# Image features: Histograms, Aliasing, Filters, Orientation and HOG. D.A. Forsyth

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1 Image features: Histograms, Aliasing, Filters, Orientation and HOG D.A. Forsyth

2 Simple color features Histogram of image colors in a window Opponent color representations R-G B-Y=B-(R+G)/2 Intensity=(R+G+B)/3 Blue pixel map Percentage of blue pixels

3 Matlab slide

4 Scaled representations Represent one image with many different resolutions Why? find bigger, smaller swimming pools

5 Obtained pyramid of images by subsampling Carelessness causes aliasing

6 Matlab slide: subsampling

7 Aliasing and fast changing signals

8 More aliasing examples Undersampled sine wave -> Color shimmering on striped shirts on TV Wheels going backwards in movies temporal aliasing

9 Another aliasing example location of a sharp change is known poorly

10 Fundamental facts A sine wave will alias if sampled less often than twice per period

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19 Fundamental facts Sample(A+B)=Sample(A)+Sample(B) if a signal contains a high frequency sine wave, it will alias

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23 Weapons against aliasing Filtering or smoothing take the signal, reduce the fast-changing/high-frequency content can do this by weighted local averaging

24 Prefiltering (Ideal case) Continuous Image Filter Sampling Discrete Samples Reconstruction Continuous Image 1 1 p Sampling Function p Reconstruction Kernel

25 Smoothing by Averaging N ij = 1 N Σ uvo i+u,j+v where u, v, is a window of N pixels in total centered at 0, 0

26 Smoothing with a Gaussian Notice ringing apparently, a grid is superimposed Smoothing with an average actually doesn t compare at all well with a defocussed lens what does a point of light produce? A Gaussian gives a good model of a fuzzy blob

27 Gaussian filter kernel K uv = 1 u 2 + v 2 2πσ 2 exp 2σ 2 We re assuming the index can take negative values

28 Smoothing with a Gaussian N ij = uv O i u,j v K uv Notice the curious looking form

29 Matlab slide: convolution in 2D

30 Linear Filters Example: smoothing by averaging form the average of pixels in a neighbourhood Example: smoothing with a Gaussian form a weighted average of pixels in a neighbourhood Example: finding a derivative form a weighted average of pixels in a neighbourhood

31 Finding derivatives N ij = 1 x (I i+1,j I ij )

32 Convolution Each of these involves a weighted sum of image pixels The set of weights is the same we represent these weights as an image, H H is usually called the kernel Operation is called convolution it s associative Any linear shift-invariant operation can be represented by convolution linear: G(k f)=k G(f) shift invariant: G(Shift(f))=Shift(G(f)) Examples: smoothing, differentiation, camera with a reasonable, defocussed lens system N ij = uv H uv O i u,j v

33 Filters are templates N ij = uv H uv O i u,j v At one point output of convolution is a (strange) dot-product Filtering the image involves a dot product at each point Insight filters look like the effects they are intended to find filters find effects they look like

34 Smoothing reduces noise Generally expect pixels to be like their neighbours surfaces turn slowly relatively few reflectance changes Expect noise to be independent from pixel to pixel Scale the parameter in the symmetric Gaussian as this parameter goes up, more pixels are involved in the average and the image gets more blurred and noise is more effectively suppressed Implies that smoothing suppresses noise, for appropriate noise models K uv = 1 2πσ 2 exp u 2 + v 2 2σ 2

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36 Representing image changes: Edges Idea: points where image value change very sharply are important changes in surface reflectance shadow boundaries outlines Finding Edges: Estimate gradient magnitude using appropriate smoothing Mark points where gradient magnitude is Locally biggest and big

37 Matlab slide: gradients

38 Matlab slide: smoothed gradients

39 Scale affects derivatives 1 pixel 3 pixels 7 pixels

40 Scale affects gradient magnitude

41 Smoothing and Differentiation Issue: noise smooth before differentiation two convolutions to smooth, then differentiate? actually, no - we can use a derivative of Gaussian filter

42 Matlab slide: orientations and arrow plots

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45 Matlab slide: rose plots

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47 Hog features Take a window subdivide into boxes, each with multiple pixels these might overlap for each box, build a histogram of gradient orientations possibly weighting by distance from center possibly normalizing by intensity over the box string these histograms together to a vector Extremely strong at spatial coding

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49 Vlfeat pointer

50 Image HOG features Positive terms in linear classifier Negative terms in linear classifier

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