1. References Jain (a text book?; IP per se; available)
Castleman (a real text book ; image analysis; less available)
Lim (unavailable?)
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2. Image Transforms – Why? Simplicity Applications
Image compression (JPEG
Image enhancement (e.g., filtering)
Image analysis (e.g., feature extraction)
Simplicity (1. uniform background image 2. Convolution theorem)
Image compression
Image enhancement (filtering) – next week
Image analysis (feature extraction) – next talks
Cover foundations thoroughly rather than too many transforms superficially
Notation is changing
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3. Image Transforms Preliminary definitions Orthogonal matrix
Unitary matrix
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4. Preliminary Definitions (cont’)
Real orthogonal matrix is unitary
Unitary matrix need not be orthogonal
Columns (rows) of an NxN unitary matrix are orthogonal and form a complete set of basis vectors in an N-dimensional vector space
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5. Preliminary Definitions (cont’)
Examples (Jain, 1989)
orthogonal & unitary
not unitary
unitary
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6. Image Transforms (cont’)
... are a class of unitary matrices used to facilitate image representation
Representation using a discrete set of basis images (similar to orthogonal series expansion of a continuous function)
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7. Image Transforms (cont’)
For a 1D sequence , a unitary transformation is written as
where (unitary). This gives
u(n) for a specific n and v(k) for a specific k are scalars; a(k,n) a*(k,n) are vectors. v(k) & u(n) are the result of an inner product.
A series representation of the sequence u(n) using a series coefficients v(k).
The columns of A*T, that is, a*(k,n), are called the basis vectors of A.
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8. Basic Vectors of 8x8 Orthogonal Transforms
Jain, 1989
8 8-dimensional basis vectors. Vectors are orthogonal to each other.
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9. 2D Orthogonal & Unitary Transformations
A general orthogonal series expansion for an NxN image u(m,n) is a pair of transformations
where is called an image transform, the elements v(k,l) are called the transform coefficients and is the transformed image.
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10. 2D Orthogonal & Unitary Transformations (cont’)
is a set of complete orthonormal discrete basis functions satisfying
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11. 2D Orthogonal & Unitary Transformations (cont’)
The orthonormality property assures that any truncated series expansion of the form
will minimise the sum-square-error
for v(k,l) as above, and the completeness property guarantees that this error will be zero for P=Q=N.
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12. Basis Images
Define the matrices , where is the kth column of , and the matrix inner product of two NxN matrices F and G as
Then Equations 2 & 1 provide series representation for the image as
Any image U is expressed as a linear combination of the N2 matrices Akl*, l=0…N-1, which are called the basis images.
v(k,l), the transform coefficients, are the inner product of the (k,l)th basis image with the image.
Thus, any NxN image can be expanded in a series using a complete set of N2 basis images.
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13. Basic Images of the 8x8 2D Transforms
Jain, 1989
Basic Images of the 8x8 2D Transforms
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14. The Continuous 1D Fourier Transform
The Fourier transform pair
The Fourier transform is a linear integral transformation that takes a complex function of n real variables into another complex function of n real variables.
The only difference between the direct and inverse transformation is the sign of the exponent.
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