gC1(Ω̅). For any ε>0, define two open subsets Ωε={xΩ:dist(x,Ω)<ε},ΩεΓ={xΩΩε:mini<jdist(x,Γij)<ε}. Here, dist(x,D)=min{|x-y|:yD} is the distance between x and a closed set D. Obviously, for some constant c>0, meas(Ωε)+meas(ΩεΓ)<cε. We start with a function ψε(x)C(Ω̅)d satisfying |ψε(x)|1forxΩ,|ψε(x)|=0forxΩε/4,ψε(x)=sgn(ci-cj)υi(x)|υi(x)|forxΓij(ΩΩε/2),ψε(x)·g(x)=-|g(x)|forxΩ(Ωε/2Ωε/2Γ) and then apply the well-known mollification technique in the theory of Sobolev space  to define ϕε,δ(x)=Bδηδ(x-y)ψε(y)dy, where Bδ is the ball of radius δ centered at the origin, ηδ(x)=η(x/δ)/δd, and η(x)={c0e1/(|x|2-1)if|x|<1,0if|x|1,c0-1=|x|<1e1/(|x|2-1)dx. Then |ϕε,δ(x)|1 for xΩ, and for δ sufficiently small, ϕε,δ(x)C0(Ω)d. Moreover, as δ0, ϕε,δ(x) converges uniformly to ψε(x) for xΩ(Ωε/2Ωε/2Γ). Thus from (11), Ω|D(f+g)|i<j(ci-cj)Γijϕε,δ(x)·υi(x)ds-Ωg(x)·ϕε,δ(x)dx and as δ0, we obtain, with some constant c1>0Ω|D(f+g)|i<j(ci-cj)Γijψε(x)·υi(x)ds-Ωg(x)·ψε(x)dx-c1ε. Using the defining properties of ψε, we further have Ω|D(f+g)|Ω|Df|+Ω|g|dx-c2ε for some other constant c2>0. Since ε>0 is arbitrary, we obtain from the above relation that Ω|D(f+g)|Ω|Df|+Ω|g|dx. Combining (12) and (21), we conclude (7) for gC1(Ω̅).

For gW1,1(Ω), we use the density ofC1(Ω̅) in W1,1(Ω)  and choose {gn}C1(Ω̅) such that gnginW1,1(Ω),asn. Since (3) defines a seminorm on BV(Ω), we have |Ω|D(f+g)|-Ω|D(f+gn)||Ω|(g-gn)|dx. Thus, taking this limit n in (7) for gnC1(Ω̅), we obtain (7) for gW1,1(Ω).

As an example of (8), let Ω=Br02 be a disk of radius r0 centered at the origin. Consider a piecewise constant, radial function f(r) defined on Br0 such that it has a jump jm at rm, 1mM, where 0<r1<<rM<r0. Then by (8), we have Br0|Df|=2πm=1M|jm|rm (cf. [3, Theorem 2.2]).

3. Discussions and Conclusion

Some comments on the name of our approach “CS-based interior tomography” are in order. In the strict sense, compressed sensing refers to situations where the sampling scheme is built (often with random techniques) to achieve specific properties for satisfactory recovery of an underlying signal, rather than imposed by a specific detector arrangement as in limited data tomography. However, in a broad sense, compressed sensing can be interpreted as achieving better reconstruction from less data relative to the common practice. Hence, while the current name is not far off, an alternative phrase for our approach can be “total variation minimization-based interior tomography.”

In conclusion, we have extended the total variation minimization property of a piecewise function from two-dimensions to any dimensionality in the Sobolev space, which can be used for exact reconstruction of any piecewise function on an ROI by minimizing its total variation under the constraint of the truncated projection data through the ROI. Previously, we implemented an alternating iterative reconstruction algorithm to minimize the total variation, which is time-consuming and needs improvement. Under the guidance of the theoretical finding presented here, we are working to develop a multidimensional ROI reconstruction algorithm for better performance. Clearly, major efforts are still needed in this direction.

Acknowledgments

This work is partially supported by NIH/NIBIB Grants (EB002667, EB004287, EB007288) and a Grant from Toshiba Medical Research Institute USA, Inc.

HamakerC.SmithK. T.SolomonD. C.WagnerS. L.The divergent beam X-ray transformRocky Mountain Journal of Mathematics198010125328310.1216/RMJ-1980-10-1-253NattererF.The Mathematics of Computerized Tomography2001Philadelphia, Pa, USASociety for Industrial and Applied MathematicsYuH.WangG.Compressed sensing based interior tomographyPhysics in Medicine and Biology2009549279128052-s2.0-6764989257310.1088/0031-9155/54/9/014YuH.YangJ.JiangM.WangJ.Supplemetnal analysis on compressed sensing based interior tomographyPhysics in Medicine and Biology20095418N425N43210.1088/0031-9155/54/18/N04EvansL. C.GariepyR. F.Measure Theory and Fine Properties of Functions1992Boca Raton, Fla, USACRC PressEvansL. C.Partial Differential Equations1998Providence, RI, USAAmerican Mathematical Society