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- 1. Medical Imaging • Simultaneous measurements on a spatial grid. • Many modalities: mainly EM radiation and sound.
- 2. “To invent you need a good imagination and a pile of junk.” Thomas Edison 1879
- 3. Bremsstrahlung Electron rapidly decelerates at heavy metal target, giving off X-Rays.
- 4. 1896
- 5. X-Ray and Fluoroscopic Images Projection of X-Ray silhouette onto a piece of film or detector array, with intervening fluorescent screen.
- 6. Computerized Tomography From a series of projections, a tomographic image is reconstructed using Filtered Back Projection.
- 7. Mass Spectrometer Radioactive isotope separated by difference in inertia while bending in magnetic field.
- 8. Nuclear Medicine Gamma camera for creating image of radioactive target. Camera is rotated around patient in SPECT (Single Photon Emission Computed Tomography).
- 9. Phased Array Ultrasound Ultrasound beam formed and steered by controlling the delay between the elements of the transducer array.
- 10. Real Time 3D Ultrasound
- 11. Positron Emission Tomography Positron-emitting organic compounds create pairs of high energy photons that are detected synchronously.
- 12. Other Imaging Modalities • MRI (Magnetic Resonance Imaging) • OCT (Optical Coherence Tomography)
- 13. Current Trends in Imaging • 3D • Higher speed • Greater resolution • Measure function as well as structure • Combining modalities (including direct vision)
- 14. The Gold Standard • Dissection: – Medical School, Day 1: Meet the Cadaver. – From Vesalius to the Visible Human
- 15. Local Operators and Global Transforms
- 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. 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. 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. 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. • 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. • 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. • 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. 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. Histogram Equalization • A pixel-wise intensity mapping is found that produces a uniform density of pixel intensity across the dynamic range.
- 25. Adaptive Thresholding from Histogram • Assumes bimodal distribution. • Trough represents boundary points between homogenous areas.
- 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. 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. 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. 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. • 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. • Scalar • Maximum at the boundary • Orientation-invariant. Intensity Gradient Magnitude ( ) nˆ2 1 222 ⋅∇=++=∇ IIIII zyx
- 32. I xI yI I∇
- 33. Classic Edge Detection Kernel (Sobel) xI⇒ − − − 101 202 101 yI⇒ −−− 121 000 121
- 34. Isosurface, Marching Cubes (Lorensen) • 100% opaque watertight surface • Fast, 28 = 256 combinations, pre-computed
- 35. • Marching cubes works well with raw CT data. • Hounsfield units (attenuation). • Threshold calcium density.
- 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. 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. 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. 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. Texture Boundaries • Two regions with the same intensity but differentiated by texture are easily discriminated by the human visual system.
- 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. 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. Separability ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )vFyf uFxf vuFvFuFyfxf then vuFyxf yfxfyxf F F F F 22 11 2121 21 , ,, , if ⏐→← ⏐→← =⏐→← ⏐→← =
- 44. Rotation Invariance ( ) ( )θθθθ θθθθ cossin,sincos cossin,sincos vuvuF yxyxf F +−+ ⏐→←+−+ − = ′ ′ y x y x θθ θθ cossin sincos
- 45. Projection ( ) ( ) ( ) ( )0, , uFuP dyyxfxp = = ∫ +∞ ∞− Combine with rotation, have arbitrary projection.
- 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. 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. 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. 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. 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. 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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