Defining statistical perceptions with an empirical Bayesian approach.

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An idea for studying human texture perception in a probabilistic framework. Reference: S. Tajima (2013) Defining statistical perceptions with an empirical Bayesian approach. Physical Review E, 87(4):042707. https://sites.google.com/site/satohirotajima/

Technology

Defining statistical perceptions
with an empirical Bayes approach
Satohiro Tajima

Perception of image statistics is an empirical Bayes estimation.

Stimulus statistics?

Contrast

Smoothness

1

α

α=1.284
α=1.122

α=1.201

Stimulus statistics = cue for recognition

1

α

α=1.284
α=1.122

α=1.201

Stimulus statistics = cue for recognition
Texture

(Hansen & Hess, J. Vis., 2006)

Blur

(Liu et al., CVPR., 2008)

Scene category

(Torralba & Oliva, Network, 2003)

Idea

Image engineering “Image restoration”
Empirical Bayes

Vision science “Perception of stimulus statistics”

Problem in perceiving stimulus statistics

Statistics
(smoothness)

Stimulus
(image)

θ
Stochasticity

Neural
response

s

“Percepts”

r
Noise

^

^
s
Estimate

θ

Bayesian framework
Statistics Stimulus Response

θ

θ

s

s

Goal of:

r

Image
restoration

r

Visual
recognition

Bayesian framework
Statistics Stimulus Response
“hyperparameter”

θ

s

r
Bayes
Empirical Bayes

θ

s

r

Bayesian framework
Statistics Stimulus Response

θ

s

r

θ

s

r

Different goals of estimation
Image
restoration

Visual
recognition

Statistics

Stimulus
for

Stimulus

Statistics
for

… But both are mathematically equal manipulations.

Different criteria
Image
restoration

Visual
recognition

Mean square error
E [ s

ˆ
s

2

/N]

Variance of
estimate

Fisher information
E [-

ln P ( r | θ )]

ˆ ] -1
Var [

(Ideal observer)

Different criteria
Image
restoration

Visual
recognition

Mean square error
E [ s

ˆ
s

2

/N]
Signal detection
theory

Fisher information
E [-

ln P ( r | θ )]

(d ' )

(Ideal observer)

2

Decoding retinal codes
θ

s

r

Statistics
Cortex

θ

Estimate (Percept)

Natural image statistics
• Smoothness
• Contrast

Statistics

θ

Cortex

Neural response model

Receptive field
Statistics

θ

Cortex

Estimation of stimulus

Receptive field

s

As

r

^

s

(Tajima, Inoue & Okada, J. Phys. Soc. Jpn., 2008)

Estimation of statistics

- ｌｎ P(r|θ)

Receptive field

(Tajima, Inoue & Okada, J. Phys. Soc. Jpn., 2008)

Optimal receptive field size?
Image restoration
Criteria:

Visual recognition

Mean square error

Fisher information
of smoothness

(Noise level)

Receptive field size

Receptive field size

(Natural image model: Power-law model)

Receptive field size

Optimal receptive field size?

(Natural image model: Power-law model)

Receptive field size

Optimal receptive field size?

Intensity-dependent RF changes:
Retina (Barlow et al., 1957)
V1 (Polat & Norcia, 1996)
MT (Hunter & Born, 2011)

Receptive field size

Optimal receptive field size?

Parasol

Bistratified
Midget

Retinal ganglion cells

Which cell type is the best?

Receptive field
Parasol
(Magno)

SIZE

Midget
(Parvo)

SHAPE

Bistratified
(Konio)

(Field et al., Nature, 2010)

Parasol (Magno)
Midget (Parvo)
Bistratified (Konio)

Fisher information

Which cell type is the best? - (prediction)

(Natural image model: Power-law model)

Message

Image engineering “Image restoration”
Empirical Bayes!

Vision science “Perception of stimulus statistics”

Vision science  Image engineering

What criteria should we use?
Purpose:

•

Mean square error Restoration

•

Fisher information

Human recognition

Image engineering  Vision science

What is the function of statistics perception?
Rolls of statistics:

•

Compression

Cue for recognition

•

Denoising

Prior for stimulus estimation

•

Prediction

Hidden variables of system

Perception of image statistics is empirical Bayes estimation.

Satohiro Tajima.
Defining statistical perceptions with an empirical Bayesian approach.
Physical Review E, 87(4):042707, (2013).
https://sites.google.com/site/satohirotajima/

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Defining statistical perceptions with an empirical Bayesian approach.

1. Defining statistical perceptions with an empirical Bayes approach Satohiro Tajima

2. Perception of image statistics is an empirical Bayes estimation.

3. Stimulus statistics? Contrast Smoothness 1 α α=1.284 α=1.122 α=1.201

4. Stimulus statistics = cue for recognition 1 α α=1.284 α=1.122 α=1.201

5. Stimulus statistics = cue for recognition Texture (Hansen & Hess, J. Vis., 2006) Blur (Liu et al., CVPR., 2008) Scene category (Torralba & Oliva, Network, 2003)

6. We can perceive statistics!

7. Idea Image engineering “Image restoration” Empirical Bayes Vision science “Perception of stimulus statistics”

8. Problem in perceiving stimulus statistics Statistics (smoothness) Stimulus (image) θ Stochasticity Neural response s “Percepts” r Noise ^ ^ s Estimate θ

9. Bayesian framework Statistics Stimulus Response θ θ s s Goal of: r Image restoration r Visual recognition

10. Bayesian framework Statistics Stimulus Response “hyperparameter” θ s r Bayes Empirical Bayes θ s r

11. Bayesian framework Statistics Stimulus Response θ s r θ s r

12. Different goals of estimation Image restoration Visual recognition Statistics Stimulus for Stimulus Statistics for … But both are mathematically equal manipulations.

13. Different criteria Image restoration Visual recognition Mean square error E [ s ˆ s 2 /N] Variance of estimate Fisher information E [- ln P ( r | θ )] ˆ ] -1 Var [ (Ideal observer)

14. Different criteria Image restoration Visual recognition Mean square error E [ s ˆ s 2 /N] Signal detection theory Fisher information E [- ln P ( r | θ )] (d ' ) (Ideal observer) 2

15. Application

16. Decoding retinal codes

17. Decoding retinal codes θ s r Statistics Cortex θ Estimate (Percept)

18. Natural image statistics • Smoothness • Contrast Statistics θ Cortex

19. Neural response model Receptive field Statistics θ Cortex

20. Estimation of stimulus Receptive field s As r ^ s (Tajima, Inoue & Okada, J. Phys. Soc. Jpn., 2008)

21. Estimation of statistics - ｌｎ P(r|θ) Receptive field (Tajima, Inoue & Okada, J. Phys. Soc. Jpn., 2008)

22. Optimal receptive field size? Image restoration Criteria: Visual recognition Mean square error Fisher information of smoothness (Noise level) Receptive field size Receptive field size (Natural image model: Power-law model)

23. Receptive field size Optimal receptive field size? (Natural image model: Power-law model)

24. Receptive field size Optimal receptive field size? Intensity-dependent RF changes: Retina (Barlow et al., 1957) V1 (Polat & Norcia, 1996) MT (Hunter & Born, 2011)

25. Receptive field size Optimal receptive field size? Parasol Bistratified Midget Retinal ganglion cells

26. Which cell type is the best? Receptive field Parasol (Magno) SIZE Midget (Parvo) SHAPE Bistratified (Konio) (Field et al., Nature, 2010)

27. Parasol (Magno) Midget (Parvo) Bistratified (Konio) Fisher information Which cell type is the best? - (prediction) (Natural image model: Power-law model)

28. Parasol (Magno) Midget (Parvo) Bistratified (Konio) Fisher information Which cell type is the best? - (prediction) (Natural image model: Power-law model)

29. Implementation of empirical Bayes?

30. Message Image engineering “Image restoration” Empirical Bayes! Vision science “Perception of stimulus statistics”

31. Vision science  Image engineering What criteria should we use? Purpose: • Mean square error Restoration • Fisher information Human recognition

32. Image engineering  Vision science What is the function of statistics perception? Rolls of statistics: • Compression Cue for recognition • Denoising Prior for stimulus estimation • Prediction Hidden variables of system

33. Perception of image statistics is empirical Bayes estimation.

34. Satohiro Tajima. Defining statistical perceptions with an empirical Bayesian approach. Physical Review E, 87(4):042707, (2013). https://sites.google.com/site/satohirotajima/

Defining statistical perceptions with an empirical Bayesian approach.

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