References Jain (a text book?; IP per se; available) Castleman (a real text book ; image analysis; less available) Lim (unavailable?) MSP

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 MSP

Image Transforms Preliminary definitions Orthogonal matrix Unitary matrix MSP

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 MSP

Preliminary Definitions (cont’) Examples (Jain, 1989) orthogonal & unitary not unitary unitary MSP

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) MSP

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. MSP

Basic Vectors of 8x8 Orthogonal Transforms Jain, 1989 8 8-dimensional basis vectors. Vectors are orthogonal to each other. MSP

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. MSP

2D Orthogonal & Unitary Transformations (cont’) is a set of complete orthonormal discrete basis functions satisfying MSP

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. MSP

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. MSP

Basic Images of the 8x8 2D Transforms Jain, 1989 Basic Images of the 8x8 2D Transforms MSP

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. MSP