Continuous and Discrete Image Reconstruction

Transcription

1 25 th SSIP Summer School on Image Processing 17 July 2017, Novi Sad, Serbia Continuous and Discrete Image Reconstruction Péter Balázs Department of Image Processing and Computer Graphics University of Szeged, HUNGARY

2 Steps of Machine Vision Image acquisition Preprocessing Segmentation Feature extraction Classification, interpretation Actuation 2

3 Image Acquisition by visible light by X-rays 3

4 Wilhelm Conrad Röntgen describes the properties of X-rays X-rays Kind of electromagnetic radiation (similar to light but having more energy) Attenuation of X-rays depends on tissue Shadow of the object from one direction

5 X-rays are Useful in Radiology 5

6 X-rays are Useful in Radiology (in some cases) 6

7 and in Security 7

8 and in Industrial Quality Control 8

9 and in Food Industry and Metals, Stone, Glass, Rubber, Bone, Void 9

10 1000 year old Buddha Statue 10

11 and Its Inhabitant 11

12 Tomography Tomos = part, section Grapho = to write Tomos + Grapho imaging by cross-sections (slices) 12

13 Computerized Tomography A technique for imaging the 2D cross-sections of 3D objects (human organs) without seriously damaging them Take X-ray images from many angles and combine them in a clever way X-ray tube beams detectors 13

14 Slide: Attila Kuba

15 A Modern CT Scanner Scanner CT image 15

16 Image Quality: Then and Now first CT scanners modern CT scanners 16

17 or Even on-line scanning of how a fly tries to fly 17

18 The Mathematics of CT y X-rays Reconstruct f(x,y) from its f (x,y) projections where a projection in direction u (defined by angle ϴ) can be obtained by calculating the line integrals along each line parallel to u. ϴ x g ( l, ) f ( l co s u sin, l sin u co s ) du 18

19 Sinogram Object Sinogram of the Object Sinogram: image of g(l, ϴ) with l and ϴ as rectilinear coordinates Reconstruction: sinogram image 19

20 Backprojection Summation Image (laminogram) One backprojection image Backprojection summation image (blur!) 20

21 Filtered Backprojection with 240 Projections 21

22 Filtered Backprojection The FBP reconstruction process 2D sinogram (projections) high pass filtered for all angles sinogram is backprojected into the image domain Works well only when 180 is equiangularly and densly covered MATLAB functions: radon, iradon Movie: 22

23 ART Algebraic Reconstruction Technique The interaction of the projection rays and the image pixels can be written as a system of equations Direct inverse methods are not applicable: big system underdetermined (#equations << #unknowns) possibly no solution (if there is noise) Solve it iteratively satisfying just one projection in each step 23

24 Digital Images position along the line 24

25 Projection = Line Sums

26 Projection = Line Sums

27 Projection = Line Sums

28 Reconstruction

29 An Example 29

30 An Example a+b=12 c+d=8 30

31 An Example a+c=11 b+d=9 31

32 An Example a+d=5 b+c=15 32

33 Discrete/Binary Tomography FBP and ART need several hundreds of projections time consuming expensive may damage the object not possible In certain applications the range of the function to be reconstructed is discrete and known DT (only few (2-10) projections are needed) Binary Tomography: the range of the function is {0,1} (absence or presence of material) 33

34 Tomography from a Few Projections projections unknown image 34

35 Tomography from a Few Projections projections continuous reconstruction 35

36 Tomography from a Few Projections projections binary tomography 36

37 Binary Reconstruction from 2 Projections?? 37

38 Binary Reconstruction from 2 Projections? 38

39 Nonograms 39

40 Example for Uniqueness 40

41 Example for Inconsistency 41

42 Classification 42

43 Two Main Problems Reconstruction: Construct a binary image from its projections. Uniqueness: Is a binary image uniquely determined by a given set of projections? 43

44 Reconstruction Ryser, 1957 from row sums R and column sums S Order the elements of S in a non-increasing way by π S Fill the rows from left to right B (canonical matrix) Shift elements from the rightmost columns of B to the columns where S(B) < S Reorder the colums by applying the inverse of π 44

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61 Uniqueness and Switching Components The presence of a switching component is necessary and sufficient for non-uniqueness 61

62 62 Ambiguity Due to the presence of switching components there can be many solutions with the same two (or even more) projections Use prior information (convexity, smoothness, etc.) of the binary image to be reconstructed

63 Reconstruction as Optimization x 1 x 2 x 3 x 4 x 5 x 6 63

64 Optimization Problems: P x binary variables big system b x { 0,1 } underdetermined (#equations << #unknowns) possibly no solution (if there is noise) C ( x) x {0,1} Px b 2 m n g ( x) m min n Term for prior information: smoothness, similarity to a model image, etc. 64

65 Solving the Optimzation Task Problem: Classical hill-climbing algorithms can become trapped in local minima. Idea: Allow some changes that increase the objective function. p = 1 0 < p < 1 65

66 Simulated Annealing Annealing: a thermodinamical process in which a metal cools and freezes. Due to the thermical noise the energy of the liquid in some cases grows during the annealing. By carefully controlling the cooling temperature the fluid freezes into a minimum energy crystalline. Simulated annealing: a random-search technique based on the above observation. 66

67 Outline of SA Set inital solution x and temperature T 0 Modify x act x Calculate C(x ) C(x ) < C(x act )? x act = x Y x act = x with probability p=e - C/T N Lower temperature N Termination? Modify x act x Y Stop Greater uphill steps are less probable to be accepted in later phases of the process 67

68 Finding the Optimum Tuning the parameters appropriately SA finds the global optimum Fine-tuning of the parameters for a given optimization problem can be rather delicate Sourse: 68

69 69 SA in Pixel Based Reconstruction A binary matrix describes the binary image Initial suggestion: random binary matrix Randomly invert matrix value(s)

70 Nondestructive Testing: Pipe Corrosion, Deposit, Crack, etc. Study 32 fan beam X-ray projections 70

71 Results with and without Noise no noise 10 % Gaussian noise Source: A. Nagy 71

72 Neutron Tomography Gas pressure controller 18 projections, also multilevel FBP DT Source: A. Nagy 72

73 Further Applications pebble beds metal-, plastic foams cracks air bubbles, metal alloy defects 73

74 Electron Microscopy QUANTITEM: a method which provides quantitative information for the number of atoms lying in a single atomic column from HRTEM images Possible to detect crystal defects (e.g. missing atoms) Source: Batenburg, Palenstijn 74

75 and Many More /microct/gallery/index.aspx 75