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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

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COMP 546

Lecture 14

Maximum likelihood models

  • Tues. Feb. 27, 2018
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Overview of today

  • Informal notion of likelihood
  • Formal definition of likelihood as conditional

probability

  • Maximum likelihood problems (sketch)

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Scene Image Estimated Scene

image formation vision

luminance

  • rientation

disparity motion surface slant, tilt … image intensity filter responses

S = s 𝐽 = 𝑗 S = 𝑑

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luminance

  • rientation

disparity motion surface slant, tilt …

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Task: detecting an intensity increment

percent correct

𝐽0 𝐽0 + βˆ†π½

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𝐽0 𝐽0 + βˆ†π½

likelihood of intensity in center (solid) and background (dashed)

𝐽0 𝐽0 + βˆ†π½

If βˆ†π½ is small and noise is big, then the task becomes more difficult.

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Task: estimate orientation

likelihood of

  • rientation

πœ„

90

πœ„

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Task: estimate velocity of black dots

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𝑀𝑦 𝑀𝑧

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𝑀𝑦 𝑀𝑧

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𝑀𝑦 𝑀𝑧

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Task: estimate disparity of patch

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left eye right eye

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We could add noise by independently randomizing B&W value of bits in left and right images.

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likelihood of disparities of background (dashed) and central (solid) square

π‘’π‘‘π‘“π‘œπ‘’π‘“π‘  π‘’π‘π‘π‘‘π‘™π‘•π‘ π‘π‘£π‘œπ‘’

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Task: estimate surface slant

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Is this an ellipse on a frontoparallel plane,

  • r a disk on a slanted plane?
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Task: estimate the slant from texture

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Random distribution of disk shapes and sizes (rather than pixel noise).

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likelihood

  • f slant Οƒ

Οƒ Οƒ

[Knill, 1998]

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What is the formal definition of β€œlikelihood” ?

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The conditional probability π‘ž 𝐽 = 𝑗 𝑇 = 𝑑 ) is known as the β€œlikelihood” of 𝑇 = 𝑑, for a given image 𝐽 = 𝑗.

Likelihood

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𝑑

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Maximum likelihood estimation:

Given an image 𝐽 = 𝑗, choose the scene 𝑇 = 𝑑 that maximizes π‘ž(𝐽 = 𝑗 | 𝑇 = 𝑑) .

𝑑

S value that that maximizes likelihood

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Maximum likelihood estimation in vision

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image intensity filter responses luminance

  • rientation

disparity motion surface slant, tilt …

Image I = 𝑗 Estimated S = 𝑑

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Task: estimate 𝐽0, 𝐽0 + βˆ†π½ in presence of noise

𝐽0 𝐽0 + βˆ†π½

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Additive Gaussian noise π‘œ : mean 0 and variance πœπ‘œ2 .

Task: estimate 𝐽0, 𝐽0 + βˆ†π½ in presence of noise

𝐽0 𝐽0 + βˆ†π½ π½π‘‘π‘“π‘œπ‘’π‘“π‘  𝑦, 𝑧 = 𝐽0 + βˆ†π½ + π‘œ(𝑦, 𝑧) π½π‘‘π‘£π‘ π‘ π‘π‘£π‘œπ‘’ 𝑦, 𝑧 = 𝐽0 + π‘œ(𝑦, 𝑧)

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Let’s define a likelihood function for 𝐽0 :

π½π‘‘π‘£π‘ π‘ π‘π‘£π‘œπ‘’ 𝑦, 𝑧 = 𝐽0 + π‘œ(𝑦, 𝑧)

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π‘ž π½π‘‘π‘£π‘ π‘ π‘π‘£π‘œπ‘’(𝑦, 𝑧) 𝐽0 ) = π‘ž π‘œ 𝑦, 𝑧

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π‘ž π½π‘‘π‘£π‘ π‘ π‘π‘£π‘œπ‘’(𝑦, 𝑧) 𝐽0 ) = π‘ž π‘œ 𝑦, 𝑧 π½π‘‘π‘£π‘ π‘ π‘π‘£π‘œπ‘’ 𝑦, 𝑧 βˆ’ 𝐽0 π½π‘‘π‘£π‘ π‘ π‘π‘£π‘œπ‘’ 𝑦, 𝑧 = 𝐽0 + π‘œ(𝑦, 𝑧)

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π‘ž π½π‘‘π‘£π‘ π‘ π‘π‘£π‘œπ‘’(𝑦, 𝑧) 𝐽0 ) = π‘ž π‘œ 𝑦, 𝑧 1 2πœŒπœπ‘œ 𝑓

βˆ’π‘œ(𝑦,𝑧)2 2 πœπ‘œ2 Gaussian pixel noise

π½π‘‘π‘£π‘ π‘ π‘π‘£π‘œπ‘’ 𝑦, 𝑧 βˆ’ 𝐽0 π½π‘‘π‘£π‘ π‘ π‘π‘£π‘œπ‘’ 𝑦, 𝑧 = 𝐽0 + π‘œ(𝑦, 𝑧)

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Independent Random Variables

Two random variables π‘Œ1 and π‘Œ1 are independent if, for all values 𝑦1 and 𝑦2, π‘ž( π‘Œ1= 𝑦1, π‘Œ2 = 𝑦2) = π‘ž( π‘Œ1 = 𝑦1) π‘ž(π‘Œ2 = 𝑦2) The same definition holds for many random variables. The example here is pixel noise.

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π‘ž π½π‘‘π‘£π‘ π‘ π‘π‘£π‘œπ‘’ 𝐽0 ) =

(𝑦,𝑧)

1 2πœŒπœπ‘œ 𝑓

βˆ’ π½π‘‘π‘£π‘ π‘ π‘π‘£π‘œπ‘’ 𝑦,𝑧 βˆ’ 𝐽0

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2 πœπ‘œ2

Likelihood for 𝐽0 for all pixels 𝑦, 𝑧 in the surround:

π½π‘‘π‘£π‘ π‘ π‘π‘£π‘œπ‘’ 𝑦, 𝑧 = 𝐽0 + π‘œ(𝑦, 𝑧)

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𝐽0 𝑂 = 400 noise samples, 𝐽0 = 2, πœπ‘œ = 3 π‘ž( 𝐽 𝑦, 𝑧 | 𝐽0 )

Maximum likelihood estimate is an approximation only.

See exercises. http://www.cim.mcgill.ca/~langer/546/MATLAB/likelihood.m

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Task: estimate disparity of patch

p(π‘ π‘“π‘‘π‘žπ‘π‘œπ‘‘π‘“π‘‘ 𝑦, 𝑧, π‘’π‘’π‘£π‘œπ‘“π‘’ = 𝑠 | π‘’π‘—π‘‘π‘žπ‘π‘ π‘—π‘’π‘§ = 𝑒 )

It not obvious how to write down such a function.

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Task: estimate slant of surface

p( 𝐽 = 𝑗 | 𝑇 = 𝑑 )

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Given a set of image ellipses, 𝐽 = 𝑗, and assuming some probability distribution

  • f disk shapes on the surface, define the likelihood of different surface slants

𝑇 = 𝑑. (For details, see papers by David Knill in 1990’s.)

[Knill, 1998]

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The above examples take an β€œideal observer” approach. Can we model a human observer’s uncertainty, using a likelihood function ?

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100% 50% 0% Psychometric function (fit with cumulative Gaussian i.e. blurred step edge) Model of likelihood (Gaussian shape with mean s, standard deviation βˆ†s) S

s-βˆ†s s s+βˆ†s

S 75%

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25%

s-βˆ†s s s+βˆ†s

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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)

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