Bayesian inference

accounts for the

filling-in and suppression of visual perception of bars by context

Li Zhaoping1 & Li Jingling2

1University College London, 2China Medical School

Based on a publication Zhaoping & Jingling (2008) PLoS Comput. Biol.Ask me for a copy, or download from www.cs.ucl.ac.uk/staff/Zhaoping.Li/allpaper.html

Low response from V1 cells to a low

contrast bar

Higher responses from V1 cells when there is a colinear flanker

Test bar within the receptive field of a V1 cell

Contextual bar outside the receptive field

Some previously known contextual influences in vision

Colinear facilitation in V1

Nelson & Frost 1985, Kapadia et al 1995, Li, Piech, & Gilbert 2006

Human sensitivity to detect target bar enhanced by colinear flankers.

Polat & Sagi 1993, Morgan & Dresp 1995, Yu & Levi 2000, Huang et al 2006 etc

Location for the target bar to appear if it does.

Contextual bar

1st presentation interval

2st presentation interval

Humans are better at saying which interval (the 2AFC task)contains the target with the colinear contextual bars

A demo of the perception in this study: presence or absence of a vertical target in four different contexts

Bar seems present

Bar seems absent

Bar present?

Bar absent?

These three combined are contrary to what one may expect from the colinearfacilitation in V1 cells and human sensitivity for bar detection WHY???

Hints from another example of how context influence visual perception

The same input of a white patch on retina

Different perceptions inside the brain

Perception fills in the occluded part of the square

Focus of this study: Contextual influence in perceptual bias, not in input sensitivity, in visual object inference not in input image representation

Methods in the study:

Psychophysical: rather than the 2AFC method, we used one-interval method (observers answered after each one-interval stimulus presentation whether the target bar was present) to probe perceptual bias (rather than input sensitivity). Whether the perception is veridical is not an issue, since we study inference rather than input representation.

Computational: build a Bayesian inference model to understand the psychophysical data, showing that the model fits the data with fewer parameters than needed by phenomenological (e.g., logistic) models of the psychometric functions.

w

ithou

t con

text

Psychophysical investigation Ask an observer (with one interval presentation):

Is the vertical bar present? Answer: yes or no?

Target contrast Ct

P(yes| Ct)

1

Targets without context

contextual

effect

Yes rate, or psychometric function P(yes|Ct): Probability an observer answering “yes” given contrast Ct of the target bar in image

0

With context

Targets with context

Observations in an experiment: Perception of the target bar more likely when contextual contrast Cc is low

Contextual contrast Cc is lower,Yes rate P(yes|Ct) is larger

Contextual contrast Cc is higher,Yes rate P(yes|Ct) is smaller

Observations contrary to expectation, since in V1 (primary visual cortex), a neuron’s response is facilitated by the presence of colinear contextual bars. So stronger colinear context should facilitate more.

Experiment randomly interleaved trials of different target contrast Ct and contextual contrast Cc, including Cc=0 for the no context condition.

No target in image

Seeing

ghost?

target with contrast Ct

Not seen without context

seen without context

Seeing less

Perceptual suppression

Data

Perceptual

filling-in

P(yes|Ct)

In dimvague context with low Cc

Strong context with high C c

No conte

xtIn

medium

conte

xt with

C c

Two contextual configuations: colinear and orthogonal

TargetLess visible

TargetLess visible

Colinear context Orthogonal context

Two contextual configurations: colinear and orthogonal

Bright context Cc = 0.4

Dim context Cc = 0.01Colinear context

Orthogonal context

Data

suppresses perception regardless of contextual configuration

Filling-in only in co-linear contextual configuration

colinear

orthogonal

No context

No context

colinear

orthogonal

No context

No context

Dim context

Dim co

ntext

Bright c

ontext

Bright c

ontext

Again, trials of all contextual and target variations were randomly interleaved

Received visual signal: Ct: e.g.: neural activities in response

to the target bar or noise.

Making decision on: yes or no, the bar is there or not

Prior believed probabilities: P(yes), P(no) = 1-P(yes) of visual events “yes” or “no”.

Conditional Probability: P(Ct |yes), likelihood or evidence of likely contrast Ct for target present.

Decision probability: P(yes|Ct) = P(Ct|yes)P(yes)/P(Ct)

Understanding by Bayesian inference:

Decision: P(yes | Ct ) =

P(Ct |yes) P(yes)

P(Ct |yes) P(yes) + (1-P(yes)) P(Ct | no)

Note: the model above is derived from a neural level model, when input contrasts evoke neural responses, when the brain has an internal model of the likely neural responses to input contrast, and how neural responses to contextual inputs influences the priors and the “likelihood” models. Details of the derivations can be found in the published paper Zhaoping & Jingling 2008.

A Bayesian account: the priors P(yes) and the evidence (likelihood) P(Ct|yes)

Evidence P(Ct |yes) Larger Smaller

Larger prior P(yes) in aligned context

Smaller prior P(yes) in non-aligned context

Prior P(yes)Larger Smaller

When these different conditions are interleaved within the same experimental session, different priors manifest themselves in different trials ---- rapidly switching between priors!!!

Bayesian model formulation:

Context influences decision in two ways:

(1)Contextual configuration determines the prior prob. parameter P(yes)

(2) Contextual contrast Cc determines likelihood P(Ct|yes)

P(Ct |yes) ~ exp [ -|Ct –Cc|/ (k Cc) ] favouring targets that resemble context in contrast

3 Parameters: k, σn, P(yes) can completely model a given contextual configuration to give P(yes|Ct) for all Ct and Cc

P(Ct |yes) P(yes)

P(Ct |yes) P(yes) + (1-P(yes)) P(Ct | no) Decision: P(yes | Ct ) =

Additionally: P(Ct|no) ~ exp(-Ct/ σn) a model of noise contrast

Note: P(Ct|yes) is not the probability of the experimenter presenting a contrast C t for the target, nor is P(yes) the prob. of experimenter presenting a target. Both P(C t|yes) and P(yes) are internal models in the observer’s brain only, and “yes” and “no” refer the brain’s perceptions and assumptions rather than the external stimulus, see paper for more details.

Ct

Ct = Cc

P(Ct|no)

P(Ct|yes)

Evidence P(Ct|no) for non-target is higher when target contrast Ct is close to zero

Evidence P(Ct|yes) for target is higher for target contrast Ct resembling the contextual contrast Cc.

The decision P(yes|Ct) results from weighing the evidences P(Ct|yes) and P(Ct|no), for and against the target, weighted by the priors P(yes) and P(no)=1-P(yes)

Decision: P(yes | Ct ) =

P(Ct |yes) P(yes)

P(Ct |yes) P(yes) + (1-P(yes)) P(Ct | no)

Bayesian decision by combining input evidences with prior beliefs

Effect of contextual contrast Cc

Weaker Cc --- larger P(Ct | yes)

Stronger Cc

--- smaller P(Ct |yes)

Effect of prior P(yes)

Higher P(yes)

Lower P(yes)

Weaker contextual contrast Cc and/no higher prior P(yes) bias the response to “yes”

P(yes|Ct) P(yes|Ct)

The Bayesian model can fit the data well

No co-linear facilitation mechanism necessary for explaining the data!!!!If a 4th parameter for colinear facilitation is fitted, it returns a zero facilitation magnitude

Fitting data from a colinear context --- a total of 3 model parameters (k, P(yes), σn) to fit all 3 psychometric curves

Typically, 6 parameters would be needed to adequately fit 3 psychometric curves.Using only 3 parameters to fit the data, the Bayesian model demonstrates its adequacy.

Solid curves are the results of the Bayesian fit

Bayesian model fitting data from the exp. including both colinear and orthogonal contexts

Fitted Priors P(yes) Four model parameters: k, σn, P(yes)colinear, P(yes)orthogonal

used to fit all 4 psychometric curves

P(yes)colinear and P(yes)orthogonal are both quite big, reflecting an additional response bias by the subjects to respond roughly 50% “yes” in total.

Colinear context, data and fits Orthogonal context, data and fits

Comparing Bayesian and logistic fitting results:

Dashed curves: Logistic fits --- using 8 parametersSolid curves: Bayesian fits --- using 4 parameters

Mean Fitting Error in units of error bar size = 0.83 for logistic Mean Fitting Error in units of error bar size = 1.01 for Bayesian

Weak context Strong context

Another example: more subtle difference in context or P(yes)

Data fitting for 3 different contexts,3 different Contextual contrastsCc = 0.01, 0.05, 0.4.

5 Bayesian parameters,18 Logistic parameters.

Mean Fitting Error in units of error bar size = 0.54 for logistic fits (dashed curves) Mean Fitting Error in units of error bar size = 1.07 for Bayesian fits (solid curves)

Fitted Priors P(yes)

Summary:

Studied contextual influence in perceptual bias --- filling-in & suppression

Study uses simple stimuli, more easily controlled and modelled one interval tasks used to study bias rather than sensitivities.

Found context influences perception by (1) affecting prior expectation of perceptions (2) affecting likelihood model of sensory inputs

Findings (1) accountable by a Bayesian inference model (2) unexpected from colinear facilitation in V1, suggest mechanisms beyond V1

Other related works and issues:

Contextual influences in object recognition and attentional guidance

Contextual effects on mid-level vision assimilation and induction in the perception of motion, orientation, color, and lightness etc.

Effects of the input signal-to-noise on input encoding and perception.

Perceptual ambiguity

Relationship/difference between object inference and image representation

Bayesian inference in vision in many previous works, often with more complex stimuli (which can be difficult to manipulate and model)

Statistics of the natural scenes and adaptation

Decision making, and internal beliefs unchanged by input samples.

Etc, etc. … see detailed discussions in the published papers.

2AFC tasks remove the effects of the priors:

Visual Signal received: x1, x2, for time interval 1 and 2.

Making decision on: y

Prior expectation: P(y)

Conditional Probability: P(x1|y), P(x2|y)

Decision based on: P(y|x1) > ? < P(y|x2)

P(x1|y)P(y)/P(x1) > ? < P(x2|y)P(y)P(x2)