Numerical Derivatives See also T&V, Appendix A.2 Gradient = vector of partial derivatives of image I(x,y) = [di(x,y)/dx, di(x,y)/dy]

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1 I have put some Matlab image tutorials on Angel. Please take a look i you are unamiliar with Matlab or the image toolbox. Lecture : Linear Operators Administrivia I have posted Homework on Angel. It is due next Friday at beginning o class. Recall: D Gradient Numerical Derivatives See also T&V, Appendix A. Gradient = vector o partial derivatives o image I(x,y) = [di(x,y)/dx, di(x,y)/dy] Finite orward dierence Gradient vector ield indicates the direction and slope o steepest ascent (when considering image pixel values as a surace / height map). Finite backward dierence Finite central dierence Example: Spatial Image Gradients More accurate Example: Spatial Image Gradients Ix=dI(x,y)/dx Ix=dI(x,y)/dx I(x+,y) - I(x-,y) I(x,y) I(x+,y) - I(x-,y) I(x,y) Partial derivative wrt x I(x,y+) - I(x,y-) Partial derivative wrt y Partial derivative wrt x I(x,y+) - I(x,y-) Iy=dI(x,y)/dy Iy=dI(x,y)/dy Partial derivative wrt y Note: From now on we will drop the constant actor /. We can divide by it later.

2 More Speciically I, I, I, I, I, I, I, - More Speciically I, - + I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, More Speciically I, I, I, I, I, I, I, - I, I, I, I, I, + I, I, I, I, I, I, More Speciically I, I, I, I, I, I, I, + I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, - + And so on Linear Filters General process: Form new image whose pixels are a weighted sum o original pixel values, using the same set o weights at each point. Properties Output is a linear unction o the input Output is a shit-invariant unction o the input (i.e. shit the input image two pixels to the let, the output is shited two pixels to the let) Image Filtering Example: smoothing by averaging orm the average o pixels in a neighbourhood Example: smoothing with a Gaussian orm a weighted average o pixels in a neighbourhood Example: inding a derivative orm a weighted average o pixels in a neighbourhood Note: The Linear in Linear Filters means linear combination o neighboring pixel values. Freeman, MIT

3 Linear Filtering We don t want to only do this at a single pixel, o course, but want instead to run the kernel over the whole image. think o this as a weighted sum (kernel speciies the weights): *+*+*+*+*.+*+*+*+*. = Freeman, MIT Linear Filtering Freeman, MIT Convolution (D) Convolution Given a kernel (template) and image h, the convolution is deined as (p,q) (u,v) ) Note strange indexing into neighborhood o h(x,y). As a h(u,v) result, behaves as i rotated by degrees beore combining with h. (u,v) h(x-u,y-v) (x-p,y-q) (-p,-q) ) That doesn t matter i has deg symmetry ) I it *does* matter, use cross correlation instead. Integral o red area is the convolution or this x and y h(x-u,y-v) h(-u,-v) Adapted rom Roseneld and Kak, Digital Picture Processing h *+*+(-)*+* = adapted rom C. Rasmussen, U. o Delaware

4 *+*+(-)*+* = -*+*+*-*-*+*= *+*+(-)*+* = -*+*+*-*-*+*= -*+*+*-*-*+*= *+*+*-*-*+*= -*+*+*-*-*+*= -*+*-*-* = *+*+(-)*+* = - - *+*+(-)*+* = -*+*+*-*-*+*= -*+*+*-*-*+*= -*+*-*-* = - *+*+*+*-*+*= and so on *+*+(-)*+* = -*+*+*-*-*+*= -*+*+*-*-*+*= -*+*-*-* = - *+*+*+*-*+*= *+*+*-*+*+*-*-*+*= From C. Rasmussen, U. o Delaware

5 Problem: what do we do or border pixels where the kernel does not completely overlap the image Other methods... Replication replace each o-image pixel with the value rom the nearest pixel that IS in the image. Example: s Dierent border handling methods speciy dierent ways o deining values or pixels that are o the image. One o the simplest methods is zero-padding, which we used by deault in the earlier example.... Other methods... Relection relect pixel values at the border (as i there was a little mirror there) s Other methods... Wraparound when going o the right border o the image, you wrap around to the let border. Similarly, when leaving the bottom o the image you reenter at the top. Basically, the image is a big donut (or torus). s Example: s but what values do we use or pixels that are o the image or interior pixels where there is ull overlap, we know what to do. Example: Convolution in Matlab Could use conv and conv, but newer versions use: Imilter(image,template{,option,option, }) Boundary options: constant, symmetric, replicate, circular Output size options: same as image, or ull size (includes partial values computed when mask is o the image). Corr or conv option: convolution rotates the template (as we have discussed). Correlation does not). Type help imilter on command line or more details

6 Properties o Convolution Commutative: * g = g * Associative: ( * g) * g = * (g * h) Distributive: ( + g) * h = * h + g * h Linear: (a + b g) * h = a * h + b g * h Shit Invariant: (x+t) * h = ( * h)(x+t) Dierentiation rule: Linear Filtering I O Filter F O NxM output image I NxM input image F (m+)x(m+) mask a b c e.g. F= O(i, j ) You will see some o these again! h m d e g h i or m= k m I (i h, j k ) F (h, k ) h m k m O.Camps, PSU Back to the Gradient (-) * I, + () * I, + (+) * I, I, I, Finite Dierences computed using convolution kernels Vertical Edges: I, I, I, I, I, - Finite Dierence Filters Convolve with: + I, I, I, I, I, - I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, Horizontal Edges: Convolve with: - Question or class: Is this correct O.Camps, PSU Example: Spatial Image Gradients Ix=dI(x,y)/dx I(x,y) Partial derivative wrt x Iy=dI(x,y)/dy Partial derivative wrt y Note that there is a dierence between convolving with a xn row ilter and an nx col ilter.