• Laser & Optoelectronics Progress
  • Vol. 60, Issue 12, 1211001 (2023)
Jinlan Li1, Zhaoyang Xie1, Guoqi Liu2, and Jian Zou1,*
Author Affiliations
  • 1School of Information and Mathematics, Yangtze University, Jingzhou 434020, Hubei, China
  • 2College of Computer and Information Engineering, Henan Normal University, Xinxiang 453007, Henan, China
  • show less
    DOI: 10.3788/LOP221095 Cite this Article Set citation alerts
    Jinlan Li, Zhaoyang Xie, Guoqi Liu, Jian Zou. Diffusion Optical Tomography Based on Convex-Nonconvex Finite Element Total Variation Regularization[J]. Laser & Optoelectronics Progress, 2023, 60(12): 1211001 Copy Citation Text show less

    Abstract

    A reconstruction model based on convex-nonconvex finite element total variation (CNC-FETV) regularization is proposed to avoid the biased estimation of L1 regularization in diffuse optical tomography. First, the finite element method was used to divide the computational domain into a finite number of triangles, after which a continuous piecewise polynomial function was used to approximate the absorption coefficient value in each triangle. Then, the derived difference matrix was assembled element by element to obtain a representation of the FETV regularization. Subsequently, the CNC-FETV regularization was obtained by the construction method based on convex-nonconvex sparse regularization. Results theoretically proved that the nonconvex regularization term could maintain the overall convexity of the objective function under certain conditions. Finally, the alternating direction multiplier method was used to solve the proposed model. Numerical experiments show that compared with the Tikhonov and FETV regularization models, the proposed CNC-FETV regularization model has superior performances in both numerical criteria and visual effects for diffusion optical tomography reconstructions.
    δΦ=Jδμa+n,
    minδμa12Jδμa-δΦ22+λϕδμa,
    U=i=1NNμaiψi,
    ψj(Vi)=1,  i=j0, ij
    ΩxU+yUdxdy=Dxμa1+Dyμa1,
    minδμa12Jδμa-δΦ22+λDxδμa1+λDyδμa1
    minδμa12Jδμa-δΦ22+λϕβDxδμa+λϕβDyδμa,
    ϕβu=u1-mintβ22u-t22+t1,
    Hβδμa=12Jδμa-δΦ22+λϕβDxδμa+λϕβDyδμa= 12Jδμa-δΦ22+λDxδμa1-mintxβ22Dxδμa-tx22+tx1+λDyδμa1-mintyβ22Dyδμa-ty22+ty1=12Jδμa-δΦ22+λDxδμa1-λmintxβ22Dxδμa-tx22+tx1+λDyδμa1-λmintyβ22Dyδμa-ty22+ty1=maxtx,ty12Jδμa-δΦ22+λDxδμa1-λβ22Dxδμa-tx22-λtx1+λDyδμa1-λβ22Dyδμa-ty22-λty1=maxtx,ty12δμaTJTJ-λβ2DxTDx-λβ2DyTDyδμa+λDxδμa1+λDyδμa1+gδμa,tx,ty=12δμaTJTJ-λβ2DxTDx-λβ2DyTDyδμa+λDxδμa1+λDyδμa1+maxtx,tygδμa,tx,ty
    Lδμan,vxn,vyn,bxn,byn=12Jδμa-δΦ22+λϕβvx+bxn-1TDxδμa-vxn-1+θ2vxn-1-Dxδμa22+λϕβvy+byn-1TDyδμa-vyn-1+θ2vyn-1-Dyδμa22,
    δμan=argminδμa12Jδμa-δΦ22+bxn-1TDx(δμa)+byn-1TDyδμa+θ2vxn-1-Dxδμa22+θ2vyn-1-Dyδμa22=argminδμa12Jδμa-δΦ22+θ2vxn-1-Dxδμa-1θbxn-122+θ2vyn-1-Dyδμa-1θbyn-122=JTJ+θDxTDx+DyTDy-1×JTδΦ-θDxT(1θbxn-1-vxn-1)-θDyT(1θbyn-1-vyn-1)
    vxn=argminvxλϕβvx-bxn-1Tvx+θ2vx-Dxδμan22=argminvxλϕβvx+θ2vx-Dxδμan+1θbxn-122,
    vyn=argminvyλϕβvy-byn-1Tvy+θ2vy-Dyδμan22=argminvyλϕβvy+θ2vy-Dyδμan+1θbyn-122,
    vxn=proxλθϕβDxδμan+1θbxn-1=proxαλθ11-αvx+αDxδμan+1θbxn-1+αλβ2θvx-prox1β21vx,
    vyn=proxλθϕβDyδμan+1θbyn-1=proxαλθ11-αvy+αDyδμan+1θbyn-1+αλβ2θvy-prox1β21vy,
    bxn=bxn-1+θDxδμan-vxn,
    byn=byn-1+θDyδμan-vyn
    LE=Xt-Xr2
    AC=i=1Nnμar,i/Nn/μat
    PSNR=10log10maxμa2MSE,
    SSIM=2μarmμatm+c12σr,t+c2μarm2+μatm2+c1σr2+σt2+c2,
    RRA=SrSt×100%
    Jinlan Li, Zhaoyang Xie, Guoqi Liu, Jian Zou. Diffusion Optical Tomography Based on Convex-Nonconvex Finite Element Total Variation Regularization[J]. Laser & Optoelectronics Progress, 2023, 60(12): 1211001
    Download Citation