792 research outputs found
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
ํ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
Training End-to-End Unrolled Iterative Neural Networks for SPECT Image Reconstruction
Training end-to-end unrolled iterative neural networks for SPECT image
reconstruction requires a memory-efficient forward-backward projector for
efficient backpropagation. This paper describes an open-source, high
performance Julia implementation of a SPECT forward-backward projector that
supports memory-efficient backpropagation with an exact adjoint. Our Julia
projector uses only ~5% of the memory of an existing Matlab-based projector. We
compare unrolling a CNN-regularized expectation-maximization (EM) algorithm
with end-to-end training using our Julia projector with other training methods
such as gradient truncation (ignoring gradients involving the projector) and
sequential training, using XCAT phantoms and virtual patient (VP) phantoms
generated from SIMIND Monte Carlo (MC) simulations. Simulation results with two
different radionuclides (90Y and 177Lu) show that: 1) For 177Lu XCAT phantoms
and 90Y VP phantoms, training unrolled EM algorithm in end-to-end fashion with
our Julia projector yields the best reconstruction quality compared to other
training methods and OSEM, both qualitatively and quantitatively. For VP
phantoms with 177Lu radionuclide, the reconstructed images using end-to-end
training are in higher quality than using sequential training and OSEM, but are
comparable with using gradient truncation. We also find there exists a
trade-off between computational cost and reconstruction accuracy for different
training methods. End-to-end training has the highest accuracy because the
correct gradient is used in backpropagation; sequential training yields worse
reconstruction accuracy, but is significantly faster and uses much less memory.Comment: submitted to IEEE TRPM
DULDA: Dual-domain Unsupervised Learned Descent Algorithm for PET image reconstruction
Deep learning based PET image reconstruction methods have achieved promising
results recently. However, most of these methods follow a supervised learning
paradigm, which rely heavily on the availability of high-quality training
labels. In particular, the long scanning time required and high radiation
exposure associated with PET scans make obtaining this labels impractical. In
this paper, we propose a dual-domain unsupervised PET image reconstruction
method based on learned decent algorithm, which reconstructs high-quality PET
images from sinograms without the need for image labels. Specifically, we
unroll the proximal gradient method with a learnable l2,1 norm for PET image
reconstruction problem. The training is unsupervised, using measurement domain
loss based on deep image prior as well as image domain loss based on rotation
equivariance property. The experimental results domonstrate the superior
performance of proposed method compared with maximum likelihood expectation
maximazation (MLEM), total-variation regularized EM (EM-TV) and deep image
prior based method (DIP)
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