Edges and Scale From Sandlot Science Todays reading Cipolla & Gee on edge detection (available online) Szeliski 3.4.1 3.4.2 Origin of Edges surface normal discontinuity depth discontinuity surface color discontinuity illumination discontinuity
Edges are caused by a variety of factors Detecting edges Whats 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
Where is the edge? Solution: smooth first Where is the edge? Look for peaks in Associative property of convolution This saves us one operation: Laplacian of Gaussian Consider
Laplacian of Gaussian operator Where is the edge? Zero-crossings of bottom graph 2D edge detection filters Laplacian of Gaussian Gaussian
derivative of Gaussian is the Laplacian operator: The Sobel operator Common approximation of derivative of Gaussian -1 0 1 1
2 1 -2 0 2 0
0 0 -1 0 1 -1 -2 -1 The standard defn. of the Sobel operator omits the 1/8 term
doesnt make a difference for edge detection the 1/8 term is needed to get the right gradient value, however The effect of scale on edge detection larger larger Scale space (Witkin 83) Some times we want many resolutions
Known as a Gaussian Pyramid [Burt and Adelson, 1983] In computer graphics, a mip map [Williams, 1983] A precursor to wavelet transform Gaussian Pyramids have all sorts of applications in computer vision Gaussian pyramid construction filter kernel Repeat Filter (called prefiltering) Subsample (faster approach: filter only of pixels)
Until minimum resolution reached can specify desired number of levels (e.g., 3-level pyramid) The whole pyramid is only 4/3 the size of the original image! Subsampling with Gaussian pre-filtering G 1/8 G 1/4 Gaussian 1/2 Filter the image, then subsample Subsampling with Gaussian pre-filtering
Gaussian 1/2 G 1/4 Filter the image, then subsample G 1/8 Subsampling without pre-filtering 1/2
1/4 (2x zoom) 1/8 (4x zoom) 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/imagean 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? d = 1 in this example 1 2 3
4 5 Recall how a digital image is formed It is a discrete point-sampling of a continuous function If we could somehow reconstruct the original function, any new image could be generated, at any resolution and scale 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?
d = 1 in this example 1 2 3 4 5
Recall how a digital image is formed It is a discrete point-sampling of a continuous function If we could somehow reconstruct the original function, any new image could be generated, at any resolution and scale Image resampling So what to do if we dont know Answer: guess an approximation Can be done in a principled way: filtering 1 d = 1 in this
example 1 2 2.5 3 4 Image reconstruction Convert
to a continuous function Reconstruct by cross-correlation: 5 Resampling filters What does the 2D version of this hat function look like? performs linear interpolation
(tent function) performs bilinear interpolation Often implemented without cross-correlation E.g., http://en.wikipedia.org/wiki/Bilinear_interpolation Better filters give better resampled images Bicubic is common choice fit 3rd degree polynomial surface to pixels in neighborhood
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