Topics in Applied Mathematics

The objective of this course is to study the basic theory and methodsin the toolbox of the core of applied mathematics, with a central schemethat addresses “information processing” and with an emphasis on manipulationof digital image data. Linear algebra in the Saylor Foundation’s MA211and MA212 are extended to “linear analysis” with applications to principalcomponent analysis (PCA) and data dimensionality reduction (DDR). Fordata compression, the notion of entropy is introduced to quantify coding effi-ciency as governed by Shannon’s Noiseless Coding theorem. Discrete Fouriertransform (DFT) followed by an efficient computational algorithm, called fastFourier transform (FFT), as well as a real-valued version of the DFT, calleddiscrete cosine transform (DCT) are discussed, with application to extractingfrequency content of the given discrete data set that facilitates reduction ofthe entropy and thus significant improvement of the coding efficiency. DFTcan be viewed as a discrete version of the Fourier series, which will be studiedin some depth, with emphasis on orthogonal projection, the property of11positive approximate identity of Fejer’s kernels, Parseval’s identity and theconcept of completeness. The integral version of the sequence of Fourier coef-ficients is called the Fourier transform (FT). Analogous to the Fourier series,the formulation of the inverse Fourier transform (IFT) is derived by applyingthe Gaussian function as sliding time-window for simultaneous time-frequencylocalization, with optimality guaranteed by the Uncertainty Principle. Localtime-frequency basis functions are also introduced in this course by discretizationof the frequency-modulated sliding time-window function at the integerlattice points. Replacing the frequency modulation by modulation with thecosines avoids the Balian-Low stability restriction on the local time-frequencybasis functions, with application to elimination of blocky artifact caused byquantization of tiled DCT in image compression. Gaussian convolution filteringalso provides the solution of the heat (partial differential) equation withthe real-line as the spatial domain. When this spatial domain is replaced bya bounded interval, the method of separation of variables is applied to separatethe PDE into two ordinary differential equations (ODEs). Furthermore,when the two end-points of the interval are insulated from heat loss, solutionof the spatial ODE is achieved by finding the eigenvalue and eigenvectorpairs, with the same eigenvalues to govern the exponential rate of decay ofthe solution of the time ODE. Superposition of the products of the spatialand time solutions over all eigenvalues solves the heat PDE, when the Fouriercoefficients of the initial heat content are used as the coefficients of the termsof the superposition. This method is extended to the two-dimensional rectangularspatial domain, with application to image noise reduction. The methodof separation of variables is also applied to solving other typical linear PDEs.Finally, multi-scale data analysis is introduced and compared with the Fourierfrequency approach, and the architecture of multiresolution analysis (MRA)is applied to the construction of wavelets and formulation of the multi-scalewavelet decomposition and reconstruction algorithms. The lifting scheme isalso introduced to reduce the computational complexity of these algorithms,with applications to digital image manipulation for such tasks as progressivetransmission, image edge extraction, and image enhancement.