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PET Reconstruction With an Anatomical MRI Prior Using Parallel Level Sets.
The combination of positron emission tomography (PET) and magnetic resonance imaging (MRI) offers unique possibilities. In this paper we aim to exploit the high spatial resolution of MRI to enhance the reconstruction of simultaneously acquired PET data. We propose a new prior to incorporate structural side information into a maximum a posteriori reconstruction. The new prior combines the strengths of previously proposed priors for the same problem: it is very efficient in guiding the reconstruction at edges available from the side information and it reduces locally to edge-preserving total variation in the degenerate case when no structural information is available. In addition, this prior is segmentation-free, convex and no a priori assumptions are made on the correlation of edge directions of the PET and MRI images. We present results for a simulated brain phantom and for real data acquired by the Siemens Biograph mMR for a hardware phantom and a clinical scan. The results from simulations show that the new prior has a better trade-off between enhancing common anatomical boundaries and preserving unique features than several other priors. Moreover, it has a better mean absolute bias-to-mean standard deviation trade-off and yields reconstructions with superior relative l2-error and structural similarity index. These findings are underpinned by the real data results from a hardware phantom and a clinical patient confirming that the new prior is capable of promoting well-defined anatomical boundaries.This research was funded by the EPSRC (EP/K005278/1) and EP/H046410/1 and supported by the National Institute for Health Research University College London Hospitals Biomedical Research Centre. M.J.E was supported by an IMPACT studentship funded jointly by Siemens and the UCL Faculty of Engineering Sciences. K.T. and D.A. are partially supported by the EPSRC grant EP/M022587/1.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/TMI.2016.254960
ํด๋ถํ์ ์ ๋ PET ์ฌ๊ตฌ์ฑ: ๋งค๋๋ฝ์ง ์์ ์ฌ์ ํจ์๋ถํฐ ๋ฅ๋ฌ๋ ์ ๊ทผ๊น์ง
ํ์๋
ผ๋ฌธ (๋ฐ์ฌ) -- ์์ธ๋ํ๊ต ๋ํ์ : ์๊ณผ๋ํ ์๊ณผํ๊ณผ, 2021. 2. ์ด์ฌ์ฑ.Advances in simultaneous positron emission tomography/magnetic resonance imaging (PET/MRI) technology have led to an active investigation of the anatomy-guided regularized PET image reconstruction algorithm based on MR images. Among the various priors proposed for anatomy-guided regularized PET image reconstruction, Bowsherโs method based on second-order smoothing priors sometimes suffers from over-smoothing of detailed structures. Therefore, in this study, we propose a Bowsher prior based on the l1 norm and an iteratively reweighting scheme to overcome the limitation of the original Bowsher method. In addition, we have derived a closed solution for iterative image reconstruction based on this non-smooth prior. A comparison study between the original l2 and proposed l1 Bowsher priors were conducted using computer simulation and real human data. In the simulation and real data application, small lesions with abnormal PET uptake were better detected by the proposed l1 Bowsher prior methods than the original Bowsher prior. The original l2 Bowsher leads to a decreased PET intensity in small lesions when there is no clear separation between the lesions and surrounding tissue in the anatomical prior. However, the proposed l1 Bowsher prior methods showed better contrast between the tumors and surrounding tissues owing to the intrinsic edge-preserving property of the prior which is attributed to the sparseness induced by l1 norm, especially in the iterative reweighting scheme. Besides, the proposed methods demonstrated lower bias and less hyper-parameter dependency on PET intensity estimation in the regions with matched anatomical boundaries in PET and MRI.
Moreover, based on the formulation of l1 Bowsher prior, the unrolled network containing the conventional maximum-likelihood expectation-maximization (ML-EM) module was also proposed. The convolutional layers successfully learned the distribution of anatomically-guided PET images and the EM module corrected the intermediate outputs by comparing them with sinograms. The proposed unrolled network showed better performance than ordinary U-Net, where the regional uptake is less biased and deviated. Therefore, these methods will help improve the PET image quality based on the anatomical side information.์์ ์๋ฐฉ์ถ๋จ์ธต์ดฌ์ / ์๊ธฐ๊ณต๋ช
์์ (PET/MRI) ๋์ ํ๋ ๊ธฐ์ ์ ๋ฐ์ ์ผ๋ก MR ์์์ ๊ธฐ๋ฐ์ผ๋ก ํ ํด๋ถํ์ ์ฌ์ ํจ์๋ก ์ ๊ทํ ๋ PET ์์ ์ฌ๊ตฌ์ฑ ์๊ณ ๋ฆฌ์ฆ์ ๋ํ ์ฌ๋์๋ ํ๊ฐ๊ฐ ์ด๋ฃจ์ด์ก๋ค. ํด๋ถํ ๊ธฐ๋ฐ์ผ๋ก ์ ๊ทํ ๋ PET ์ด๋ฏธ์ง ์ฌ๊ตฌ์ฑ์ ์ํด ์ ์ ๋ ๋ค์ํ ์ฌ์ ์ค 2์ฐจ ํํํ ์ฌ์ ํจ์์ ๊ธฐ๋ฐํ Bowsher์ ๋ฐฉ๋ฒ์ ๋๋๋ก ์ธ๋ถ ๊ตฌ์กฐ์ ๊ณผ๋ํ ํํํ๋ก ์ด๋ ค์์ ๊ฒช๋๋ค. ๋ฐ๋ผ์ ๋ณธ ์ฐ๊ตฌ์์๋ ์๋ Bowsher ๋ฐฉ๋ฒ์ ํ๊ณ๋ฅผ ๊ทน๋ณตํ๊ธฐ ์ํด l1 norm์ ๊ธฐ๋ฐํ Bowsher ์ฌ์ ํจ์์ ๋ฐ๋ณต์ ์ธ ์ฌ๊ฐ์ค์น ๊ธฐ๋ฒ์ ์ ์ํ๋ค. ๋ํ, ์ฐ๋ฆฌ๋ ์ด ๋งค๋๋ฝ์ง ์์ ์ฌ์ ํจ์๋ฅผ ์ด์ฉํ ๋ฐ๋ณต์ ์ด๋ฏธ์ง ์ฌ๊ตฌ์ฑ์ ๋ํด ๋ซํ ํด๋ฅผ ๋์ถํ๋ค. ์๋ l2์ ์ ์ ๋ l1 Bowsher ์ฌ์ ํจ์ ๊ฐ์ ๋น๊ต ์ฐ๊ตฌ๋ ์ปดํจํฐ ์๋ฎฌ๋ ์ด์
๊ณผ ์ค์ ๋ฐ์ดํฐ๋ฅผ ์ฌ์ฉํ์ฌ ์ํ๋์๋ค. ์๋ฎฌ๋ ์ด์
๋ฐ ์ค์ ๋ฐ์ดํฐ์์ ๋น์ ์์ ์ธ PET ํก์๋ฅผ ๊ฐ์ง ์์ ๋ณ๋ณ์ ์๋ Bowsher ์ด์ ๋ณด๋ค ์ ์ ๋ l1 Bowsher ์ฌ์ ๋ฐฉ๋ฒ์ผ๋ก ๋ ์ ๊ฐ์ง๋์๋ค. ์๋์ l2 Bowsher๋ ํด๋ถํ์ ์์์์ ๋ณ๋ณ๊ณผ ์ฃผ๋ณ ์กฐ์ง ์ฌ์ด์ ๋ช
ํํ ๋ถ๋ฆฌ๊ฐ ์์ ๋ ์์ ๋ณ๋ณ์์์ PET ๊ฐ๋๋ฅผ ๊ฐ์์ํจ๋ค. ๊ทธ๋ฌ๋ ์ ์ ๋ l1 Bowsher ์ฌ์ ๋ฐฉ๋ฒ์ ํนํ ๋ฐ๋ณต์ ์ฌ๊ฐ์ค์น ๊ธฐ๋ฒ์์ l1 ๋
ธ๋ฆ์ ์ํด ์ ๋๋ ํฌ์์ฑ์ ๊ธฐ์ธํ ํน์ฑ์ผ๋ก ์ธํด ์ข
์๊ณผ ์ฃผ๋ณ ์กฐ์ง ์ฌ์ด์ ๋ ๋์ ๋๋น๋ฅผ ๋ณด์ฌ์ฃผ์๋ค. ๋ํ ์ ์๋ ๋ฐฉ๋ฒ์ PET๊ณผ MRI์ ํด๋ถํ์ ๊ฒฝ๊ณ๊ฐ ์ผ์นํ๋ ์์ญ์์ PET ๊ฐ๋ ์ถ์ ์ ๋ํ ํธํฅ์ด ๋ ๋ฎ๊ณ ํ์ดํผ ํ๋ผ๋ฏธํฐ ์ข
์์ฑ์ด ์ ์์ ๋ณด์ฌ์ฃผ์๋ค.
๋ํ, l1Bowsher ์ฌ์ ํจ์์ ๋ซํ ํด๋ฅผ ๊ธฐ๋ฐ์ผ๋ก ๊ธฐ์กด์ ML-EM (maximum-likelihood expectation-maximization) ๋ชจ๋์ ํฌํจํ๋ ํผ์ณ์ง ๋คํธ์ํฌ๋ ์ ์๋์๋ค. ์ปจ๋ณผ๋ฃจ์
๋ ์ด์ด๋ ํด๋ถํ์ ์ผ๋ก ์ ๋ ์ฌ๊ตฌ์ฑ๋ PET ์ด๋ฏธ์ง์ ๋ถํฌ๋ฅผ ์ฑ๊ณต์ ์ผ๋ก ํ์ตํ์ผ๋ฉฐ, EM ๋ชจ๋์ ์ค๊ฐ ์ถ๋ ฅ๋ค์ ์ฌ์ด๋
ธ๊ทธ๋จ๊ณผ ๋น๊ตํ์ฌ ๊ฒฐ๊ณผ ์ด๋ฏธ์ง๊ฐ ์ ๋ค์ด๋ง๊ฒ ์์ ํ๋ค. ์ ์๋ ํผ์ณ์ง ๋คํธ์ํฌ๋ ์ง์ญ์ ํก์์ ๋์ด ๋ ํธํฅ๋๊ณ ํธ์ฐจ๊ฐ ์ ์ด, ์ผ๋ฐ U-Net๋ณด๋ค ๋ ๋์ ์ฑ๋ฅ์ ๋ณด์ฌ์ฃผ์๋ค. ๋ฐ๋ผ์ ์ด๋ฌํ ๋ฐฉ๋ฒ๋ค์ ํด๋ถํ์ ์ ๋ณด๋ฅผ ๊ธฐ๋ฐ์ผ๋ก PET ์ด๋ฏธ์ง ํ์ง์ ํฅ์์ํค๋ ๋ฐ ์ ์ฉํ ๊ฒ์ด๋ค.Chapter 1. Introduction 1
1.1. Backgrounds 1
1.1.1. Positron Emission Tomography 1
1.1.2. Maximum a Posterior Reconstruction 1
1.1.3. Anatomical Prior 2
1.1.4. Proposed l_1 Bowsher Prior 3
1.1.5. Deep Learning for MR-less Application 4
1.2. Purpose of the Research 4
Chapter 2. Anatomically-guided PET Reconstruction Using Bowsher Prior 6
2.1. Backgrounds 6
2.1.1. PET Data Model 6
2.1.2. Original Bowsher Prior 7
2.2. Methods and Materials 8
2.2.1. Proposed l_1 Bowsher Prior 8
2.2.2. Iterative Reweighting 13
2.2.3. Computer Simulations 15
2.2.4. Human Data 16
2.2.5. Image Analysis 17
2.3. Results 19
2.3.1. Simulation with Brain Phantom 19
2.3.2.Human Data 20
2.4. Discussions 25
Chapter 3. Deep Learning Approach for Anatomically-guided PET Reconstruction 31
3.1. Backgrounds 31
3.2. Methods and Materials 33
3.2.1. Douglas-Rachford Splitting 33
3.2.2. Network Architecture 34
3.2.3. Dataset and Training Details 35
3.2.4. Image Analysis 36
3.3. Results 37
3.4. Discussions 38
Chapter 4. Conclusions 40
Bibliography 41
Abstract in Korean (๊ตญ๋ฌธ ์ด๋ก) 52Docto
A function space framework for structural total variation regularization with applications in inverse problems
In this work, we introduce a function space setting for a wide class of
structural/weighted total variation (TV) regularization methods motivated by
their applications in inverse problems. In particular, we consider a
regularizer that is the appropriate lower semi-continuous envelope (relaxation)
of a suitable total variation type functional initially defined for
sufficiently smooth functions. We study examples where this relaxation can be
expressed explicitly, and we also provide refinements for weighted total
variation for a wide range of weights. Since an integral characterization of
the relaxation in function space is, in general, not always available, we show
that, for a rather general linear inverse problems setting, instead of the
classical Tikhonov regularization problem, one can equivalently solve a
saddle-point problem where no a priori knowledge of an explicit formulation of
the structural TV functional is needed. In particular, motivated by concrete
applications, we deduce corresponding results for linear inverse problems with
norm and Poisson log-likelihood data discrepancy terms. Finally, we provide
proof-of-concept numerical examples where we solve the saddle-point problem for
weighted TV denoising as well as for MR guided PET image reconstruction
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