Medical Imaging Simultaneous measurements on a spatial grid.

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Medical Imaging Simultaneous measurements on a spatial grid.

  1. 1. Medical Imaging • Simultaneous measurements on a spatial grid. • Many modalities: mainly EM radiation and sound.
  2. 2. “To invent you need a good imagination and a pile of junk.” Thomas Edison 1879
  3. 3. Bremsstrahlung Electron rapidly decelerates at heavy metal target, giving off X-Rays.
  4. 4. 1896
  5. 5. X-Ray and Fluoroscopic Images Projection of X-Ray silhouette onto a piece of film or detector array, with intervening fluorescent screen.
  6. 6. Computerized Tomography From a series of projections, a tomographic image is reconstructed using Filtered Back Projection.
  7. 7. Mass Spectrometer Radioactive isotope separated by difference in inertia while bending in magnetic field.
  8. 8. Nuclear Medicine Gamma camera for creating image of radioactive target. Camera is rotated around patient in SPECT (Single Photon Emission Computed Tomography).
  9. 9. Phased Array Ultrasound Ultrasound beam formed and steered by controlling the delay between the elements of the transducer array.
  10. 10. Real Time 3D Ultrasound
  11. 11. Positron Emission Tomography Positron-emitting organic compounds create pairs of high energy photons that are detected synchronously.
  12. 12. Other Imaging Modalities • MRI (Magnetic Resonance Imaging) • OCT (Optical Coherence Tomography)
  13. 13. Current Trends in Imaging • 3D • Higher speed • Greater resolution • Measure function as well as structure • Combining modalities (including direct vision)
  14. 14. The Gold Standard • Dissection: – Medical School, Day 1: Meet the Cadaver. – From Vesalius to the Visible Human
  15. 15. Local Operators and Global Transforms
  16. 16. Images are n dimensional signals. • 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.
  17. 17. 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       2 n
  18. 18. Orthographic Transform Matrix • Capable of geometric, similarity, or affine. • Homogeneous coordinates. • Multiply in reverse order to combine • SGI “graphics engine” 1982, now standard.                     =           ′ ′ 11001 3,22,21,2 3,12,11,1 y x aaa aaa y x
  19. 19. Translation by (tx , ty)                     =           ′ ′ 1100 10 01 1 y x t t y x y x Scale x by sx and y by sy                     =           ′ ′ 1100 00 00 1 y x s s y x y x
  20. 20. • 2 x 2 rotation portion is orthogonal (orthonormal vectors). • Therefore only 1 degree of freedom, . Rotation in 2D                     − =           ′ ′ 1100 0cossin 0sincos 1 y x y x θθ θθ θ
  21. 21. • 3 x 3 rotation portion is orthogonal (orthonormal vectors). • 3 degree of freedom (dotted circled), , as expected. Rotation in 3D                         =             ′ ′ ′ 11000 0 0 0 1 3,32,31,3 3,22,21,2 2,12,11,1 z y x aaa aaa aaa z y x       2 n
  22. 22. • 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. Non-Orthographic Projection in 3D                         =             ′ ′ ′ 1100 0100 0010 0001 1 z y x k z y x
  23. 23. Point Operators • f is usually monotonic, and shift invariant. • Inverse may not exist due to discrete values of intensity. • Brightness/contrast, “windowing”. • Thresholding. • Color Maps. • f may vary with pixel location, eg., correcting for inhomogeneity of RF field strength in MRI. ( ) ( )[ ]yxIfyxI ,,' =
  24. 24. Histogram Equalization • A pixel-wise intensity mapping is found that produces a uniform density of pixel intensity across the dynamic range.
  25. 25. Adaptive Thresholding from Histogram • Assumes bimodal distribution. • Trough represents boundary points between homogenous areas.
  26. 26. Algebraic Operators • Assumes registration. • Averaging multiple acquisitions for noise reduction. • Subtracting sequential images for motion detection, or other changes (eg. Digital Subtractive Angiography). • Masking. ( ) ( ) ( )[ ]yxIyxIgyxI ,,,,' 21=
  27. 27. 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.
  28. 28. 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.
  29. 29. Neighborhood PDE Operators • 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.
  30. 30. • Vector • Direction of maximum change of scalar intensity I. • Normal to the boundary. • Nicely n-dimensional. Intensity Gradient zyxzyx ˆˆˆˆˆˆ zyx III dz dI dy dI dx dI I ++=++=∇
  31. 31. • Scalar • Maximum at the boundary • Orientation-invariant. Intensity Gradient Magnitude ( ) nˆ2 1 222 ⋅∇=++=∇ IIIII zyx
  32. 32. I xI yI I∇
  33. 33. Classic Edge Detection Kernel (Sobel) xI⇒           − − − 101 202 101 yI⇒           −−− 121 000 121
  34. 34. Isosurface, Marching Cubes (Lorensen) • 100% opaque watertight surface • Fast, 28 = 256 combinations, pre-computed
  35. 35. • Marching cubes works well with raw CT data. • Hounsfield units (attenuation). • Threshold calcium density.
  36. 36. • Ixy = Iyx = curvature • Orientation-invariant. • What about in 3D? Jacobian of the Intensity Gradient       =             yyyx xyxx II II dy Id dxdy Id dydx Id dx Id 2 22 2 2 2
  37. 37. Laplacian of the Intensity • Divergence of the Gradient. • Zero at the inflection point of the intensity curve. 222 2 2 22 2 22 2 2 2 zzyyxx III dz Id dy Id dx Id I ++=      +      +      =∇ I Ix Ixx           −−− −− −−− 111 181 111
  38. 38. Binomial Kernel • Repeated averaging of neighbors => Gaussian by Central Limit Theorem. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 1 2 4 2 3 9 9 3 4 16 24 16 4 1 2 1 3 9 9 3 6 24 36 24 6 1 3 3 1 4 16 24 16 4 1 4 6 4 1
  39. 39. 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 -1 0 1 -1 -2 0 2 1 -1 -4 -6 -4 0 4 6 4 1 -2 -4 0 4 2 -4 -16 -24 -16 0 16 24 16 4 -1 -2 0 2 1 -6 -24 -36 -24 0 24 36 24 6 -4 -16 -24 -16 0 16 24 16 4 -1 -4 -6 -4 0 4 6 4 1
  40. 40. Texture Boundaries • Two regions with the same intensity but differentiated by texture are easily discriminated by the human visual system.
  41. 41. 2D Fourier Transform ( ) ( ) ( )2 , , j ux vy f x y F u v e du dv π +∞ +∞ + −∞ −∞ = ∫ ∫ ( ) ( ) ( )2 , , j ux vy F u v f x x e dx dy π +∞ +∞ − + −∞ −∞ = ∫ ∫ analysis synthesis ( ) ( ) 2 2 , , j ux j vy F u v f x y e dx e dyπ π +∞ +∞ − − −∞ −∞ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ ∫ ∫ or
  42. 42. Properties • Most of the usual properties, such as linearity, etc. • Shift-invariant, rather than Time-invariant • Parsevals relation becoms Rayleigh’s Theorem • Also, Separability, Rotational Invariance, and Projection (see below)
  43. 43. Separability ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )vFyf uFxf vuFvFuFyfxf then vuFyxf yfxfyxf F F F F 22 11 2121 21 , ,, , if ⏐→← ⏐→← =⏐→← ⏐→← =
  44. 44. Rotation Invariance ( ) ( )θθθθ θθθθ cossin,sincos cossin,sincos vuvuF yxyxf F +−+ ⏐→←+−+             − =      ′ ′ y x y x θθ θθ cossin sincos
  45. 45. Projection ( ) ( ) ( ) ( )0, , uFuP dyyxfxp = = ∫ +∞ ∞− Combine with rotation, have arbitrary projection.
  46. 46. Gaussian ( ) 2 2 2 2 2 22 222 σσσ yxyx eee −−+− = seperable ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 3 1 2 3 F g x G u g x g x g u G u G u G u ←⏐→ ∗ = = Since the Fourier Transform is also separable, the spectra of the 1D Gaussians are, themselves, separable.
  47. 47. Hankel Transform For radially symmetrical functions ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )qFdrrderf dydxeyxfvuF vuqqFvuF yxrrfyxf r qrj r vyuxj r r =⎥ ⎦ ⎤ ⎢ ⎣ ⎡ == +== +== ∫ ∫ ∫ ∫ ∞ − +− ∞+ ∞− ∞+ ∞− ,, ,, ,, 0 2 0 cos2 2 222 222 θ π θπ π
  48. 48. Elliptical Fourier Series for 2D Shape ( ) k Fs atx ⏐→← ( ) k Fs bty ⏐→← Parametric function, usually with constant velocity. ( )00,center ba= Truncate harmonics to smooth.
  49. 49. Fourier shape in 3D • Fourier surface of 3D shapes (parameterized on surface). • Spherical Harmonics (parameterized in spherical coordinates). • Both require coordinate system relative to the object. How to choose? Moments? • Problem of poles: sigularities cannot be avoided
  50. 50. Quaternions – 3D phasors 4321 kajaiaaa +++= 1222 −==== ιϕκκϕι 4321 * kajaiaaa −−−= Product is defined such that rotation by arbitrary angles from arbitrary starting points become simple multiplication. ( )2 1 2 4 2 3 2 2 2 1 aaaaa +++= ( ) ( ) ( ) ( )44332211 bakbajbaibaba +++++++=+
  51. 51. Summary • Fourier useful for image “processing”, convolution becomes multiplication. • Fourier less useful for shape. • Fourier is global, while shape is local. • Fourier requires object-specific coordinate system.

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