Images & Objects Input Output Image Objects Image Image Processing Image Analysis Input Objects Computer Graphics Modeling?

COMP 235 Outline Sensing Display of objects and geometric entities Handling noise and discreteness Sampling and interpolation Selected 3D matters Transformations; homogenous coordinates 2D images from 3D

Discrete Representations of Objects and Images Sampled Images: pixels Objects Interior: voxels, medial Boundary: tiles and vertices Limiting damage of sampling Parametrized Images Global: I(x,y) = i ai i(x,y) Local: patches: linear combination of local basis Of interior Of boundary Global: 2D: S(u)= i ai i(u), 3D: S(u,v)=i ai i(u,v) Local: splines: linear combination of local basis Interpolation from sampled to parametrized

Object representations d)  e) f) g) a) Boundary points. b) Boundary tiles. c) Fourier harmonics. d) Atlas displacements with binary labels. e) Landmarks. f) Medial. g) Medial atoms . [Add point-normal]

Forms of Medial Representation Traditional: by disks or balls By medial atoms: hubs with two spokes, touching at tangency points of disk or ball

Questions for parametrized representations Dimensions of argument space: 1,2,3,4 Topologies of argument space Bounded [0,1]: vs. cyclic [0,2p) Number of holes Basis functions How handle different levels of detail Level(s) of locality

Global Sinusoidal Basis Functions On [0, 2p)n: Fourier basis functions On sphere [0, p]  [0,2p): spherical harmonics Different frequencies (levels of detail) in each parameter (see next slide)

2D Basis Functions Different sub-panels show differing level of detail

Laplacian of Gaussian Wider or larger aperture gives different scales

Laplacian of Gaussian, Wavelet Wavelet ties aperture size and level of detail

Parametrization with spherical harmonics 3 1 7 12

Noise & Unwanted Detail; Spatial Scale Images Idiscrete(x,y)= Idiscrete & ideal(x,y) + noise(x,y) Objects S(u,v)= Ssmooth(u,v) + d(u,v)N(u,v), with N= the normal of S Removing noise & unwanted detail

Noise & Unwanted Detail; Spatial Scale

Blurring & Spatial Distortion Idiscrete(x,y) = Sampling [Pixel integration { Distortion [Blurring {Projection [ Reflections (imaged objects)]}]}] + noise Blurring = replace each point by blur kernel Distortion: x=x'+Dx(x') Result of blurring is f(x')

Display Of objects Of geometric entities, e.g., lines & curves Coloring, shading Illumination, reflection, projection Of geometric entities, e.g., lines & curves Of images Contrast control Devices Hardware Making produced images perceptually predictable

Contrast Enhancement (CT) Adaptive Histogram Equalization (UNC) Truth? Preference -vs- Utility?

Contrast Enhancement in Mammography