defining statistical perceptions with an empirical bayesian approach

Defining statistical perceptionswith an empirical Bayes approach

Satohiro Tajima

Perception of image statistics is an empirical Bayes estimation.

Stimulus statistics?

1

α

α=1.284

α=1.122

α=1.201

Contrast Smoothness

Stimulus statistics = cue for recognition

1

α

α=1.284

α=1.122

α=1.201

Stimulus statistics = cue for recognition

Scene categoryBlurTexture

(Hansen & Hess, J. Vis., 2006) (Liu et al., CVPR., 2008) (Torralba & Oliva, Network, 2003)

We can perceive statistics!

Idea

Image engineering

“Perception of stimulus statistics”

“Image restoration”

Empirical Bayes

Vision science

Problem in perceiving stimulus statistics

θ s r

Statistics(smoothness)

Stimulus(image)

Neural response

θs

EstimateStochasticity Noise

“Percepts”

θ s r

θ s r

Stimulus

Imagerestoration

Visualrecognition

Goal of:Bayesian framework

Statistics Response

Bayesian framework

θ s r

θ s r

“hyperparameter”StimulusStatistics Response

Bayes

Empirical Bayes

Bayesian framework

θ s r

θ s r

StimulusStatistics Response

Different goals of estimation

Imagerestoration

Visualrecognition

… But both are mathematically equal manipulations.

for

StimulusStatistics

for

Stimulus Statistics

Different criteria

Imagerestoration

Visualrecognition

Mean square error

]/ˆ[E2Nss

Fisher information

)]|(ln[E θrP- -1]ˆ[Var

Variance of estimate

(Ideal observer)

Different criteria

Imagerestoration

Visualrecognition

]/ˆ[E2Nss

Mean square error

Fisher information

)]|(ln[E θrP- 2)'( d

Signal detectiontheory

(Ideal observer)

Application

Decoding retinal codes

Decoding retinal codes

θStatistics

Cortex

Estimate (Percept)

θ s r

Natural image statistics

θStatistics

Cortex

• Smoothness• Contrast

Neural response model

θStatistics

Cortex

Receptive field

Estimation of stimulus

Receptive field

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

s As r s^

Estimation of statistics

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

SmoothnessCo

ntra

st

Receptive field

- ｌ

ｎ P

(r|θ

)

Optimal receptive field size?

(Natural image model: Power-law model)

Receptive field size

Mean square error

(Noise level)

Fisher informationof smoothness

Receptive field size

Criteria:

Image restoration Visual recognition

Optimal receptive field size?

(Natural image model: Power-law model)

Rec

eptiv

e fie

ld s

ize

Optimal receptive field size?

Intensity-dependent RF changes:

Retina (Barlow et al., 1957)V1 (Polat & Norcia, 1996)MT (Hunter & Born, 2011)

Rec

eptiv

e fie

ld s

ize

Optimal receptive field size?

Retinal ganglion cells

Midget

Parasol

Bistratified

Rec

eptiv

e fie

ld s

ize

Which cell type is the best?

Midget(Parvo)

Parasol(Magno)

Bistratified(Konio)

Receptive field

(Field et al., Nature, 2010)

SIZE SHAPE

Which cell type is the best? - (prediction)

(Natural image model: Power-law model)

Midget (Parvo)Parasol (Magno)

Bistratified (Konio)

Fis

her

info

rmat

ion

Which cell type is the best? - (prediction)

(Natural image model: Power-law model)

Midget (Parvo)Parasol (Magno)

Bistratified (Konio)

Fis

her

info

rmat

ion

Implementation of empirical Bayes?

Message

Image engineering

“Perception of stimulus statistics”

“Image restoration”

Empirical Bayes!

Vision science

Vision science Image engineering

• Mean square error• Fisher information Human recognition

Restoration

Purpose:

What criteria should we use?

Image engineering Vision science

• Compression• Denoising• Prediction

Prior for stimulus estimation

Hidden variables of system

Cue for recognition

Rolls of statistics:

What is the function of statistics perception?

Perception of image statistics is empirical Bayes estimation.

https://sites.google.com/site/satohirotajima/

Satohiro Tajima.Defining statistical perceptions with an empirical Bayesian approach.Physical Review E, 87(4):042707, (2013).