Basics of vector-space formalism for linear approximations: from
orthonormal bases to redundant frames for global and local weighted
least-squares approximations. Coupled frames and principled design of
operators and (convolution) kernels for the interpolation, resampling,
smoothing, and differentiation of regularly and irregularly sampled
data.
Course Content:
- Vector and inner-product spaces
- Bases, Analysis and Synthesis
- Orthonormalization, Approximations
- Continuous-variable models for discrete data
- Global vs Local Processing
- Windowing and weighted norms
- Moving least squares
- Frames and Dual frames
- Singular Value Decomposition, Principal components and KLT transforms
- Smoothing operators
- Discrete differential operators
- Color spaces and color and multispectral image processing
- Popular transforms
- Noise propagation
All the above is explained and demonstrated through a vast set of working examples in Matlab and Python.
After completing this course, the student will
- be able to model and solve signal processing problems through a wide range of vector-space methods
- be able to extend the geometrical intuition from familiar 2D and 3D to very high-dimensional settings within a rigorous framework
- be able to implement a wide range of fixed and adaptive signal transforms and understand their general properties
- master the geometrical structures inherent to basic signal processing techniques, including color transforms, convolution filters, sliding window methods.
- be able to design and implement consistent multi-dimensional differential operators for discrete signals
- be able to analyze the propagation of signal distortions (e.g., noise) across transformations
- Opettaja: Alessandro Foi