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

2. 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

3. 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

4. Object representationsd)  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]

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

6. 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

7. 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)

8. 2D Basis Functions Different sub-panels show differing level of detail

9. Laplacian of Gaussian Wider or larger aperture gives different scales

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

11. Parametrization with spherical harmonics3 1 7 12

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

13. Noise & Unwanted Detail; Spatial Scale

14. 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')

15. 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

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

17. Contrast Enhancement in Mammography