For today you should:
1) Case study
Today:
1) Convolution theorem
2) LTI system theory
3) Image feature detection
For next time:
1) There is no next time :(
Case study report due on LDOI, Thursday December 11 by the end of the day.
Poster session Monday December 15 from 1-3pm in AC326 (note the room change).
Let's prove (finally) that
1) The complex exponentials are eigenvectors of convolution operations.
2) The corresponding eigenvalues are the elements of the DFT(g), where g is the convolution window.
LTI System Theory
"System" in this context refers to anything that can take a signal as input and produce a signal as output.
Example: In the seawater problem from Chapter 1, the ocean is a system that takes an input signal and produces an output signal.
In addition, an LTI system has these properties:
Linearity: If the input x1 produces output y1 and input x2 produces y2, then a x1 + b x2 produces a y1 + b y2.
Time invariance: if x1 and x2 differ by a shift in time, y1 and y2 differ by the same shift.
Consequence: You can break a signal into components (any way you like), compute the output for each of the components, and then add up the outputs.
Example: Idealized LRC circuits.
Example: Idealized spring-mass-dashpot mechanical systems.
Things that are not LTI:
Example: Circuits with non-linear elements like diodes and transistors.
Example: Mechanical systems with friction or air resistance.
One other consequence: The output contains the same frequency components as the input.
So Day-Glo paint is not LTI, because the output contains frequency components are not in the input.
If a system is LTI, the convolution theorem provides a framework for
1) Characterizing the system by finding its eigenvalues.
2) Finding the response to any signal by expressing the signal as a linear combination of eigenvectors, finding the response to each component, and adding up the responses.
So how do we characterize an LTI system? By kicking it.
(Jump to 1:20)
Then check out chap10.ipynb.
Check out 1_image_filters_soln.ipynb
Then read and work through skimage-tutorial/lectures/2_feature_detection.ipynb