Notes 20110118 CIS 6320 Image Processing

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Class 2 -- Lecturer: Dr. Matsakis

Previously ...

3 segmentation
4 representation, description
5 object recognition
segmentation: the number of partitions is a combinatorics problem where each partition :D
remember: a partition is a scheme for subsetting an image
a "4-partition" image is partition containing four sets

region based approach: starting at a point, and finding continuous values
edge based approach: look for neighbouring values that are discontinuous (threshold)

texture analysis: characterize objects based on matrices representing textures
example: let's try to characterize compactness
best compact shape: is a disk
geometric definition: a disk has compactness 1, a oblong object has compactness 0.
parameters: Area, Perimeter
A / (pi R^2) = 1 for a disk
R is the maximum distance between two points
disk ...
P = 2 pi r
A = pi r ^2
2 pi r / pi r ^ 2 ?
2 / r = A / P ?
2/rP = A ?
compactness is a dimensionless unit
solution: 4 pi A / (P ^2) = 1 -- a standard degree of compactness
test: disk: 4 pi pi R^2 / ((2 pi R)^2)
perimeter can be arbitrarily large
in the denominator of the above expression, the above expression can get close to zero
convecity --
S subset of R^2
S is convex iff:
- for all (x, y) in S, for all (x', y') in S, for all t in [0, 1],
  - (tx + (1-t)x', ty + (1-t)y') in S
could also say ... (we agree p = (x,y); q = (x',y');
- for all p in S, for all q in S, for all t in [0, 1],
  - (tp + (1-t)q) in S
this says: pick two point: p,q in S -- any pixel on the line segment between them is also in S.

Low level: output is an image -- assists the human
High level: is a description of an image -- autonomous machine perception
DIP -- characterized by specific solutions
Techniques are task and problem specific
examples ...
low level: image enhancement -- adjustment of grey levels four human viewing
low level: image restoration -- adjustment of faulty image back to what the image should have looked back
high level: segmentation -- edge detection to select the regions of the photograph that are parts of animals
high level: object recognition -- selecting out an individual's face

a 3-tuple (A, B, G) where
A (domain) is a non-empty set
B (codomain) is a non-empty set
G (graph) is a subset of A × B (the Cartesian product)
for all a in A, for all b in B, for any b' in B, (a, b) in G or (a, b') in G → b = b'
-- an element of the domain may only go to one element in the co-domain
-- many elements from the domain may go to the same element in the co-domain
this is a partial function
what is a total function?
let f = (A, B, G) be a function
every element of A is mapped to something in B
every a in A, exists b in B, (a, b) in G
-- every total function is a partial function
the domain of definition of a function (A, B, G) is A' defined by ...
- A' = {a in A | exists b in B, (a, b) in G}
- example -- f(x) = 1/x -- domain is R but domain of definition is R - {0}
the range of (A, B, G) is the set B' defined by ...
- B' = {b in B | exists a in A, (a, b) in G}
the function (A, B, G) is injective (is an injectin) iff:
- every a in A, every a' in A, every b in B, (a, b) in G and (a', b) in G → a = a'
- there is only one arrow arriving in B
surjective iff:
- every b in B, every a in A, (a, b) in G
bijective iff:
- total injective and surjective function
in a function (A, B, G)
- f(a) = b is the image of a under f
- a is the pre-image
an example of an analogue image
- f_a | R² → R₊ -- R₊ is all positive real numbers