perception as inference
TRANSCRIPT
lens
ImageWorld Model
Perception as inference
Natural scenes are full of ambiguity
180 200 220 240
100
120
140
100
120
140
180 200 220 240
180 200 220 240
100
120
140
100
120
140
180 200 220 240180 200 220 240
100
120
140
100
120
140
180 200 220 240
Natural scenes are full of ambiguity
What do these edges mean?
What do these edges mean?
What do these edges mean?
What is this?
What do these edges mean?
offers a complete description.)Luminance, illuminance, and reflectance, are physical
quantities that can be measured by physical devices. Thereare also two subjective variables that must be discussed.
Lightness is defined as the perceived reflectance of a sur-face. It represents the visual system’s attempt to extractreflectance based on the luminances in the scene.
Brightness is defined as the perceived intensity of lightcoming from the image itself, rather than any property of theportrayed scene. Brightness is sometimes defined as per-ceived luminance.
These terms may be understood by reference to figure 24.7.The block is made of a 2x2 set of cubes, each colored eitherlight or dark gray. We call this the “checker- b l o c k . ”Illumination comes from an oblique angle, lighting differentfaces differently. The luminance image can be considered tobe the product of two other images: the reflectance imageand the illuminance image, shown below. These underlyingimages are termed intrinsic images in machine vision(Barrow and Tenenbaum, 1978). Intrinsic image decomposi-tions have been proposed for understanding lightness per-ception (Arend, 1994; Adelson and Pentland, 1996)
Patches p and q have the same reflectance, but differentluminances. Patches q and r have different reflectances andd i fferent luminances; they share the same illuminance.Patches p and r happen to have the same luminance, becausethe lower reflectance of p is counterbalanced by its higher
illuminance. Faces p and q appear to be painted with the same gray,
and thus they have the same lightness. However, it is clearthat p has more luminance than q in the image, and so thepatches differ in brightness. Patches p and r differ in bothlightness and brightness.
The problem of lightness constancy
From a physical point of view, the problem of lightness con-stancy is as follows. An illuminance image, E(x,y), and areflectance image, R(x,y), are multiplied to produce a lumi-nance image, L(x,y):
An observer is given L at each pixel, and attempts todetermine the two numbers E and R that were multiplied tomake it. Unfortunately, unmultiplying two numbers isimpossible. If E(x,y) and R(x,y) are arbitrary functions, thenfor any E(x,y) there exists an R(x,y) that produces theobserved image. The problem appears impossible, buthumans do it pretty well. This must mean that illuminanceand reflectance images are not arbitrary functions. They areconstrained by statistical properties of the world, as pro-posed by Land and McCann.
Note that Land and McCann’s constraints fail whenapplied to the checker-block image. Figure 24.8(a) showstwo light-dark edges. They are exactly the same in the
342 SENSORY SYSTEMS
FIGURE 24.6 Variants on the Koffka ring. (a) The ring appears aboutuniform. (b) When split, the two half-rings appear distinctly differ-ent. (c) When shifted, the two half-rings appear quite different. FIGURE 24.7 The “checker-block” and its analysis into two intrinsic
images.
L(x,y) = E(x,y)R(x,y).
offers a complete description.)Luminance, illuminance, and reflectance, are physical
quantities that can be measured by physical devices. Thereare also two subjective variables that must be discussed.
Lightness is defined as the perceived reflectance of a sur-face. It represents the visual system’s attempt to extractreflectance based on the luminances in the scene.
Brightness is defined as the perceived intensity of lightcoming from the image itself, rather than any property of theportrayed scene. Brightness is sometimes defined as per-ceived luminance.
These terms may be understood by reference to figure 24.7.The block is made of a 2x2 set of cubes, each colored eitherlight or dark gray. We call this the “checker- b l o c k . ”Illumination comes from an oblique angle, lighting differentfaces differently. The luminance image can be considered tobe the product of two other images: the reflectance imageand the illuminance image, shown below. These underlyingimages are termed intrinsic images in machine vision(Barrow and Tenenbaum, 1978). Intrinsic image decomposi-tions have been proposed for understanding lightness per-ception (Arend, 1994; Adelson and Pentland, 1996)
Patches p and q have the same reflectance, but differentluminances. Patches q and r have different reflectances andd i fferent luminances; they share the same illuminance.Patches p and r happen to have the same luminance, becausethe lower reflectance of p is counterbalanced by its higher
illuminance. Faces p and q appear to be painted with the same gray,
and thus they have the same lightness. However, it is clearthat p has more luminance than q in the image, and so thepatches differ in brightness. Patches p and r differ in bothlightness and brightness.
The problem of lightness constancy
From a physical point of view, the problem of lightness con-stancy is as follows. An illuminance image, E(x,y), and areflectance image, R(x,y), are multiplied to produce a lumi-nance image, L(x,y):
An observer is given L at each pixel, and attempts todetermine the two numbers E and R that were multiplied tomake it. Unfortunately, unmultiplying two numbers isimpossible. If E(x,y) and R(x,y) are arbitrary functions, thenfor any E(x,y) there exists an R(x,y) that produces theobserved image. The problem appears impossible, buthumans do it pretty well. This must mean that illuminanceand reflectance images are not arbitrary functions. They areconstrained by statistical properties of the world, as pro-posed by Land and McCann.
Note that Land and McCann’s constraints fail whenapplied to the checker-block image. Figure 24.8(a) showstwo light-dark edges. They are exactly the same in the
342 SENSORY SYSTEMS
FIGURE 24.6 Variants on the Koffka ring. (a) The ring appears aboutuniform. (b) When split, the two half-rings appear distinctly differ-ent. (c) When shifted, the two half-rings appear quite different. FIGURE 24.7 The “checker-block” and its analysis into two intrinsic
images.
L(x,y) = E(x,y)R(x,y).
reflectance shading (Adelson, 2000)
What are the letters?
What letter is this?
What is this?
What is this?
How are these different percepts represented in cortex?(Scott Murray - Ph.D. thesis)
BOLD signal: LOC vs. V1
“non-diamond”
“non-diamond”“diamond”
“diamond”
Time (s) Time (s)
MR
I Sig
nal
V1 LOC
BOLD signal in V1 and LOC
Picket-fence effect with speech(from Bregman ‘Auditory Scene Analysis’)
Sinewave speech
Please say what this word is
sill shook rust wed pass lark jaw
coop beak
Sinewave speech
Bayes’ rule
You observe y,what is x?
Simple example
y = x+ n
P (x|y) � P (y|x)P (x)
0 4 8 12 16 20 24
P
x, yobserved value
y = 14estimated value
x̂
prior
P (x) =1
Zxe� (x�µx)2
2�2x
<latexit sha1_base64="bOAisQ47RcwmY5A7sKVKgVGMJYM=">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</latexit><latexit sha1_base64="bOAisQ47RcwmY5A7sKVKgVGMJYM=">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</latexit><latexit sha1_base64="bOAisQ47RcwmY5A7sKVKgVGMJYM=">AAACIHicbZBNS8MwGMfT+Tbn29Sjl+IQtsNGO4R5EYZePE5wL7h2Jc3SLSxpS5JKR+lH8eJX8eJBEb3ppzHbetDNPwR++T/PQ/L83ZASIQ3jS8utrW9sbuW3Czu7e/sHxcOjjggijnAbBTTgPRcKTImP25JIinshx5C5FHfdyfWs3n3AXJDAv5PTENsMjnziEQSlspxio1WOK5eWxyFKzDS5d+IUD5LqwijHVYtFTlwZ1NOkbgkyYtCJ1SUtOMWSUTPm0lfBzKAEMrWc4qc1DFDEsC8RhUL0TSOUdgK5JIjitGBFAocQTeAI9xX6kGFhJ/MFU/1MOUPdC7g6vtTn7u+JBDIhpsxVnQzKsViuzcz/av1Iehd2QvwwkthHi4e8iOoy0Gdp6UPCMZJ0qgAiTtRfdTSGKhupMp2FYC6vvAqdes1UfHteal5lceTBCTgFZWCCBmiCG9ACbYDAI3gGr+BNe9JetHftY9Ga07KZY/BH2vcPGFKi7A==</latexit><latexit sha1_base64="bOAisQ47RcwmY5A7sKVKgVGMJYM=">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</latexit>
likelihood
P (y|x) = 1
Zne� (y�x)2
2�2n
<latexit sha1_base64="pVvY6Or+J6MJ9Bli1v3e/p9grwE=">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</latexit><latexit sha1_base64="pVvY6Or+J6MJ9Bli1v3e/p9grwE=">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</latexit><latexit sha1_base64="pVvY6Or+J6MJ9Bli1v3e/p9grwE=">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</latexit><latexit sha1_base64="pVvY6Or+J6MJ9Bli1v3e/p9grwE=">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</latexit>
P (x|y) � P (y|x)P (x)
How to compute ?
=1
Zne� (y�x)2
2�2n
1
Zxe� (x�µx)2
2�2x
<latexit sha1_base64="OIadET3ve5/rHOwzQHQfOj1Wlxk=">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</latexit><latexit sha1_base64="OIadET3ve5/rHOwzQHQfOj1Wlxk=">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</latexit><latexit sha1_base64="OIadET3ve5/rHOwzQHQfOj1Wlxk=">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</latexit><latexit sha1_base64="OIadET3ve5/rHOwzQHQfOj1Wlxk=">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</latexit>
� logP (x|y) = (y � x)2
2�2n
+(x� µx)2
2�2x
+ const.<latexit sha1_base64="Xiw52cekiagCiuvkb7M4kTQSOb0=">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</latexit><latexit sha1_base64="Xiw52cekiagCiuvkb7M4kTQSOb0=">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</latexit><latexit sha1_base64="Xiw52cekiagCiuvkb7M4kTQSOb0=">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</latexit><latexit sha1_base64="Xiw52cekiagCiuvkb7M4kTQSOb0=">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</latexit>
� @
@xlogP (x|y) = � (y � x)
�2n
+(x� µx)
�2x
= 0<latexit sha1_base64="bSDqQRu07buEYTpI6OjxVht7hB4=">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</latexit><latexit sha1_base64="bSDqQRu07buEYTpI6OjxVht7hB4=">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</latexit><latexit sha1_base64="bSDqQRu07buEYTpI6OjxVht7hB4=">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</latexit><latexit sha1_base64="bSDqQRu07buEYTpI6OjxVht7hB4=">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</latexit>
x̂<latexit sha1_base64="KKaXq1DRH3bprdfdkqAYBXzc5rw=">AAAB73icbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjxWsB/QhrLZbtqlm03cnYgl9E948aCIV/+ON/+NmzYHbX1h4eGdGXbmDRIpDLrut7Oyura+sVnaKm/v7O7tVw4OWyZONeNNFstYdwJquBSKN1Gg5J1EcxoFkreD8U1ebz9ybUSs7nGScD+iQyVCwShaq9MbUcyepuV+perW3JnIMngFVKFQo1/56g1ilkZcIZPUmK7nJuhnVKNgkk/LvdTwhLIxHfKuRUUjbvxstu+UnFpnQMJY26eQzNzfExmNjJlEge2MKI7MYi03/6t1Uwyv/EyoJEWu2PyjMJUEY5IfTwZCc4ZyYoEyLeyuhI2opgxtRHkI3uLJy9A6r3mW7y6q9esijhIcwwmcgQeXUIdbaEATGEh4hld4cx6cF+fd+Zi3rjjFzBH8kfP5A+dfj90=</latexit><latexit sha1_base64="KKaXq1DRH3bprdfdkqAYBXzc5rw=">AAAB73icbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjxWsB/QhrLZbtqlm03cnYgl9E948aCIV/+ON/+NmzYHbX1h4eGdGXbmDRIpDLrut7Oyura+sVnaKm/v7O7tVw4OWyZONeNNFstYdwJquBSKN1Gg5J1EcxoFkreD8U1ebz9ybUSs7nGScD+iQyVCwShaq9MbUcyepuV+perW3JnIMngFVKFQo1/56g1ilkZcIZPUmK7nJuhnVKNgkk/LvdTwhLIxHfKuRUUjbvxstu+UnFpnQMJY26eQzNzfExmNjJlEge2MKI7MYi03/6t1Uwyv/EyoJEWu2PyjMJUEY5IfTwZCc4ZyYoEyLeyuhI2opgxtRHkI3uLJy9A6r3mW7y6q9esijhIcwwmcgQeXUIdbaEATGEh4hld4cx6cF+fd+Zi3rjjFzBH8kfP5A+dfj90=</latexit><latexit sha1_base64="KKaXq1DRH3bprdfdkqAYBXzc5rw=">AAAB73icbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjxWsB/QhrLZbtqlm03cnYgl9E948aCIV/+ON/+NmzYHbX1h4eGdGXbmDRIpDLrut7Oyura+sVnaKm/v7O7tVw4OWyZONeNNFstYdwJquBSKN1Gg5J1EcxoFkreD8U1ebz9ybUSs7nGScD+iQyVCwShaq9MbUcyepuV+perW3JnIMngFVKFQo1/56g1ilkZcIZPUmK7nJuhnVKNgkk/LvdTwhLIxHfKuRUUjbvxstu+UnFpnQMJY26eQzNzfExmNjJlEge2MKI7MYi03/6t1Uwyv/EyoJEWu2PyjMJUEY5IfTwZCc4ZyYoEyLeyuhI2opgxtRHkI3uLJy9A6r3mW7y6q9esijhIcwwmcgQeXUIdbaEATGEh4hld4cx6cF+fd+Zi3rjjFzBH8kfP5A+dfj90=</latexit><latexit sha1_base64="KKaXq1DRH3bprdfdkqAYBXzc5rw=">AAAB73icbZBNS8NAEIYnftb6VfXoZbEInkoigh6LXjxWsB/QhrLZbtqlm03cnYgl9E948aCIV/+ON/+NmzYHbX1h4eGdGXbmDRIpDLrut7Oyura+sVnaKm/v7O7tVw4OWyZONeNNFstYdwJquBSKN1Gg5J1EcxoFkreD8U1ebz9ybUSs7nGScD+iQyVCwShaq9MbUcyepuV+perW3JnIMngFVKFQo1/56g1ilkZcIZPUmK7nJuhnVKNgkk/LvdTwhLIxHfKuRUUjbvxstu+UnFpnQMJY26eQzNzfExmNjJlEge2MKI7MYi03/6t1Uwyv/EyoJEWu2PyjMJUEY5IfTwZCc4ZyYoEyLeyuhI2opgxtRHkI3uLJy9A6r3mW7y6q9esijhIcwwmcgQeXUIdbaEATGEh4hld4cx6cF+fd+Zi3rjjFzBH8kfP5A+dfj90=</latexit>
) x̂ =�2x y + �2
n µx
�2x + �2
n<latexit sha1_base64="54T9utMlH0S3Bbo4vShYLLYh0qU=">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</latexit><latexit sha1_base64="54T9utMlH0S3Bbo4vShYLLYh0qU=">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</latexit><latexit sha1_base64="54T9utMlH0S3Bbo4vShYLLYh0qU=">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</latexit><latexit sha1_base64="54T9utMlH0S3Bbo4vShYLLYh0qU=">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</latexit>
Generative models
observed dataD
causesα
inference
modelM
explanation or prediction
inference
P (D|�,M)
P (�|D,M)
Inference:
Explanation or prediction:
with
Objective for learning: