Image analysis – Applications – 3-d or stereo imaging analysis
Reexamination Certificate
1999-12-08
2002-05-21
Ahmed, Samir (Department: 2623)
Image analysis
Applications
3-d or stereo imaging analysis
C345S653000, C345S679000
Reexamination Certificate
active
06393143
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
This invention pertains generally to a technique for recognizing and locating objects and more specifically a technique for estimating the pose of an object from a range image containing the object.
2. Description of the Related Art
In recent decades, a wide variety of instruments have been built to obtain range images; a range image being a two-dimensional array of numbers which gives the depth of a scene along many directions from a central point in the instrument. Instead of measuring the brightness of many points in a scene, as in a television camera, these instruments measure where each point is in a three-dimensional space. Both range images and the more conventional intensity images from digital cameras have been used in the computer vision research community to determine the pose of observed objects. The term “object”, as used herein, means a particular surface shape. “Pose” means a complete description of an object's position and orientation. For a rigid object this requires six numbers, such as X, Y, Z, pitch, yaw and roll, or six equivalent coordinates. The previous methods for pose estimation all suffer from either a lack of generality or from time inefficiency.
It is possible to do pose estimation using tripod operators (TO). See, Pipitone; TRIPOD OPERATORS FOR THE INTERPRETATION OF RANGE IMAGES; NRL Memorandum Report 6780, February 1991 for a crude and incomplete discussion. Tripod operators are a versatile class of feature extraction operators for surfaces. They are useful for recognition and/or localization (pose estimation) based on range or tactile data. They extract a few sparse point samples in a regimented way so that N surface points yield only N−3 independent scalar features containing all the pose-invariant surface shape information in these points and no other information. They provide a powerful index into sets or prestored surface representations. A TO consists of three points in 3-space fixed at the vertices of a triangle and a procedure for making several “depth” measurements in the coordinate frame of the triangle, which is placed on the surface like a surveyor's tripod. TOs can be embedded in a vision system in many ways and applied to almost any surface shape.
As stated above, a TO consists of three points in space fixed at the vertices of a triangle of fixed edge lengths and a procedure for making several depth measurements in the coordinate frame of the triangle, which is placed on the surface like a surveyor's tripod. These measurements take the form of arc-lengths along “probe curves” at which the surface is intersected.
FIGS. 1
a
through
1
c
shows three examples of TO's.
FIG. 1
a
shows a very simple TO with one line probe fixed symmetrically with respect to the rigid triangle ABC. The single scalar feature is the distance from the plane of ABC at which the probe intersects the surface. This resembles a mechanical optician's tool called a spherometer. The number d of scalar features is called the order of the operator.
FIGS. 1
b
and
1
c
show TO's that can be viewed as a set of equilateral triangles hinged together so that all d+3 points can be made to contact a surface. The angles of the d hinges are the features. This type, called linkable TO's, is preferred because of their symmetry and uniform sensitivity to noise. The application if this TO to a planar surface yields &phgr;≡0 for all the hinges. Many variations of these TO's could be constructed. Feature noise is related to range noise n by the approximate expression n
&phgr;
≈51×n/e, where n
&phgr;
is the feature error in degrees, and n is expressed in the same distance units as the edge length e.
From an N-point TO, the N sampled surface points yield only N−3 independent scalar features, and the order d is N−3. These features contain all the surface shape information in the 3N components of the points since they suffice to reconstruct the relative positions of the N points. They contain no other information. For example, they have complete six DOF invariance under rigid motions, the group R
3
×SO(
3
). Thus, they depend upon where the tripod lies on the surface, but upon nothing else. A key property is that for any dimensionality d of feature vector only a 3 (or fewer)-dimensional manifold of feature space points can be generated from a given surface, since the tripod can be moved in only 3 DOF on a surface. This allows objects to be densely sampled with TOs at preprocessing time with a manageable number of operator applications, typically a few thousand, to obtain almost all of the possible feature vector values obtainable from any range image of the object. This set is a kind of invariant signature. For brevity, this is called the signature of the object or surface, with respect to a particular type TO. It can be stored in an array of bins in feature space, e.g., of dimension
3
or
4
, for later efficient access of near neighbors to TO features measured at recognition time. These bins can optionally contain precomputed probability densities, analytic expressions for distances to nearby signature manifolds, and partial or complete descriptions of the relative poses of tripods and models, all to serve various purposes in a recognition system.
Since in some applications of the tripod operator, the computation consists only of placement and a little indexing, the cost of placing the operator should be kept small. This can be done by efficiently implementing a procedure similar to the following. Consider placing the TO's of
FIGS. 1
b
or
1
c
on a dense range map. Point A can be chosen as any point on the image surface. Interpolation is to be done locally as needed, e.g., using piecewise triangular facets. Point B can be found by moving along a line at orientation &agr; in image coordinates, pixel indices, until the 3D distance |AB|≡e. This can be done in logarithmic time, essentially constant here, using binary search. Then the circle of the radius 0.53 e oriented coaxially around the center of the segment AB, using binary search, to find a point C close to the surface. A similar circular search yields each remaining point. A key step in the circular search is the mapping, specific to a range scanner's geometry, from a point (x, y, z) to the two indices of the range pixel whose ray (x, y, z) lies on. This allows the front/behind decision required by the binary search. In the case of a sequential random access range scanner, it may be efficient to monotonically search elliptical paths in image coordinates until the two distances being enforced, e.g., |AC| and |BC|; are both correct. The ellipses here are the projections of the previously described circles onto image coordinates. Finally, in the case of a tactile TO, the computation is mechanical; the feature values are to be read from position transducers, e.g., from linear potentiometers by an analog-to-digital (A/D) converter.
The following are a few of the symmetry properties of TOs of the types of
FIGS. 1
b
and
1
c.
Surfaces with one symmetry, such as extrusions, surfaces of revolution, and helical projections produce only a 2-dimensional manifold in feature space (for
FIGS. 1
b
and
1
c
). Cylinders, having two symmetries, produce only a nearly circular 1-dimensional curve, and spheres a single point. Scaling a TO by changing its edgelength does not affect the signature of surfaces swept by a line with one point fixed, e.g., cones, planar n-hedral vertices, and planar dihedral edges. Regardless of the surface, an operator with a 3-fold symmetry, e.g., those in
FIGS. 1
b
and
1
c,
produces signatures unchanged by cyclically permuting each triple of corresponding features. In
FIG. 1
c,
the three 3-cycles (1, 2, 3), (4, 5, 6) and (7, 8, 9) show this property, for features &phgr;
1
through &phgr;
9
, respectively. This allows a 3-fold storage reduction, e.g., by permuting the features so that &phgr;
1
is the largest.
Ahmed Samir
Karasek John J.
Mills III John Gladstone
The United States of America as represented by the Secretary of
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