STATISTICAL ANALYSIS OF SECOND-ORDER RELATIONS OF 3D

STRUCTURES

Sinan Kalkan, Florentin W

org

otter

Bernstein Centre for Computational Neuroscience, Unv. of G

ottingen, Germany

Norbert Kr

uger

Cognitive Vision Group, Unv. of Southern Denmark, Denmark

Keywords:

Range Data Statistics, Indirect Depth Estimation.

Abstract:

Algorithmic 3D reconstruction methods like stereopsis or structure from motion fail to extract depth at ho-

mogeneous image structures where the human visual system succeeds and is able to estimate depth. In this

paper, using chromatic 3D range data, we analyze in which way depth in homogeneous structures is related to

the depth at the bounding edges. For this, we ﬁrst extract the local 3D structure of regularly sampled points,

and then, analyze the coplanarity relation between these local 3D structures. We can statistically show that the

likelihood to ﬁnd a certain depth at a homogeneous image patch depends on the distance between the image

patch and its edges. Furthermore, we ﬁnd that this prediction is higher when there is a second edge which

is proximate to and coplanar with the ﬁrst edge. These results allow deriv ing statistically based prediction

models for depth extrapolation into homogeneous image structures. We present initial results of a model that

predicts depth based on these statistics.

1 INTRODUCTION

Depth estimation relies on the extraction of 3D struc-

ture from 2D images which is realized by a set of

inverse problems including structure from motion,

stereo vision, shape from shading, linear perspective,

texture gradients and occlusion (Bruce et al., 2003).

In methods which make use of multiple views (i.e.,

stereo and structure from motion), correspondences

between different 2D views of the scene are required.

In contrast, monocular or pictorial cues such as shape

from shading, utilization of texture gradients or linear

perspective use statistical and geometrical relations in

one image to make statements about the underlying

3D structure.

Many surfaces have only weak texture or no tex-

ture at all, and as a consequence, the correspondence

problem is very hard or not at all resolvable for these

surfaces. Nevertheless, humans are able to recon-

struct 3D information for these surfaces, too. This

gives rise to the assumption that in the human visual

system, an interpolation process is realized that start-

ing with the local analysis of edges, corners and tex-

tures, computes depth also in areas where correspon-

dences cannot easily be found.

In ﬁgure 1, the relation between the depth of ho-

mogeneous image structures and edges is shown. In

ﬁgure 1(a), we see that the depth of homogeneous im-

age structures is directly related to the depth of the

bounding edges; however, this relation does not al-

ways exist as shown in ﬁgure 1(b,c) where the depth

is cued in shading.

With the notion that the human visual system is

adapted to the statistics of the environment (Brunswik

and Kamiya, 1953; Knill and Richards, 1996; Kr

uger,

1998; Kr

uger and W

org

otter, 2004; Olshausen and

Field, 1996; Rao et al., 2002; Purves and Lotto, 2002)

and its successful applications to grouping, object

recognition and stereo (Elder and Goldberg, 2002; El-

der et al., 2003; Pugeault et al., 2004; Zhu, 1999), the

analysis, and the usage of natural image statistics has

become an important focus of vision research. More-

over, with the advances in technology, it has been also

possible to analyze the underlying 3D world using 3D

range scanners (Howe and Purves, 2004; Huang et al.,

2000; Potetz and Lee, 2003; Yang and Purves, 2003).

In this paper, by making use of chromatic range

data (see ﬁgure 3 for examples), we investigate

Kalkan S., Wörgötter F. and Krüger N. (2007).

STATISTICAL ANALYSIS OF SECOND-ORDER RELATIONS OF 3D STRUCTURES.

In Proceedings of the Second International Conference on Computer Vision Theory and Applications - IU/MTSV, pages 13-20

 SciTePress