MSP4 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
MSP6 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)
MSP7 Image Transforms (cont’) For a 1D sequence, a unitary transformation is written as where (unitary). This gives
MSP8 Basic Vectors of 8x8 Orthogonal Transforms Jain, 1989
MSP9 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.
MSP10 2D Orthogonal & Unitary Transformations (cont’) is a set of complete orthonormal discrete basis functions satisfying
MSP11 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.
MSP12 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
MSP13 Basic Images of the 8x8 2D Transforms Jain, 1989
MSP14 The Continuous 1D Fourier Transform The Fourier transform pair