10/06/20041 Resolution Enhancement in MRI By: Eyal Carmi Joint work with: Siyuan Liu, Noga Alon, Amos Fiat & Daniel Fiat

2 Lecture Outline Introduction to MRI The SRR problem (Camera & MRI) Our Resolution Enhancement Algorithm Results Open Problems

3 Introduction to MRI Magnetic resonance imaging (MRI) is an imaging technique used primarily in medical settings to produce high quality images of the inside of the human body.

4 Introduction to MRI The nucleus of an atom spins, or precesses, on an axis. Hydrogen atoms – has a single proton and a large magnetic moment.

5 Magnetic Resonance Imaging Uniform Static Magnetic Field – Atoms will line up with the direction of the magnetic field.

6 Magnetic Resonance Imaging Resonance – A state of phase coherence among the spins. Applying RF pulse at Larmor frequency When the RF is turned off the excess energy is released and picked up.

7 Magnetic Resonance Imaging Gradient Magnetic Fields – Time varying magnetic fields (Used for signal localization) x y z

8 Magnetic Resonance Imaging Gradient Magnetic Fields: 1-D X Y B0B0 B1B1 B2B2 B3B3 B4B4

9 Signal Localization slice selection Gradient Magnetic Fields for slice selection z ω z1z1 z2z2 z3z3 G z,1 G z,2 z4z4 ω1ω1 ω2ω2 FT B 1 (t)

10 Signal Localization frequency encoding Gradient Magnetic Fields for in-plane encoding t x B0B0 B=B 0 +G x (t)x t G x (t)

11 Signal Localization phase encoding Gradient Magnetic Fields for in-plane encoding t x B0B0 B=B 0 +G x (t)x t G x (t)

12 k-space interpretation y x WxWx WyWy 1-D path k-space Δk x Sampling Points DFT

13 Magnetic Resonance Imaging Collected Data (k-space) 2-D DFT

14 The Super Resolution Problem Definition: SRR (Super Resolution Reconstruction): The process of combining several low resolution images to create a high-resolution image.

15 SRR – Imagery Model The imagery process model: Y k – K-th low resolution input image. G k – Geometric trans. operator for the k-th image. B k – Blur operator of the k-th image. D k – Decimation operator for the k-th image. E k – White Additive Noise.

16 SRR – Main Approaches Frequency Domain techniques Tsai & Huang [1984] Kim [1990] Frequency Domain

17 SRR – Main Approaches Iterative Algorithms Irani & Peleg [1993] : Iterative Back Projection Current HR Best Guess Back Projection Back Projected LR images Original LR images Iterative Refinement

18 SRR – Main Approaches Patti, Sezan & Teklap [1994] POCS : Elad & Feuer [1996 & 1997] ML : MAP : POCS & ML

19 SRR – In MRI Peled & Yeshurun [2000] 2-D SRR, IBP, single FOV, problems with sub-pixel shifts. Greenspan, Oz, Kiryati and Peled [2002] 3-D SRR (slice-select direction), IBP.

20 Resolution Enhancement Alg. A Model for the problem Reconstruction using boundary values 1-D Algorithm 2-D Algorithm

21 Modeling The Problem Subject Area: 1x1 rectilinearly aligned square grid. (0,0)

22 Modeling The Problem True Image : A matrix of real values associated with a rectilinearly aligned grid of arbitrary high resolution (0,0)

23 Modeling The Problem A Scan of the image: Pixel resolution – Offset – True Image Image Scan, m=2

24 Modeling The Problem Definitions: Maximal resolution – Pixel resolution = Maximal Offset resolution – We can perform scans at offsets where with pixel resolution

25 Modeling The Problem Goal: Compute an image of the subject area with pixel resolution while the maximal measured pixel resolution is Errors: 1. Errors ~ Pixel Size & Coefficients 2. Immune to Local Errors => localized errors should have localized effect.

26 Multiple offsets of a single resolution scan? 2x2

27 Multiple offsets of a single resolution scan? 3x3

28 Multiple offsets of a single resolution scan? 4x2

29 Using boundary value conditions Assumption:or Reconstruct using multiple scans with the same pixel resolution. Introduce a variable for each HR pixel of physical dimension. Algorithm: Perform c 2 scans at all offsets & Solve linear equations (Gaussian elimination) CCC C C C 0C

30 Using boundary value conditions Example: 4 Scans, PD=2x2 Second sample

31 Using boundary value conditions Example: 4 Scans, PD=2x

32 Using boundary value conditions Example: 4 Scans, PD=2x2

33 Problems using boundary values Example: 4 Scans, PD=2x2 (0,0)

34 Problems using boundary values Example: 4 Scans, PD=2x2 (-1,-1)

35 Problems using boundary values Example: 4 Scans, PD=2x2 Add more information Solve using LS Propagation problem

36 Demands On the algorithm No assumptions on the values of the true image. Over determined set of equations  Use LS to reduce errors: Error propagation will be localized. ???? ? ? ? ? ? ? A x b

37 The One dimensional algorithm Input: Pixel of dimension Notation: gcd(x,y) – greatest common divisor of x & y. (Extended Euclidean Algorithm)

38 The One dimensional algorithm (Extended Euclidean Algorithm)

39 One dimensional reconstruction

40 One dimensional reconstruction

41 The One dimensional algorithm Algorithm: Given w.l.g, let: a>0 & b<0 To compute, compute:

42 The One dimensional algorithm Localized Reconstruction Localized Reconstruction Effective Area: x+y high-resolution pixels

43 One dimensional reconstruction

44 Two and More Dimensions Given pixels of size: Where x,y & z are relatively prime. Reconstruct 1x1 pixels.  Error Propagation is limited to an area of O(xyz) HR pixels. 1-D Algorithm Stack

45 Two and More Dimensions 1-D Algorithm gcd(xy,xz)=xgcd(xy,zy)=y

46 Example Two dimensional reconstruction PD=5x5 PD=3x3 PD=15x1

47 Example Two dimensional reconstruction PD=4x4 PD=3x3 PD=12x1

48 Example Two dimensional reconstruction PD=5x5 PD=4x4 PD=20x1

49 Example Two dimensional reconstruction

50 Larger Dimensions Generalize to Dimension k Using k+1 relatively prime Low-Resolution pixels

51 Results Model Results Experiment Results Problems…

52 Model Design HR Scene Blur Sampling LR Image Noise LR Image LR Image SRR Algorithm HR Image

53 Results

54 Experiment GE clinical 1.5T MRI scanner was used. Phantom: - plastic frames - filled with water Three FOV: 230.4, & 384 mm.

55 Experiment

56 Experiment Results Modeled DataMRI Data

57 Experiment Results Modeled DataMRI Data

58 Problems Homogeneity of the phantom Phantom Orientation

59 Problems Homogeneity of the phantom Phantom Orientation Rectangular Blur Vs. Gaussian-like Blur

60 Problems Homogeneity of the phantom Phantom Orientation Rectangular Blur Vs. Gauss-like Blur Truncated

61 Problems Homogeneity of the phantom Phantom Orientation Rectangular Blur Vs. Gauss-like Blur k-space and Fourier based MRI

62 k-space and Fourier based MRI y x WxWx WyWy 1-D path k-space Δk x Sampling Points DFT Δk x

63 Problems ProblemNoise# Samples One scan is enoughNoInfinite SNR too lowYesInfinite No perfect reconstructionYesFinite Apply manual shifts → Different experiment

64 Open Problems Optimization problems: “What is the smallest number of scans we can do to reconstruct the high resolution image?“

65 Scan Selection Problem

66 Scan Selection Problem S 5x and… 

67 Scan Selection Problem S 3x and…  

68 Open Problems Optimization problems: “What is the smallest number of scans we can do to reconstruct the high resolution image?“ Decision/Optimization problem: Given a set of scans, what can we reconstruct? Design problem: Plan a set of scans for “good” error localization.

69 Questions