Part VII: Appendix
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Last updated: September 22, 2025
The appendices provide a more formal mathematical foundation for the concepts used throughout the book. They are designed as a reference for advanced undergraduate and graduate students who wish to deepen their understanding of the linear systems theory that underpins modern image systems engineering.
A summary of the topics that we will cover in the Sensors section of the book should be placed here.
- General linear systems
- Shift invariant systems
- Software Examples
Appendix topics
General Linear Systems
This appendix establishes the fundamental properties and mathematical representations of general linear systems.
- Defines the core principles of linearity: superposition and scaling.
- Introduces the matrix representation of a linear system, where the system’s response is calculated through matrix-vector multiplication.
- Explains the concept of representing signals (images) using different sets of basis functions, such as the point basis and harmonic basis.
- Covers the properties of orthogonality and orthonormality, showing how they provide a simple method for calculating the basis function weights for any given signal.
Shift-Invariant Systems
This appendix specializes the discussion to the crucial subclass of linear shift-invariant (LSI) systems, which are widely used to model optics and other imaging components.
- Defines shift-invariance and shows how it imposes a special circulant structure on the system matrix.
- Introduces convolution as the fundamental operation of LSI systems in the spatial domain.
- Establishes the most important property of LSI systems: that complex harmonics are their eigenvectors.
- Introduces the Discrete Fourier Transform (DFT) as the practical tool for analyzing signals in terms of these harmonic eigenvectors.
- Presents the Convolution Theorem, which states that computationally intensive convolution in the spatial domain is equivalent to simple multiplication in the frequency domain.
Software Examples
This appendix is planned, but not implemented.
It will provide practical software examples, likely using ISETCam and MATLAB/Python, to illustrate the key concepts from the main chapters. It will serve as a hands-on resource for simulating optical systems, sensor behavior, and image processing algorithms.