1	Edges and Scale Today’s reading Cipolla & Gee on edge detection (available online) Szeliski 3.4.1 – 3.4.2
2	Origin of Edges Edges are caused by a variety of factors
3	Detecting edges What’s an edge? intensity discontinuity (= rapid change) How can we find large changes in intensity? gradient operator seems like the right solution
4	Effects of noise Consider a single row or column of the image Plotting intensity as a function of position gives a signal
5	Solution: smooth first
6	Associative property of convolution This saves us one operation:
7	Laplacian of Gaussian Consider
8	2D edge detection filters
9	The Sobel operator Common approximation of derivative of Gaussian
10	The effect of scale on edge detection
11	Some times we want many resolutions
12	Gaussian pyramid construction
13	Subsampling with Gaussian pre-filtering
14	Subsampling with Gaussian pre-filtering
15	Subsampling without pre-filtering
16	Sampling and the Nyquist rate Aliasing can arise when you sample a continuous signal or image occurs when your sampling rate is not high enough to capture the amount of detail in your image Can give you the wrong signal/image—an alias formally, the image contains structure at different scales called “frequencies” in the Fourier domain the sampling rate must be high enough to capture the highest frequency in the image To avoid aliasing: sampling rate ≥ 2 * max frequency in the image said another way: ≥ two samples per cycle This minimum sampling rate is called the Nyquist rate
17	Image resampling So far, we considered only power-of-two subsampling What about arbitrary scale reduction? How can we increase the size of the image?
18	Image resampling So far, we considered only power-of-two subsampling What about arbitrary scale reduction? How can we increase the size of the image?
19	Image resampling So what to do if we don’t know
20	Resampling filters What does the 2D version of this hat function look like?