Notes 20110127 CIS 6320 Image Processing

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image transformations continued

pixel operations are the simplest
- mostly for image enhancements
then neighbourhood operations
- require definition of a distance (is a function)
- the notion of a neighbour -- where distances == 1
- most often used: city block, chess board distance
then full image operations

rotating an image
resizing an image
we must consider image coordinates
for the transformation that takes image f|Z²→R to g|Z²→R ...
we first consider the inverse transformation
T^-1_S(g|Z²→R)→(f|Z²→R)
g(u,v) depends on the grey levels from the same neighbourhood of (x,y) in f.
example -- resize an image
- we want T to double the size of f()
- we get f from g by using a scaling ratio of 0.5
- we thus ask "given (u,v) in g -- which neighbourhood of (x,y) in f do we want?"
My thought ... why? allows us to interpolate when our coordinates do not undergo a integer scaler multiplier -- solved?
- solution: we do this because we want to define the transformation by the result
- we are interested in the pixels in the dimensions of g()
image scaling continued ...
- g(0,0) = f(0,0)
- g(0,Q-1) = f(0,N-1)
- g(P-1,0) = f(M-1,0)
- g(P-1,Q-1) = f(M-1,N-1)
- x = u(M-1)/(P-1)
- y = v(N-1)/(Q-1)
we actually aren't using f(x,y) since x, y aren't guaranteed to be integers
g(u,v) = f_bar(x,y)
examples of possible f_bar
f_bar(x,y) = f(X,Y) = f(nint(x),nint(y)) -- nearest neighbour
-- bilinear interpolation -- given four pixels, we use a weighted average
-- bicubic interpolation -- looking even further

these are the most interesting transformations to us

input image f|Z²→R (domain = x,y)
→ T_d (domain = ρ,θ)
x = ρcosθ
y = φsinθ
ρ = √(x²+y²)
θ = ...
θ ∈ [0,2π)
ρ ∈ R₊
ρ is the length of the moment arm
θ is the angle going from the positive x-axis to the positive y-axis
note: θ is allowed to continue past the rectangular area
going from the spatial domain to the transform domain is called a forward transform (T_d)
we now perform the transformation T (our neat calculations etc.)
we then perform a the inverse transformation T_d^-1
this brings us to the output image -- g|Z²→R
why do we want to use the transform domain?
- there's something nice about the math there that makes our work easier or more efficient
- think of the transform domain as a magic door

the logical operator (¬) can be seen as a mapping
{0;1} into {0;1} -- classical logic
[0;1] into [0;1] -- fuzzy logic
[0;255] into [0;255] etc ...
defining a fuzzy negation ...
must be consistent with classical logic ...
¬(x) = 1-x -- the standard negation
properties:
∀(a,b)∈[0,1]²,a≤b→¬(a)≥¬(b) -- monotonicity (non-increasing)
continuity
∀a∈[0,1],¬(¬(a))=a -- involutive
fuzzy negations are useful in image processing
HOMEWORK: Find the definition for continuity
defining a logical and (∧).
∧|{0,1}²→{0,1}
- functional notation: ∧(0,1) = 0
- logical notation: 0∧1 = 0
defining a fuzzy and ...
∧|[0,1]²→[0,1]
∀(a,b)∈[0,1]² ...
- min(a,b) -- standard fuzzy and
- ∧(a,b)=ab -- algebraic product