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This article explores the simplicity and applications of MSP image transforms, such as image compression, enhancement, and analysis. It discusses the basic definitions of orthogonal and unitary matrices, and their role in image representation. The article also introduces 2D orthogonal and unitary transformations, and their properties. Additionally, it discusses the basis images and the continuous 1D Fourier transform.

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

  2. Image Transforms – Why? • Simplicity • Applications • Image compression (JPEG • Image enhancement (e.g., filtering) • Image analysis (e.g., feature extraction) MSP

  3. Image Transforms • Preliminary definitions • Orthogonal matrix • Unitary matrix MSP

  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 MSP

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

  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) MSP

  7. Image Transforms (cont’) • For a 1D sequence , a unitary transformation is written as where (unitary). This gives MSP

  8. Basic Vectors of 8x8 Orthogonal Transforms Jain, 1989 MSP

  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. MSP

  10. 2D Orthogonal & Unitary Transformations (cont’) is a set of complete orthonormal discrete basis functions satisfying MSP

  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. MSP

  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 MSP

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

  14. The Continuous 1D Fourier Transform • The Fourier transform pair MSP

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