A statistical analysis of visual space

1. A statistical analysis of visual space Zhiyong Yang & Dale Purves Sung-Ho Woo Interdisciplinary Programs in Cognitive Science at SNU

2. 1. Introduction 2. Results 1) A probabilistic concept of visual space 2) Probabilistic distributions of distances in natural scenes 3) Perceived distances in impoverished settings 4) Perceived distances in more complex circumstances 5) The effect of terrain 3. Discussion 1) Other approaches to rationalizing visual space 2) Other studies of natural scence statistics

3. Ex) A single line segment on the retina can be the projection of an infinite variety of lines in the environment S E Palmer Vsion Science p 23 1. Introduction * Three dimensional spatial relationships in the physical world are transformed into two dimensions in the retinal image * A multitude of different scene geometries could underlie any particular configuration in the retinal image

4. 1. Introduction * A number of examples in perceived distance show that the apparent distance of objects bears no simple relation to their physical distance from observer (a) Specific distance tendency : When a simple object is presented in an otherwise dark environment, observers usually judge it to be at a distance of 2 – 4 m, regardless of its actual distance (b) Equidistance tendency : Under these same condition, an object is usually judged to be at about the same distance from the observer as neighboring objects, even when their physical distances differ

5. 1. Introduction (c) Perceived distance of objects at eye level : The distances of nearby objects presented at eye level tend to overestimated, whereas the distances of farther objects tend to be underestimated (d) Perceived distance of objects on the ground : An object on the ground a few meters away tends to appear closer and slightly elevated with respect to its physical location. Moreover, the perceived location becomes increasingly elevated and relatively closer to the observer as the angle of the line of sight approaches the horizontal plane at eye level

6. 1. Introduction (e) Effects of terrain on distance perception : When the terrain is disrupted by a dip, the object appears to be farther away; conversely when the ground-plane is disrupted by a hump, the object tends to appear closer than it is

7. 1. Introduction * Hypothesis : These anomalies of perceived distance are all manifestation of a probabilistic strategy for generating visual percepts in response to inevitably ambiguous visual stimuli(retinal image) * A straightforward way to examine this hypothesis : Analyzing statistical relationship between geometrical features in the retinal image plane and the corresponding physical geometry in representative visual scene * Result : Perceived distance is always biased toward the most probable physical distance underlying the stimulus

8. 2. Result 1) A probabilistic concept of visual space * The relationship between any projected image(retinal image) and its source is inherently ambiguous * The distribution of the distances of unoccluded object surfaces from observer and their spatial relationships in normal viewing must have a potentially informative statistical structure * It seems likely that visual systems have evolved to take advantage of such statistical structure, or probabilistic information, in generating perception of physical space

9. 1) A probabilistic concept of visual space * Probabilistic strategy of this sort can be formalized in terms of Bayesian 'optimal observer theory' * The probability distribution of physical sources underlying a visual stimuli P(S|I) = P(I|S)P(S)/P(I) S : the parameters of physical geometry I : Visual image(retinal image) P(S) : the probability distribution of scene geometry in typical visual environment(the prior probability) P(I|S) : the probability distribution of stimulus I generated by the scene geometry S(the likelihood function) P(I) : A normalization constant

10. 2) Probability distribution of distances in natural scenes * P(S|I) = P(I|S)P(S)/P(I) * P(S) : Probability distribution of distances in natural scenes (figure 3a, 3b, 3c, figure 5a, 5b, figure 6a, 6b, figure 7b, figure 8a) * Acquiring image database : High-precision range scanner(LMS-Z210 3D Laser Scanner)

11. 2) Probability distribution of distances in natural scenes * Detect surfaces at distances of 2 - 300m with an accuracy  25mm at a resolution of 0.144 * The gained information : distances of objects from scanner, elevation and azimuth angle of objects, reflectivity of objects at the laser wavelength, etc * 23 images in fully natural setting at Duke Forest and 51 images in ourdoor settings at Duke Univ were taken * The height of center of scanner : 1.65m(defined eye level)

12. 2) Probability distribution of distances in natural scenes * A representative range image taken from one of the wide-field images acquired by laser range scanning * The information at each pixel in the range image : distance, elevation and azimuth angle relative to the laser scanner, etc

13. 2) Probability distribution of distances in natural scenes * Dimension of raw images used in analysis : 326(horizontal)  72(vertical) * Obtaining of distribution of distances in the images : counting the frequency of occurence of all the measured ranges (the radial distance to the center of the scanner) * Obtaining the distribution of horizontal distances at eye level : counting the samples within 2 relative to the horizontal plane at this height(1.65m) * Obtaining the distribution of horizontal distances within or at different height above or below at eye level : counting samples taken within 2 relative to the spaces centered 0.8, 1, 1.2, 1.4, 1.6m above or below eye level

14. 2) Probability distribution of distances in natural scenes * Obtaining the distribution of the differences between the distances of any two locations : random sampling of pairs of locations that were separated horizontally or vertically and counting the occurrences of the absolute difference of their distances from the image plane * Obtaining the distribution of distances at any elevation angle with respect to eye level : Talling the distances of all the physical locations in the scenes that spanned a particular elevation angle relative to the horizontal plane at the level of the scanner

15. 2) Probability distribution of distances in natural scenes * The probability distribution of the radial distances from the scanner to physical locations in the scene(fig 3a) * Black line : Distribution of the radial distances from the center of scanner to all the physical locations in the range database * Red line : Distribution of the radial distances from a simple model * Maximum probability of radial distances : 3m 3m

16. 2) Probability distribution of distances in natural scenes * The probability distribution of differences in the radial distance form the observer(scanner) to any two physical locations(fig 3b) * Red, green, blue : Three different angles of two locations in the horizontal plane * Higly skewed, and maximum probability of radial difference is near zero even for angular separations as large as 30

17. 2) Probability distribution of distances in natural scenes * The probability distribution of horizontal distances from the scanner to physical locations(fig 3c) * Red to violoet : Different heights above and below eye level * The probability distribution of horizontal distances from the scanner to physical location changes relatively little with height in the scene * Maximum probability of horizontal distances at eye level : about 4. 7m * Maximum probability of horizontal distances at different height above and below eye level : about 3m

18. 2) Probability distribution of distances in natural scenes * The probability distribution of physical distances at different elevation angle(fig 5) * The average distance as function of elevation angle based on the data in a * The vertical axis : the height relative to eye level * The horizontal axis : horizontal distance * The curve below eye level : slant of 1.5 from a distance of 3-15m, and about 5 from15-24m away

19. 2) Probability distribution of distances in natural scenes * The probability distribution of physical distances below eye level when terrain has a local dip or a hump * All physical locations at elevations within [-30.8, -26.8] were at least 0.15m below the ideal ground(dip, a) or 0.15m above(hump, b) *

20. 2) Probability distribution of distances in natural scenes * Average profile of the ground obtained from probability distribution in fig 6a(green line, dip) and fig 6b(blue line, hump) respectively * For comparison, the black line is the average ground derived from the probability of all range measurement(fig 5b) below eye level * Local variations in the terrain exert a global influence on the statistical configuration of the rest of the ground *

21. 2) Probability distribution of distances in natural scenes * Diagram showing how terrain in the range images was analyzed fig 7a * Dip at elevation in [ -1, +1] in an otherwise more or less flat ground plane * The probability distribution of the horizontal distances of the physical location at elevations on the ground within [ +  - 2,  +  + 2](next slide)

22. 2) Probability distribution of distances in natural scenes * The probability distribution of the horizontal distances of the physical location at elevations on the ground within [ +  - 2,  +  + 2] given a dip * Black line : distribution when the ground is flat * Red line : distribution when the ground is disrupted by the dip * Left panel : When the dip is closer to the observer( = -26) * Right panel : When the dip is farther to the observer( = -14.4)

23. 2) Probability distribution of distances in natural scenes * The probability distribution of the horizontal distances of the physical location at elevations on the ground within [ +  - 2,  +  + 2] given a dump * Black line : distribution when the ground is flat * Red line : distribution when the ground is disrupted by the hump * Left panel : When the hump is closer to the observer( = -26) * Right panel : When the hump is farther to the observer( = -14.4)

24. 3) Perceived distances in impoverished setting (Specific distance and,Equidistance tendency) * P(S|I) = P(I|S)P(S)/P(I) * In the absence of any distance cues, the likelihood function, P(I|S) is flat * P(I) is constant * P(S) is figure 3a * The apparent distance of a point in physical space should accord with the probability distribution of the distances of all points in typical visual scene * Maximum probability is at about 3m

25. 3) Perceived distances in impoverished setting (Specific distance and,Equidistance tendency) * P(S|I) = P(I|S)P(S)/P(I) * In the absence of additional information about differences in the distances of two nearby locations , the likelihood function, P(I|S) is again more or less flat * P(I) is constant * P(S) is figure 3b * maximum probability of radial difference between two locations with relatively small angular separations is near zero

26. 4) Perceived distances in more complex circumstances Distance perception at eye level * P(S|I) = P(I|S)P(S)/P(I) * Observation were made under conditions that involved some degree of contextual visual information. Thus the likelihood function, P(I|S) is no longer flat, and their form is unknown. To approximate the likelihood function P(I|S), Gaussian distribution were used * P(I) is constant * P(S) is figure 3c(black line)

27. 4) Perceived distances in more complex circumstances Distance perception at eye level * P(S|I) = P(I|S)P(S)/P(I)

28. 4) Perceived distances in more complex circumstances Perceived distance of objects on the ground Fig 5c * If the portion of the curve at height below eye level in Fig 5c is taken as an index of the average ground, it is apparent that the average ground is a curved surface * Perceived location of an object on the ground without much additional information abuout its actual distances varies according to the declination of the line of sight

29. 4) Perceived distances in more complex circumstances Perceived distance of objects on the ground * P(S|I) = P(I|S)P(S)/P(I) * P(I|S) : The likelihood function at an angular declination  was a Gaussan function : f() = exp(-( - 0)2/22), where 0 = sin-1(H/R),  = 8, R = radial distance and H = 1.65m. The prior was the distribution of distance at angular declination within [ - 8,  + 8] Fig 4b. Predicted perceptual locations of objects on the ground predicted from probablistic distribution in fig5a and the prior

30. 5) The effects of terrain * P(S|I) = P(I|S)P(S)/P(I)

31. 5) The effects of terrain * P(S|I) = P(I|S)P(S)/P(I)

32. 3. Discussion