Images Objects Output Image Objects Image Processing Image
Images & Objects Output Image Objects Image Processing Image Analysis Objects Computer Graphics Modeling? Input
COMP 235 Outline • • • Sensing Display of objects and geometric entities Handling noise and discreteness Sampling and interpolation Selected 3 D matters – Transformations; homogenous coordinates – 2 D images from 3 D
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 – Objects • Of interior • Of boundary – Global: 2 D: S(u)= i ai i(u), 3 D: S(u, v)= i ai i(u, v) – Local: splines: linear combination of local basis • Interpolation from sampled to parametrized
Object representations a) b) c) 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, 2 p) – Number of holes • Basis functions – How handle different levels of detail – Level(s) of locality
Global Sinusoidal Basis Functions • On [0, 2 p)n: Fourier basis functions • On sphere [0, p] [0, 2 p): spherical harmonics • Different frequencies (levels of detail) in each parameter (see next slide)
2 D Basis Functions Different sub-panels show differing level of detail
Laplacian of Gaussian Wider or larger aperture gives different scales
Laplacian of Gaussian, Wavelet ties aperture size and level of detail
Parametrization with spherical harmonics 1 3 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 – 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
- Slides: 17