Today s Topics. Lecture 11: LoG and DoG Filters. Recall: First Derivative Filters. Second-Derivative Filters

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1 Today s Topics Lecture : LoG and DoG Filters Laplacian of Gaussian (LoG) Filter - useful for finding edges - also useful for finding blobs! approimation using Difference of Gaussian (DoG) Recall: First Derivative Filters Second-Derivative Filters Sharp changes in gray level of the input image correspond to peaks or valleys of the first-derivative of the input signal. Peaks or valleys of the first-derivative of the input signal, correspond to zero-crossings of the second-derivative of the input signal. F() F () F() F () F () (D eample) Taylor Series epansion Numerical Derivatives See also T&V, Appendi A. I(,y) Eample: Second Derivatives [ - ] nd Partial deriv wrt I =d I(,y)/d add - nd Partial deriv wrt y I yy =d I(,y)/dy - Central difference appro to second derivative

2 Eample: Second Derivatives I Iyy I Compare: st vs nd Derivatives Iyy I Iy benefit: you get clear localization of the edge, as opposed to the smear of high gradient magnitude values across an edge Finding Zero-Crossings An alternative appro to finding edges as peaks in first deriv is to find zero-crossings in second deriv. In D, convolve with [ - ] and look for piels where response is (nearly) zero? Problem: when first deriv is zero, so is second. I.e. the filter [ - ] also produces zero when convolved with regions of constant intensity. step edge st deriv I() D di() d F() Edge Detection Summary D > Th I(,y) I(,y) =(I (,y) + I y (,y)) / > Th tan θ = I (,y)/ I y (,y) y So, in D, convolve with [ - ] and look for piels where response is nearly zero AND magnitude of first derivative is large enough. nd deriv d I() d = 0 I(,y) =I (,y) + I yy (,y)=0 Laplacian Finite Difference Laplacian Eample: Laplacian I(,y) I + Iyy Laplacian filter I(,y)

3 Eample: Laplacian Notes about the Laplacian: I Iyy I(,y) is a SCALAR Can be found using a SINGLE mask Orientation information is lost I(,y) is the sum of SECOND-order derivatives But taking derivatives increases noise Very noise sensitive! It is always combined with a smoothing operation: I+Iyy I(,y) I(,y) Smooth Laplacian O(,y) LoG Filter D Gaussian and Derivatives First smooth (Gaussian filter), Then, find zero-crossings (Laplacian filter): O(,y) = (I(,y) * G(,y)) g ( ) = e g' ( ) = e = e Laplacian of Gaussian-filtered image Do you see the distinction? Laplacian of Gaussian (LoG) -filtered image g' '( ) ) e = ( 43 Second Derivative of a Gaussian Effect of LoG Operator g' '( ) = ( ) e 43 Original LoG-filtered D analog LoG Meican Hat Band-Pass Filter (suppresses both high and low frequencies) Why? Easier to eplain in a moment. 3

4 Zero-Crossings as an Edge Detector Raw zero-crossings (no contrast thresholding) Zero-Crossings as an Edge Detector Raw zero-crossings (no contrast thresholding) LoG sigma =, zero-crossing LoG sigma = 4, zero-crossing Zero-Crossings as an Edge Detector Raw zero-crossings (no contrast thresholding) Note: Closed Contours You may have noticed that zero-crossings form closed contours. It is easy to see why Think of equal-elevation contours on a topo map. Each is a closed contour. Zero-crossings are contours at elevation = 0. LoG sigma = 8, zero-crossing remember that in our case, the height map is of a LoG filtered image - a surface with both positive and negative elevations Other uses of LoG: Blob Detection Pause to Think for a Moment: How can an edge finder also be used to find blobs in an image? Lindeberg: ``Feature detection with automatic scale selection''. International Journal of Computer Vision, vol 30, number, pp ,

5 Eample: LoG Etrema LoG Etrema, Detail maima LoG sigma = maima minima LoG sigma = LoG Blob Finding LoG filter etrema locates blobs maima = dark blobs on light background minima = light blobs on dark background Scale of blob (size ; radius in piels) is determined by the sigma parameter of the LoG filter. convolve with LoG result LoG sigma = LoG sigma = 0 maima LoG looks a bit like an eye. LoG Derivative of Gaussian Looks like dark blob on light background Looks like vertical and horizontal step edges and it responds maimally in the eye region! Recall: Convolution (and cross correlation) with a filter can be viewed as comparing a little picture of what you want to find against all local regions in the mage. 5

6 Key idea: Cross correlation with a filter can be viewed as comparing a little picture of what you want to find against all local regions in the image. Efficient Implementation Approimating LoG with DoG LoG can be approimate by a Difference of two Gaussians (DoG) at different scales Maimum response: dark blob on light background Minimum response: light blob on dark background Maimum response: vertical edge; lighter on left Minimum response: vertical edge; lighter on right D eample M.Hebert, CMU Efficient Implementation Back to Blob Detection LoG can be approimate by a Difference of two Gaussians (DoG) at different scales. Separability of and cascadability of Gaussians applies to the DoG, so we can achieve efficient implementation of the LoG operator. DoG appro also eplains bandpass filtering of LoG (think about it. Hint: Gaussian is a low-pass filter) Lindeberg: blobs are detected as local etrema in space and scale, within the LoG (or DoG) scale-space volume. Other uses of LoG: Blob Detection Other uses for LOG: Image Coding Coarse layer of the Gaussian pyramid predicts the appearance of the net finer layer. The prediction is not eact, but means that it is not necessary to store all of the net fine scale layer. Laplacian pyramid stores the difference. Gesture recognition for the ultimate couch potato 6

7 Other uses for LOG: Image Coding takes less bits to store compressed versions of these than to compress the original full-res greyscale image The Laplacian Pyramid as a Compact Image Code Burt, P., and Adelson, E. H., IEEE Transactions on Communication, COM-3: (983). 7

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