Spiral CT Image Reconstruction Using Alternating Minimization Methods

Transcription

1 Spiral CT Image Reonsruion Using Alernaing Minimizaion Mehods Shenu Yan Thesis Advisor: Dr. O Sullivan Washingon Universi S. Louis Missouri leroni Ssems & Signals Researh Laboraor Ma 9 24

2 Conen CT inroduion Alernaing Minimizaion mehod Pose searh of high-densi obje Inomplee projeion daa Conlusion and fuure work Ma 9 24

3 CT Developmen Ma 9 24

4 Aenuaion funion µ µ Z µ µ Z Z ρ ρ Z is he aomi number of he maerial or issue Ma 9 24

5 Deeors Daa olleion Y 768 Soure Posiion 48 X Ma 9 24

6 Ma 9 24 Aenuaion Funion and Beer s Law Z µ µ θ p sin os d d p + + θ θ δ µ θ θ p e I I Radon Transform Beer s Law D spae 3D spae θ

7 Sandard reonsruion mehod Cenral slie heorem D FT of projeion daa P ρ θ 2D FT of aenuaion funion j 2πρ osθ + 2 sinθ p θ e d j 2π u + 2u2 Μ u u2 µ 2 e dd2 P ρ θ Μ u u Filered bak projeion 2 µ π 2 P ρ θ ρ e j2πρ dρ dθ Ma 9 24

8 Muli-slie Spiral CT deeor β γ fous Pih: p d S d: he able feed per roaion S: he oal widh of he ollimaed beam q 3 2 η q Advanages VS onvenional CT z Rapid sanning Ma 9 24 Redue Moion Arifas

9 Image and Measuremen Spaes 2-D 3-D Spiral Image Spae X: X X X 2 X X X 2 X 3 Daa Spae Y: Y S D Soure S posiion is speified b fan angle β. Deeor D speified onl b γ. Y S Soure S is speified b β and Pih D Deeor D speified b γ and deeor row inde q. Ma 9 24

10 Poin Spread Funion D D 3 D 2 D 4 z fous F poin spread funion h : average pah of he onebeam hrough he voel D 4 Ma 9 24 indees he image voel 2 3 indees he soure-deeor pair β γ q D D 3 D 2

11 Convenional reonsruion mehod for spiral CT daa Addiional sep: Z-inerpolaion Reonsru he images in 2D plane Cause he sair-sep arifas Nonlinear effes Ma 9 24

12 Conen CT inroduion Alernaing Minimizaion mehod Pose searh of high-densi obje Inomplee projeion daa Conlusion and fuure Work Ma 9 24

13 Referenes J. A. Fessler. Saisial image reonsruion mehods for ransmission omograph. In Handbook of Medial Imaging Volume 2. Medial Image Proessing and Analsis h. 5 SPI 2. J. A. O Sullivan and J. Bena. Alernaing minimizaion algorihms for ransmission omograph. Submied o I Trans. Med. Imaging. Ma 9 24

14 Model for Transmission CT Soure I I Obje µ Deeor g : nergies ranging from 9-2 kev I : Mean number of soure phoons µ: Aenuaion funion; he image we are ring o esimae Ma 9 24 g I ep h µ +σ

15 Ma 9 24 Opimizaion Problem: min Id g Two families: Ε h I q q ep : : µ : d p p d L AM Algorihm g q : : Saisial Model { } [ ] Y d g g d d l! ln : : ln : Y g d g d d g d I : : ln : Rewrie problem min min min q p I g d I d L p q 5

16 Ma 9 24 AM Algorihm X k k h I q ˆ ep : ˆ µ X k h I ˆ ep µ + ' ': ˆ : ˆ : ˆ k k k q d q p σ Y k k p h b : ˆ ~ µ Y k k q h b : ˆ ˆ µ Ieraive updae of he esimae ˆ ~ ln ˆ ˆ b b Z k k k k + where X Z h µ

17 perimens for Spiral CT Image spae: mm Projeion daa spae per roaion: 58 soure 68 deeor Deeor row number: 8 Collimaionslie widh:.75 mm Pih: 2 Travel of able per san roaion: 2 mm Roaion number: 2 PMMA Clinder wih Low-densi Objes - PMMA:.229/mm 2- Nlon:.295/mm 3- Teflon:.423/mm Viewing window size [.6.2] Telflon PMMA Ma 9 24 Nlon

18 Reonsrued Resuls from Low-densi noiseless Snhei daa afer AM 2 8 OS Ieraions 5h 6h 9h 2h 7h 8h 2h 22h Ma 9 24

19 Reonsrued Resuls from Low-densi Real daa afer AM 5 Ieraions 5h 6h 7h Ma h 9h 2h Too oarse sampling inerval and high pih value Voel sphere no real sphere in snhei daa

20 Reonsrued Resuls from High-densi noiseless Snhei daa afer AM 8/5 8 OS Ieraions 7h 8h 7h 8h 9h 2h 9h 2h Ma 9 24

21 Conen CT inroduion Alernaing Minimizaion mehod Pose searh of high-densi obje Inomplee projeion daa Conlusion and fuure work Ma 9 24

22 Referenes D. L. Snder J. A. O Sullivan R. J. Murph e al. Deblurring subje o nonnegaivi onsrains when known funions are presen wih appliaion o objeonsrained ompuerized omograph. I Trans. Med. Imaging 2: 9-7 O. 2. Ma 9 24

23 Inorporaing Prior Knowledge Prior Knowledge Assumpion: Aenuaion oeffiiens/geomer are known a pose posiion and orienaion is known One rigid obje individual pars are fied Appliaion amples Hip prosheses Brahherap appliaors Denal fillings Prosae seeds e. Ma 9 24

24 Orale Tes for 3D Snhei Daa wih High-densi Objes Orale mehod: A eah ieraion use AM algorihm o solve for he image hen subsiue in he known voel values. AM 8 Ieraion 8 OS AM Orale 8 8 OS Ma 9 24

25 AM wih Pose Searh Define rue piel value: a known :θ+ -a unknown ; a :θ + b Rederive algorihm wih onsrain: * a Soluion: AM ˆ k Z ln ~ b bˆ k k For some θ : ˆ k + : θ ma [ : θ ] AM a Searh he opimal θ whih resuls in he bes image Ma 9 24 θ arg min θ S2 I [ ˆ : ] k + d g θ

26 Mahemaial Desripion of he Pose 2-dimensional Case Roae & Translae θ 2 ϕ 2 R SO 2 ϖ ~ T {} T { R} ϖ ϖ R + where os ϕ : R sin ϕ ϖ T 2 ϖ R θ V 3 3 mari sin ϕ os ϕ Two onseuive operaions: Ma 9 24 { B} T{ a} T{ A} T{ b} T ϖ B A + a + b ϖ BA + Ba + b B b A a BA Ba + b

27 Ma 9 24 Calulae he gradiens of he pose parameers in S2 spae Searh direion Gradien Searh Mehod os sin sin os e R ϕ ϕ ϕ ϕ ϕ where h F h F F f i i h i i lim + ε ε ϕ ε ϕ ε ε ε R e F e F R F F f + lim Re lim 2 R R f f f f f F θ

28 Ma 9 24 Sele he sep size for eah parameer wrien in mari S Gradien Searh Mehod Coninued R R S S S S S S 2 Ge he hange for pose T f S f S e R R θ Updae he new pose new θ old θ θ old old T f S new new R f S e R R R

29 Pose Searh Resuls Ma 9 24

30 AMPS Resuls True Phanom FBP AM 2 22 OS SYN Ma 9 24 AMPS 2 22 OS SYN AMPS 2 22 OS RAL

31 Conen CT inroduion Alernaing Minimizaion mehod Pose searh of high-densi obje Inomplee projeion daa Conlusion and fuure work Ma 9 24

32 Mask for high-densi obje Ma 9 24 AMPS 2 22 OS RAL AM 2 22 OS Inomplee daa RAL

33 Sanned obje ouside he sanner FOV FBP AM 5 22 OS Ma 9 24

34 Sanned obje ouside of FOV Ma 9 24 FBP AM 5 22 OS

35 Conen CT inroduion Alernaing Minimizaion mehod Pose searh of high-densi obje Inomplee projeion daa Conlusion and fuure work Ma 9 24

36 Conlusion and fuure work Avoid he inerpolaion sep when reonsruing he image from spiral CT daa Grea improvemen in onvergene rae when he AM algorihm inlude prior informaion pose searh in hree-dimensional image reonsruion More real daa eperimens for spiral CT Ma 9 24

37 Ma 9 24 Thank You!