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## Medical Imaging

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**Simultaneous measurements on a spatial grid.**Many modalities: mainly EM radiation and sound. Medical Imaging**“To invent you need a good imagination and a pile of**junk.” Thomas Edison 1879**Electron rapidly decelerates at heavy metal target, giving**off X-Rays. Bremsstrahlung**Projection of X-Ray silhouette onto a piece of film or**detector array, with intervening fluorescent screen. X-Ray and Fluoroscopic Images**From a series of projections, a tomographic image is**reconstructed using Filtered Back Projection. Computerized Tomography**Radioactive isotope separated by difference in inertia while**bending in magnetic field. Mass Spectrometer**Gamma camera for creating image of radioactive target.**Camera is rotated around patient in SPECT (Single Photon Emission Computed Tomography). Nuclear Medicine**Ultrasound beam formed and steered by controlling the delay**between the elements of the transducer array. Phased Array Ultrasound**Positron-emitting organic compounds create pairs of**high energy photons that are detected synchronously. Positron Emission Tomography**MRI (Magnetic Resonance Imaging)**OCT (Optical Coherence Tomography) Other Imaging Modalities**3D**Higher speed Greater resolution Measure function as well as structure Combining modalities (including direct vision) Current Trends in Imaging**Dissection:**Medical School, Day 1: Meet the Cadaver. From Vesalius to the Visible Human The Gold Standard**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. Images are n dimensional signals.**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 Global Transforms in n dimensions**Capable of geometric, similarity, or affine.**Homogeneous coordinates. Multiply in reverse order to combine SGI “graphics engine” 1982, now standard. Orthographic Transform Matrix**Translation by (tx , ty)**Scale x by sx and y by sy**Rotation in 2D**• 2 x 2 rotation portion is orthogonal (orthonormal vectors). • Therefore only 1 degree of freedom, .**Rotation in 3D**• 3 x 3 rotation portion is orthogonal (orthonormal vectors). • 3 degree of freedom (dotted circled), , as expected.**Non-Orthographic Projection in 3D**• 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.**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. Point Operators**A pixel-wise intensity mapping is found that produces a**uniform density of pixel intensity across the dynamic range. Histogram Equalization**Assumes bimodal distribution.**Trough represents boundary points between homogenous areas. Adaptive Thresholding from Histogram**Assumes registration.**Averaging multiple acquisitions for noise reduction. Subtracting sequential images for motion detection, or other changes (eg. Digital Subtractive Angiography). Masking. Algebraic Operators**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. Re-Sampling on a New Lattice**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. Convolution and Correlation**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. Neighborhood PDE Operators**Vector**Direction of maximum change of scalar intensity I. Normal to the boundary. Nicely n-dimensional. Intensity Gradient**Scalar**Maximum at the boundary Orientation-invariant. Intensity Gradient Magnitude**100% opaque watertight surface**Fast, 28 = 256 combinations, pre-computed Isosurface, Marching Cubes (Lorensen)**Marching cubes works well with raw CT data.**• Hounsfield units (attenuation). • Threshold calcium density.**Ixy = Iyx= curvature**Orientation-invariant. What about in 3D? Jacobian of the Intensity Gradient**Divergence of the Gradient.**Zero at the inflection point of the intensity curve. Laplacian of the Intensity I Ix Ixx**Repeated averaging of neighbors => Gaussian by Central Limit**Theorem. Binomial Kernel**Not the conventional concentric DOG**Subtracting pixels displaced along the x axis after repeated blurring with binomial kernel yields Ix Binomial Difference of Offset Gaussian (DooG)**Two regions with the same intensity but differentiated by**texture are easily discriminated by the human visual system. Texture Boundaries**2D Fourier Transform**analysis or synthesis**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) Properties**Projection**Combine with rotation, have arbitrary projection.**Gaussian**seperable Since the Fourier Transform is also separable, the spectra of the 1D Gaussians are, themselves, separable.**Hankel Transform**For radially symmetrical functions**Elliptical Fourier Series for 2D Shape**Parametric function, usually with constant velocity. Truncate harmonics to smooth.**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 Fourier shape in 3D**Quaternions – 3D phasors**Product is defined such that rotation by arbitrary angles from arbitrary starting points become simple multiplication.