Edges and Scale

	Today’s reading
		Cipolla & Gee on edge detection (available online)
		Szeliski 3.4.1 – 3.4.2

Origin of Edges

	Edges are caused by a variety of factors

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

Effects of noise

	Consider a single row or column of the image
		Plotting intensity as a function of position gives a signal

Solution:  smooth first

Associative property of convolution

	This saves us one operation:

Laplacian of Gaussian

	Consider

2D edge detection filters

The Sobel operator

	Common approximation of derivative of Gaussian

The effect of scale on edge detection

Some times we want many resolutions

Gaussian pyramid construction

Subsampling with Gaussian pre-filtering

Subsampling with Gaussian pre-filtering

Subsampling without pre-filtering

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

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?

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?

Image resampling

	So what to do if we don’t know

Resampling filters

	What does the 2D version of this hat function look like?