Computer Vision Foundations · free preview
Integrating Back
Writing the derivative as a matrix exposes integration as its inverse, and with only valid differences the pseudoinverse recovers every image up to its mean.
1. The derivative as a matrix
The derivative as a matrix
2. The boundary cheat
The boundary cheat
3. What the mean has to do with it
What the mean has to do with it
4. Why the cumulative sum is the inverse
With a zero boundary condition assumed, the inverse of the difference matrix turns out to be the cumulative-sum matrix. Why does summing differences rebuild the signal?
5. What survives reconstruction
A valid-differences pseudoinverse reconstruction is compared with the original signal. What is the guaranteed relationship between the two?
6. The constant that went missing
A three-sample signal reads 1 then 3 then 8. Its mean is 4, its valid differences are 2 then 5, and the pseudoinverse returns minus 3 minus 1 then 4. What single constant added back restores the original signal?
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