COMP 546 Lecture 14 Maximum likelihood models Tues. Feb. 27, 2018 1
Overview of today • Informal notion of likelihood • Formal definition of likelihood as conditional probability • Maximum likelihood problems (sketch) 2
Scene Image Estimated Scene image formation vision S = s 𝐽 = 𝑗 S = 𝑡 luminance image intensity luminance orientation filter responses orientation disparity disparity motion motion surface slant, tilt surface slant, tilt … … 3
Task: detecting an intensity increment 𝐽 0 𝐽 0 + ∆𝐽 percent correct 4
If ∆𝐽 is small and noise is big, then the task becomes more difficult. 𝐽 0 𝐽 0 + ∆𝐽 likelihood of intensity in center (solid) and background (dashed) 5 𝐽 0 𝐽 0 + ∆𝐽
Task: estimate orientation likelihood of orientation 𝜄 𝜄 6 90
Task: estimate velocity of black dots 7
𝑤 𝑧 𝑤 𝑦 8
𝑤 𝑧 𝑤 𝑦 9
𝑤 𝑧 𝑤 𝑦 10
Task: estimate disparity of patch 11
left eye right eye We could add noise by independently randomizing B&W value of bits in left and right images. 12
likelihood of disparities of background (dashed) and central (solid) square 𝑒 𝑑𝑓𝑜𝑢𝑓𝑠 𝑒 𝑐𝑏𝑑𝑙𝑠𝑝𝑣𝑜𝑒 13
Task: estimate surface slant Is this an ellipse on a frontoparallel plane, or a disk on a slanted plane? 14
Task: estimate the slant from texture Random distribution of disk shapes and sizes (rather than pixel noise). 15
[Knill, 1998] likelihood of slant σ σ σ 0 0 16
What is the formal definition of “likelihood” ? 17
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Likelihood The conditional probability 𝑞 𝐽 = 𝑗 𝑇 = 𝑡 ) is known as the “likelihood” of 𝑇 = 𝑡, for a given image 𝐽 = 𝑗 . 𝑡 25
Maximum likelihood estimation: Given an image 𝐽 = 𝑗 , choose the scene 𝑇 = 𝑡 that maximizes 𝑞(𝐽 = 𝑗 | 𝑇 = 𝑡 ) . S value that that maximizes likelihood 𝑡 26
Maximum likelihood estimation in vision Image I = 𝑗 Estimated S = 𝑡 image intensity luminance filter responses orientation disparity motion surface slant, tilt … 27
Task: estimate 𝐽 0, 𝐽 0 + ∆𝐽 in presence of noise 𝐽 0 𝐽 0 + ∆𝐽 28
Task: estimate 𝐽 0, 𝐽 0 + ∆𝐽 in presence of noise 𝐽 0 𝐽 0 + ∆𝐽 Additive Gaussian noise 𝑜 : mean 0 and variance 𝜏 𝑜2 . 𝐽 𝑑𝑓𝑜𝑢𝑓𝑠 𝑦, 𝑧 = 𝐽 0 + ∆𝐽 + 𝑜(𝑦, 𝑧) 𝐽 𝑡𝑣𝑠𝑠𝑝𝑣𝑜𝑒 𝑦, 𝑧 = 𝐽 0 + 𝑜(𝑦, 𝑧) 29
𝐽 𝑡𝑣𝑠𝑠𝑝𝑣𝑜𝑒 𝑦, 𝑧 = 𝐽 0 + 𝑜(𝑦, 𝑧) Let’s define a likelihood function for 𝐽 0 : 𝑞 𝐽 𝑡𝑣𝑠𝑠𝑝𝑣𝑜𝑒 (𝑦, 𝑧) 𝐽 0 ) = 𝑞 𝑜 𝑦, 𝑧 30
𝐽 𝑡𝑣𝑠𝑠𝑝𝑣𝑜𝑒 𝑦, 𝑧 = 𝐽 0 + 𝑜(𝑦, 𝑧) 𝐽 𝑡𝑣𝑠𝑠𝑝𝑣𝑜𝑒 𝑦, 𝑧 − 𝐽 0 𝑞 𝐽 𝑡𝑣𝑠𝑠𝑝𝑣𝑜𝑒 (𝑦, 𝑧) 𝐽 0 ) = 𝑞 𝑜 𝑦, 𝑧 31
𝐽 𝑡𝑣𝑠𝑠𝑝𝑣𝑜𝑒 𝑦, 𝑧 = 𝐽 0 + 𝑜(𝑦, 𝑧) 𝐽 𝑡𝑣𝑠𝑠𝑝𝑣𝑜𝑒 𝑦, 𝑧 − 𝐽 0 𝑞 𝐽 𝑡𝑣𝑠𝑠𝑝𝑣𝑜𝑒 (𝑦, 𝑧) 𝐽 0 ) = 𝑞 𝑜 𝑦, 𝑧 −𝑜(𝑦,𝑧) 2 1 2 𝜏 𝑜2 𝑓 Gaussian pixel noise 2𝜌𝜏 𝑜 32
Independent Random Variables Two random variables 𝑌 1 and 𝑌 1 are independent if, for all values 𝑦 1 and 𝑦 2 , 𝑞( 𝑌 1 = 𝑦 1 , 𝑌 2 = 𝑦 2 ) = 𝑞( 𝑌 1 = 𝑦 1 ) 𝑞(𝑌 2 = 𝑦 2 ) The same definition holds for many random variables. The example here is pixel noise. 33
𝐽 𝑡𝑣𝑠𝑠𝑝𝑣𝑜𝑒 𝑦, 𝑧 = 𝐽 0 + 𝑜(𝑦, 𝑧) Likelihood for 𝐽 0 for all pixels 𝑦, 𝑧 in the surround: 2 − 𝐽 𝑡𝑣𝑠𝑠𝑝𝑣𝑜𝑒 𝑦,𝑧 − 𝐽 0 1 2 𝜏 𝑜2 𝑞 𝐽 𝑡𝑣𝑠𝑠𝑝𝑣𝑜𝑒 𝐽 0 ) = 𝑓 2𝜌𝜏 𝑜 (𝑦,𝑧) 34
𝑂 = 400 noise samples, 𝐽 0 = 2, 𝜏 𝑜 = 3 Maximum likelihood estimate is an | 𝐽 0 ) approximation only. 𝑞( 𝐽 𝑦, 𝑧 𝐽 0 See exercises. http://www.cim.mcgill.ca/~langer/546/MATLAB/likelihood.m 35
Task: estimate disparity of patch p(𝑠𝑓𝑡𝑞𝑝𝑜𝑡𝑓𝑡 𝑦, 𝑧, 𝑒 𝑢𝑣𝑜𝑓𝑒 = 𝑠 | 𝑒𝑗𝑡𝑞𝑏𝑠𝑗𝑢𝑧 = 𝑒 ) It not obvious how to write down such a function. 36
Task: estimate slant of surface p( 𝐽 = 𝑗 | 𝑇 = 𝑡 ) Given a set of image ellipses, 𝐽 = 𝑗 , and assuming some probability distribution of disk shapes on the surface , define the likelihood of different surface slants 𝑇 = 𝑡 . (For details, see papers by David Knill in 1990’s.) [Knill, 1998] 37
The above examples take an “ideal observer” approach. Can we model a human observer’s uncertainty, using a likelihood function ? 38
Psychometric function (fit with cumulative Gaussian i.e. blurred step edge) 100% 75% 50% 25% 0% S s- ∆ s s s+ ∆ s Model of likelihood (Gaussian shape with mean s, standard deviation ∆ s) S 39 s- ∆ s s s+ ∆ s
Q: How can a such a likelihood model explain or predict how a vision system estimates a scene parameter? A: It can tell us how people combine different cues. (Next lecture) 40
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