Methods in Image Analysis – Lecture 2 Local Operators and Global Transforms

1. Methods in Image Analysis – Lecture 2Local Operators and Global Transforms George Stetten, M.D., Ph.D. CMU Robotics Institute 16-725 U. Pitt Bioengineering 2630 Spring Term, 2006

2. Preface • Some things work in n dimensions, some don’t. • It is often easier to present a concept in 2D. • I will use the word “pixel” for n dimensions.

3. Point Operators • f is usually monotonic, and shift invariant. • Inverse may not exist. • Brightness/contrast, “windowing”. • Thresholding. • Color Maps. • f may vary with pixel location, eg., correcting for inhomogeneity of RF field strength in MRI.

4. Histogram Equalization • A pixel-wise intensity mapping is found that produces a uniform density of pixel intensity across the dynamic range.

5. Adaptive Thresholding from Histogram • Assumes bimodal distribution. • Trough represents boundary points between homogenous areas.

6. Algebraic Operators • Assumes registration. • Averaging multiple acquisitions for noise reduction. • Subtracting sequential images for motion detection, or other changes (eg. Digital Subtractive Angiography). • Masking.

10. Re-Sampling on a New Lattice • Can result in denser or sparser pixels. • Two general approaches: • Forward Mapping (Splatting) • Backward Mapping (Interpolation) • Nearest Neighbor • Bilinear • Cubic • 2D and 3D texture mapping hardware acceleration.

11. Neighborhood Operators • Kernels • Cliques • Markov Random Fields • Must limit the relationships to be practical.

12. Convolution and Correlation • Template matching uses correlation, the primordial form of image analysis. • Kernels are mostly used for “convolution” although with symmetrical kernels equivalent to correlation. • Convolution flips the kernel and does not normalize. • Correlation subtracts the mean and generally does normalize.

13. Neighborhood PDE Operators • For discrete images, always requires a specific scale. • “Inner scale” is the original pixel grid. • Size of the kernel determines scale. • Concept of Scale Space, Course-to-Fine.

14. Intensity Gradient • Vector • Direction of maximum change of scalar intensity I. • Normal to the boundary. • Nicely n-dimensional.

15. Intensity Gradient Magnitude • Scalar • Maximum at the boundary • Orientation-invariant.

17. Classic Edge Detection Kernel (Sobel)

18. Isosurface, Marching Cubes (Lorensen) • 100% opaque watertight surface • Fast, 28 = 256 combinations, pre-computed

19. Marching cubes works well with raw CT data. • Hounsfield units (attenuation). • Threshold calcium density.

20. Other Useful 3D Rendering Methods Direct shading from gradient (Levoy, Drebin) • Voxels are blended (translucent). • Opacity proportional to gradient magnitude. • Rendering uses gradient direction as “surface” normal. Maximum Intensity Projection (MIP) • Maximum Intensity Pixel along each projected line of sight. • Good for vasculature

21. Drebin, R. A., Carpenter, L., AND Hanrahan, P. Volume rendering. Computer Graphics 22, 4 (Aug. 1988), 65--74. For a nice review of graphics rendering techniques see http://accad.osu.edu/~waynec/history/lesson18.html

22. Jacobian of the Intensity Gradient • Ixy = Iyx= curvature • Orientation-invariant. • What about in 3D?

23. Viewing the intensity as “height” • Differential geometry: the surface and tangent plane • Example: cylindrical surface, curvature = 0. • Move in the y direction, no roll: Ixy= 0 • Move in the x direction, no pitch: Iyx = 0 = Iy = Ix

24. Laplacian of the Intensity • Divergence of the Gradient. • Zero at the inflection point of the intensity curve. I Ix Ixx

25. Binomial Kernel • Repeated averaging of neighbors => Gaussian by Central Limit Theorem.

26. From Insight into Images, Yoo

27. From Insight into Images, Yoo Blurring the derivative of an image = Convolving with the derivative of a Gaussian

28. Binomial Difference of Offset Gaussian (DooG) • Not the conventional concentric DOG • Subtracting pixels displaced along the x axis after repeated blurring with binomial kernel yields Ix

29. Difference of Gaussian Operators (DOG)(not offset) • Conventionally, 2 concentric Gaussians of different scale. • Acts like a Laplacian, zero at boundary. From Insight into Images, Yoo

30. From Insight into Images, Yoo DOG used for Edge Enhancement

31. Edge Enhancement (area under kernel = 0) Contrast Enhancement (area under kernel = 1)

32. Boundary Profiles (Tamburo) • Splatting in an ellipsoid along the gradient.

33. Boundary Profiles • Fitting the cumulative Gaussian

34. Boundary profiles reduce error in location due to sampling.

35. Variable Conductance Diffusion (VCD) • Attempt to get around the global nature of Fourier. • Smoothing with a Gaussian in the spatial domain lowers noise, but also blurs boundaries. • Gaussian smoothing simulates uniform heat diffusion. • VCD makes conductance an inverse function of gradient, so that “heat” does not flow well across boundaries. This homogenizes already homogenious regions while preserving boundaries.

36. Other basis sets for boundary kernel. • Wavelets • Statistical texture analysis • Conjacent spheres

37. Texture Boundaries • Two regions with the same intensity but differentiated by texture are easily discriminated by the human visual system.

38. Ridges • Attempt to tie the image structure together in a locally continual manner. • Along some manifold in less than n dimensions. • Local maximum along the normal to the ridge. • Canny edge detector is a boundariness ridge. • A “core” is a medialness ridge. • Medialness within an object is the property of being equidistant from two boundaries.

39. Global Transforms in n dimensions • Geometric (rigid body) • n translations and rotations. • Similarity • Add 1 scale (isometric). • Affine • Add n scales (combined with rotation => skew). • Parallel lines remain parallel. • Projection

40. Orthographic Transform Matrix • Homogeneous coordinates (divide vector by scalar to keep last component = 1. • Capable of geometric, similarity, or affine. • Homogeneous coordinates. • Multiply in reverse order to combine • SGI “graphics engine” 1982, now standard.

41. Translation by (tx , ty) Scale x by sx and y by sy

42. Rotation in 2D • 2 x 2 rotation portion is orthogonal (orthonormal vectors). • Therefore only 1 degree of freedom, .

43. Rotation in 3D • 3 x 3 rotation portion is orthogonal (orthonormal vectors). • 3 degree of freedom (dotted circled), , as expected.

44. Non-Orthographic Projection in 3D • When vector is made homogeneous (dividing to make last component 1), x and y are made smaller in the distance. • For X-ray or direct vision, projects onto the (x,y) plane. • Rescales x and y for “perspective” by changing the “1” in the homogeneous coordinates, as a function of z.

45. Anisotropic Scaling of Vectors(ITK Software Guide 4.25, 4.26) itkVector dy dx1 dx2 itkVector objects are used to represent distances between locations, velocities, etc. itkCovarientVector dy dx1 dx2 itkCovarientVector objects are used to represent properties such as the gradient, since stretching the image in the x dimension lowers the x gradient component (slope). dI dI dx1 dx2