This is a free online course offered by the Saylor Foundation.
'The objective of this course is to study the basic theory and methods in the toolbox of the core of applied mathematics, with a central scheme that addresses “information processing” and with an emphasis on manipulation of digital image data. Linear algebra in the Saylor Foundation’s MA211 and MA212 are extended to “linear analysis” with applications to principal component analysis (PCA) and data dimensionality reduction (DDR). For data compression, the notion of entropy is introduced to quantify coding efficiency as governed by Shannon’s Noiseless Coding theorem. Discrete Fourier transform (DFT) followed by an efficient computational algorithm, called fast Fourier transform (FFT), as well as a real-valued version of the DFT, called discrete cosine transform (DCT) are discussed, with application to extracting frequency content of the given discrete data set that facilitates reduction of the entropy and thus significant improvement of the coding efficiency. DFT can be viewed as a discrete version of the Fourier series, which will be studied in some depth, with emphasis on orthogonal projection, the property of positive approximate identity of Fejer’s kernels, Parseval’s identity and the concept of completeness. The integral version of the sequence of Fourier coefficients is called the Fourier transform (FT). Analogous to the Fourier series, the formulation of the inverse Fourier transform (IFT) is derived by applying the Gaussian function as a sliding time-window for simultaneous time-frequency localization, with optimality guaranteed by the Uncertainty Principle. Local time-frequency basis functions are also introduced in this course by discretization of the frequency-modulated sliding time-window function at the integer lattice points. Replacing the frequency modulation by modulation with the cosines avoids the Balian-Low stability restriction on the local time-frequency basis functions, with application to elimination of blocky artifact caused by quantization of tiled DCT in image compression. Gaussian convolution filtering also provides the solution of the heat (partial differential) equation with the real-line as the spatial domain. When this spatial domain is replaced by a bounded interval, the method of separation of variables is applied to separate the PDE into two ordinary differential equations (ODEs). Furthermore, when the two end-points of the interval are insulated from heat loss, solution of the spatial ODE is achieved by finding the eigenvalue and eigenvector pairs, with the same eigenvalues to govern the exponential rate of decay of the solution of the time ODE. Superposition of the products of the spatial and time solutions over all eigenvalues solves the heat PDE, when the Fourier coefficients of the initial heat content are used as the coefficients of the terms of the superposition. This method is extended to the two-dimensional rectangular spatial domain, with application to image noise reduction. The method of separation of variables is also applied to solving other typical linear PDEs. Finally, multi-scale data analysis is introduced and compared with the Fourier frequency approach, and the architecture of multiresolution analysis (MRA) is applied to the construction of wavelets and formulation of the multi-scale wavelet decomposition and reconstruction algorithms. The lifting scheme is also introduced to reduce the computational complexity of these algorithms, with applications to digital image manipulation for such tasks as progressive transmission, image edge extraction, and image enhancement.'
