Methods, systems, and computer readable media for visual odometry using rigid structures identified by antipodal transform

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First Claim
1. A method for visual odometry using rigid structures identified by an antipodal transform, the method comprising:
 receiving a sequence of images captured by a camera;
identifying rigid structures in the images using an antipodal transform;
identifying correspondence between rigid structures in different image frames; and
estimating a path of motion of the camera based on motion of corresponding rigid structures among the different image frames;
wherein identifying rigid structures using the antipodal transform includes;
generating a binary occupancy matrix for pixels in an image frame, wherein each element in the occupancy matrix indicates whether the pixel includes point on a structure or not;
linearizing the matrix;
marking occupied elements in the matrix as having a score that is different from unoccupied elements; and
for each unoccupied element in the matrix, determining a distance in each of a plurality of directions to the nearest occupied element, computing a score for each distance, and summing the scores for each distance, and identifying the rigid structures using the scores.
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Abstract
The subject matter described herein includes methods for visual odometry using rigid structures identified by an antipodal transform. One exemplary method includes receiving a sequence of images captured by a camera. The method further includes identifying rigid structures in the images using an antipodal transform. The method further includes identifying correspondence between rigid structures in different image frames. The method further includes estimating motion of the camera based on motion of corresponding rigid structures among the different image frames.
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Methods, systems, and computer readable media for visual odometry using rigid structures identified by antipodal transform  
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Current Assignee
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23 Claims
 1. A method for visual odometry using rigid structures identified by an antipodal transform, the method comprising:
receiving a sequence of images captured by a camera; identifying rigid structures in the images using an antipodal transform; identifying correspondence between rigid structures in different image frames; and estimating a path of motion of the camera based on motion of corresponding rigid structures among the different image frames; wherein identifying rigid structures using the antipodal transform includes; generating a binary occupancy matrix for pixels in an image frame, wherein each element in the occupancy matrix indicates whether the pixel includes point on a structure or not; linearizing the matrix; marking occupied elements in the matrix as having a score that is different from unoccupied elements; and for each unoccupied element in the matrix, determining a distance in each of a plurality of directions to the nearest occupied element, computing a score for each distance, and summing the scores for each distance, and identifying the rigid structures using the scores.  View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11)
 12. A system for visual odometry using rigid structures identified by an antipodal transform, the system comprising:
a line and point feature extractor for receiving a sequence of images captured by a camera, identifying rigid structures in the images using an antipodal transform, and identifying correspondence between rigid structures in different image frames; and a camera motion estimator estimating a path of motion of the camera based on motion of corresponding rigid structures among the different image frames; wherein the line and point feature extractor is configured to compute the antipodal transform by; generating a binary occupancy matrix for pixels in an image frame, wherein each element in the occupancy matrix indicates whether the pixel includes point on a structure or not; linearizing the matrix; marking occupied elements in the matrix as having a score that is different from unoccupied elements; for each unoccupied element in the matrix, determining a distance in each of a plurality of directions to the nearest occupied element, computing a score for each distance, and summing the scores for each distance, and identifying the rigid structures using the scores.  View Dependent Claims (13, 14, 15, 16, 17, 18, 19, 20, 21, 22)
 23. A nontransitory computer readable medium having stored thereon executable instructions that when executed by the processor of a computer control the computer to perform steps comprising:
receiving a sequence of images captured by a camera; identifying rigid structures in the images using an antipodal transform; identifying correspondence between rigid structures in different image frames; and estimating a path of motion of the camera based on motion of corresponding rigid structures among the different image frames; wherein identifying rigid structures using the antipodal transform includes; generating a binary occupancy matrix for pixels in an image frame, wherein each element in the occupancy matrix indicates whether the pixel includes point on a structure or not; linearizing the matrix; marking occupied elements in the matrix as having a score that is different from unoccupied elements; and for each unoccupied element in the matrix, determining a distance in each of a plurality of directions to the nearest occupied element, computing a score for each distance, and summing the scores for each distance, and identifying the rigid structures using the scores.
1 Specification
This application claims the benefit of U.S. Provisional Patent Application Ser. No. 61/990,478 filed May 8, 2014; the disclosure of which is incorporated herein by reference in its entirety.
This invention was made with government support under Grant No. 0835714 awarded by the National Science Foundation. The government has certain rights in the invention.
The subject matter described herein relates to visual odometry. More particularly, the subject matter described herein relates to methods, systems and computer readable media for visual odometry using rigid structures identified by antipodal transform.
Visual odometry refers to the estimation of the path of a camera from solely from video images taken by the camera. The term “visual odometry” was created by Nister due to its similarity to wheel odometry. Wheel odometry estimates the distance traveled by a vehicle based on rotations of the vehicle'"'"'s wheels. Visual odometry estimates motion, not only the distance traveled, but also the path or trajectory (X, Y, Z) coordinates and camera orientation at each point), traveled by a camera based on analysis of images captured by a camera in successive video frames. Such a path or trajectory can be used to retrace the path of the camera or the object to which the camera is attached. Applications of video odometry include robotics, location services, turnbyturn navigation, and augmented reality. For example, if GPS communications are not available, visual odometry can provide a trajectory to be followed if it is desirable to retrace a path.
Existing visual odometry algorithms rely on a triangulation step in order to reconstruct tracked features. The reconstructed features are then tracked between video sequences in order to maintain a uniform scale of camera trajectory. Reconstructing tracked features using triangulation for each frame is computationally intensive. Accordingly, there exists a need for improved methods for visual odometry that avoids or reduces the need for triangulation for each frame and is less computationally intensive than existing visual odometry methods.
The subject matter described herein includes methods for visual odometry using rigid structures identified by an antipodal transform. The method further includes identifying correspondence between rigid structures in different image frames. One exemplary method includes receiving a sequence of images captured by a camera. The method further includes identifying rigid structures in the images using an antipodal transform. The method further includes estimating motion of the camera based on motion of corresponding rigid structures among the different image frames.
The subject matter described herein can be implemented in software in combination with hardware and/or firmware. For example, the subject matter described herein can be implemented in software executed by a processor. In one exemplary implementation, the subject matter described herein can be implemented using a nontransitory computer readable medium having stored thereon computer executable instructions that when executed by the processor of a computer control the computer to perform steps. Exemplary computer readable media suitable for implementing the subject matter described herein include nontransitory computerreadable media, such as disk memory devices, chip memory devices, programmable logic devices, and application specific integrated circuits. In addition, a computer readable medium that implements the subject matter described herein may be located on a single device or computing platform or may be distributed across multiple devices or computing platforms.
The subject matter described herein includes methods, systems, and computer readable media for visual odometry using rigid structures identified by antipodal transform. Rather than using a triangulation step to reconstruct tracked features in each frame to maintain a uniform scale of camera trajectory, the subject matter described herein records the scale of tracked features to determine translation of the camera. An exemplary process for visual odometry using rigid structures identified by antipodal transforms will now be described.
As stated above, visual odometry is the estimation of the trajectory of motion of a camera using video as input. This trajectory is with respect to an initial coordinate system and can be visualized as a path of (X, Y, Z) coordinates in space.
The subject matter described herein includes a new approach for the computation of visual odometry based on a hybrid system using both points and line segments detected and tracked in images of the video. The present approach encompasses the following steps:
 1. A new mathematical transform referred to as the antipodal transform is used to identify rigid structures in an edge image of a scene. A method for fast computation of the antipodal transform is also described. The antipodal transform is used herein to identify unoccupied points at the center of rigid structures in image data. The term “antipodal” in mathematics means on the diametrically opposite side of a circle or sphere. However, the term “antipodal transform” as used herein refers to a transform used to identify the center point of any rigid structure in an image and is not limited to circular or spherical structures.
 2. The rigid structures that are identified are structures that do not change from scene to scene, such as architectural features like doors, windows, walls, frames, furniture, or other fixed structures that do not change from scene to scene, such as architectural features like doors, windows, walls, frames, furniture, or other fixed, immobile objects in a scene. The reason for identifying such structures is that motion of these structures between image frames indicates movement of the camera, rather than movement of the corresponding structures. The following section describes the antipodal transform and a method for its fast computation.
Antipodal Transform
The antipodal transform is closely related to the Distance Transform in that it applies a score to binary matrix elements based on their relationship to the closest occupied element and occupancy matrices, where this some distinct range of scores for all elements. The subject matter described herein used the antipodal transform to identify rigid structures in image data, where a rigid structure is a structure, such as doors, windows, walls, frames, furniture, or other fixed structures that do not change from scene to scene, that is not likely to change from one image frame to the next. Maxima and minima of the transform are used to find key points of interest in a scene, where key points will be tracked between frames. The antipodal transform biases pixels corresponding with rigid physical structures in the scene environment.
There are a number of other applications for the antipodal transform. In digital image processing the transform can be used to determine blurring effects and also for skeletonizing. The transform signature itself as well as structural correspondence between key points in a scene can be used in object detection. In robotics, where a matrix may correspond to a map where values correspond to the level or obstacles in the terrain, the transform may be applied in motion planning or path finding.
In contour analysis and segmentation, a gradient contour is often discovered by tracing an edge until a loop closure. The centroid of the contour can then be found by taking the mean of the positions of the boundary pixels. However, the antipodal transform directly computes the position of the center of gradient contours. Further, contours can be extracted by taking the region around an Antipodal Transform maxima and minima.
The Antipodal Transform (AT) can be expressed as
where f is a function that returns a value based on the occupancy of values along the vector {right arrow over (η)}. The examples shown herein use a function f that returns the coordinates of occupied elements. In the formula, p is a point in the image being tested. The function d( ) measures the distance between the point p and the nearest occupied point p+δ*η. The term δ can hold values 0 or 1 depending on the occupancy of the point in the matrix.
As stated above, the antipodal transform is similar to the distance transform. In the distance transform, given a binary matrix, the score of each point is its distance to the nearest occupied element in the matrix. The antipodal transform can also be expressed as a computation over an occupancy matrix. Each point is given a score dependent on the distance of the closest occupied element in one particular direction minus the distance of the closest element in the opposite direction
The antipodal transform is defined, for a given point, as the sum of all distances of the closest point in a particular direction minus the point closest point in the opposite direction for all directions. In our fast computation approach, we apply this summation solely along the vertical and horizontal directions in order to increase computation speed. See examples below:
The following steps outline a method for fast computation of the antipodal transform:
 a. A Fast Method for Computing the Antipodal Transform:
 i. Create a binary matrix. In this case we have the binary edge image from the scene. An edge image can be generated from a video frame captured by a camera whose motion is being tracked.
FIG. 3A illustrates an example of an image of a scene captured by a camera,FIG. 3B illustrates an intermediate image where the edges are shown in white, andFIG. 3C illustrates the corresponding edge image where the edges are shown in black and everything that is not an edge is shown in white. The binary matrix is a twodimensional matrix where each matrix element corresponds to a pixel or group of pixels in the edge image. Thus, the binary matrix for the edge image inFIG. 3C may includes a 1 or high value for pixels corresponding to edges and a 0 or low value for pixels corresponding to space that is not an edge.  ii. Given a binary matrix, reshape the matrix to onedimension (or traverse the matrix such that it is linear), where elements are organized according to the direction in which they are being evaluated. For instance when evaluating the score along the horizontal direction, the matrix traversal should be organized such that all rows of the matrix are horizontally concatenated to form a linear vector.
 iii. Mark occupied elements to have the highest score or infinity. Each matrix element is assigned a score or value depending on whether or not the corresponding pixel in the edge image is occupied by an edge or unoccupied. Thus, in a binary matrix for an edge image of a scene, matrix elements corresponding to edges are set to the highest score, and matrix elements between edges are set to zero or the lowest score
 iv. In spaces with no occupancy, find the length of the space and the midpoint of the space. Apply a score to each element determined by the y value of a parabola centered at the midpoint of the vacant occupancy space, where x is the position of the element in the vacant space. The general formula for this parabola is y=x^{2}. Thus, if the space is a rectangle, points near the center will have lower scores and points near the edges will have higher scores.
 v. Find the score for each element over all directions being evaluated. Sum the scores in each direction for all elements. Continuing with the rectangle example, scores for each point enclosed by the rectangle would have a score for the horizontal and vertical directions. The horizontal and vertical score for each point is summed to create a total (horizontal plus vertical) score for each point.
 i. Create a binary matrix. In this case we have the binary edge image from the scene. An edge image can be generated from a video frame captured by a camera whose motion is being tracked.
 3. Keypoints in the scene are identified using the antipodal transforms. The keypoint is a point at the center of a rigid structure. In the calculation of the antipodal transform, midpoints of unoccupied spaces are often identified. However, some of the midpoints may be inside of rigid structures whose motion is desired to be tracked between frames, and some midpoints are within structures and thus candidates for keypoints. The following method illustrates how to locate keypoints.
 a. Apply a suppression algorithm when computing the score for unoccupied spaces during the antipodal transform calculation such that higher values are applied when the length between the point and an occupied cell is below a determined threshold. The purpose of this step is to eliminate points that are outside of rigid structures, such as spaces between rigid structures.
 b. Extract points with the highest antipodal transform.
 a. A Fast Method for Computing the Antipodal Transform:
The following section explains the keypoint search in more detail.
Keypoint Search
 The antipodal transform is ideal for finding rigid structures in a scene. Rigid structures are often symmetric along at least one axis.
 Given the antipodal transform formula set forth above in the equation for the antipodal transform AT(p), the absolute value of the expression is negated. Thus, the highest possible value will be 0. Keypoints will be points where the antipodal transform score is highest. Given the nature of the antipodal transform, keypoints will correspond to the centers of rigid structures in the scene. In a binary edge image, there are some cases when multiple lines will be found that only correspond to one edge on a structure. However, the region between the lines can be symmetrical along two different axis and thus a maxima. In order to eliminate these maxima we employ a form of suppression. When computing the transform, only vacant areas in the linear vector (see Fast Method for Computing the Antipodal Transform above) longer than a certain threshold are given a score. Areas shorter than the threshold are given the same score as occupied values in the matrix.
 4. Once the keypoints are identified, a new image descriptor is defined and applied at keypoints in the image. The new image descriptor can be used to group edge lines in a scene. The purpose of defining an image descriptor for each keypoint is to identify correspondences between keypoints and lines. Each descriptor encodes a gradient magnitude, which is a measure of the change in pixel intensity from one pixel to neighboring pixels in different directions. An exemplary method for encoding the gradient magnitude is described in further detail below in the section entitled “Gradient Encoding over Multiple Scales”. This method is also summarized in steps ad below.
 a. A spatial grid over a region of an image is applied at multiple scales. The central region of the square grid is not computed.
 b. The mean of the gradient magnitude is computed in each grid space.
 c. Pairs of scales are binarized elementwise.
 d. Pairs of binarized scales are than subtracted from each other. This yields a ternarized vector {−1, 0, 1}.
 The following section describes how to encode the gradient magnitude for each keypoint.
GEMS: Gradient Encoding Over Multiple Scales
 The following section describes how to encode the gradient magnitude for each keypoint.
GEMS defines a family of nested feature descriptors that use the gradientmagnitude. The gradient magnitude for a given pixel is the difference in intensity from the pixel to a neighboring pixel. In one embodiment, the gradient magnitude can be computed using the following expression:
where f represents pixel intensity and x and y are pixel coordinates.
Nested descriptors are defined by their pattern and binary structure. Nested descriptors access an image at various octaves and binarize elements against symmetrically placed elements within the structure pattern.
A spatial grid over a region of an image is applied at multiple scales. The mean of the gradient magnitude is computed in each grid space. Pairs of scales are binarized. Pairs of binarized scales are than subtracted from each other. This yields a ternarized vector {−1, 0, 1}.
The grid structure is composed of adjacent polygons or circles. The grid window as shown in
 5. Once a gradient descriptor is determined for each keypoint, the descriptors for keypoints in different frames are used to determine a similarity distance between two keypoints in two different frames. In one embodiment, the similarity distance comprises a Hamming distance. An exemplary method for determining the similarity distance and using the similarity distance to identify matching lines and points in different image frames will now be described.
Line Matching and Point Matching
 5. Once a gradient descriptor is determined for each keypoint, the descriptors for keypoints in different frames are used to determine a similarity distance between two keypoints in two different frames. In one embodiment, the similarity distance comprises a Hamming distance. An exemplary method for determining the similarity distance and using the similarity distance to identify matching lines and points in different image frames will now be described.
Point Matching
The Hamming distance or an alternate similarity metric may be used to determine the distance between feature descriptors. Hamming distance is the sum of the differences between each element in the vector.
A set of matches is maintained between frame sequences. For instance, the closest matching descriptors to the features in the preceding image are found. After an outlier elimination step (described below) all matching points are kept. This is then repeated for the third frame in each frame triplet. As matched features between triplets are lost, new features are added to the tracking set.
For each triplet, one exemplary method for determining correspondence between frames utilizes a minimum of at least 5 points.
Line Matching
Once the gradients are determined, the gradients may be used to determine line matching between images.
A line is associated with a GEMS keypoint if any portion of its defining gradient is within the largest scale window. In
Three thresholds over line orientation angle are used to categorize the lines into three categories: vertical, horizontal, and diagonal lines. Sets of lines associated with matching GEMS points are matched. Only lines categorized within similar orientations are matched. For vertical and diagonal lines, lines are matched by the order of their midpoints along the horizontal axis. Horizontal lines are matched by the order of their midpoints along the vertical axis. The horizontal and vertical axis is with respect to the camera'"'"'s optical axis.
The LevenbergMarquardt algorithm can be applied to match sets of lines within orientation categories instead of matching by the order of their midpoints.
 6. Hybrid Tracking of Keypoints, Lines, and Vanishing Points.
 Once matching lines and keypoints are identified using the descriptors and the similarity distances, a confidence measure is applied based on pixel blur and drift over the camera path. At the highest confidence, groups of lines associated with matched keypoints can be matched themselves. In the instance where keypoints are identified but there are no associated lines, only the keypoints are matched. Lines, points, and vanishing points can be tracked simultaneously. Alternatively, lines and points may be tracked, and the tracking of vanishing points may only be initialized when the confidence value is low.
 a. The classification of the line segments into 3 main groups of projections of parallel lines representing the vertical and the two horizontal directions (like northsouth and eastwest) in 3D.
 b. The estimation of vanishing points from these groups. These vanishing points define a “visual compass”.
 c. The matching of feature descriptors across three consecutive frames.
 i. Matching occurs between at least one keypoint in each image, if at least three lines fall into the keypoint'"'"'s window and matches can be found between them.
 ii. Matching occurs between at least three points, if no line matches can be found between the three highest matching points.
 d. The matching of line segments associated with the gradients encoded by the feature descriptor across three consecutive frames.
 i. Lines are divided into three categories based on their major angle orientation. In one example, lines may be categorized as follows.
 1. 029 degrees: diagonal
 2. 3059 degrees: horizontal
 3. 6090 degrees: vertical
 ii. Lines are initially matched by order along the axis.
 1. Vertical and diagonal lines are matched by the associated lines midpoint distance along the horizontal axis.
 2. Horizontal lines and matched by the associated lines midpoints along the vertical axis.
 iii. If order of line segments is not sufficient for a particular frame sequence, LevenbergMarquardt can be used to match the lines.
 i. Lines are divided into three categories based on their major angle orientation. In one example, lines may be categorized as follows.
Once correspondence between keypoints is identified, outlier keypoints correspondences are filtered based on a heuristic consensus algorithm for matched featurepoints between two images. This method is applied across both pairs of images for a threeframe sequence. Structure based outlier elimination will now be explained in more detail.
Structure Based Iterative Outlier Elimination
The structure based iterative outlier elimination process described hereinbelow may be used to eliminate outliers between feature matches in images. It is an alternative to the RANSAC (Random Sample Consensus) approach most popularly used. According to this approach, N matched points in each respective frame are tracked. For all matched points within a frame, the respective distance to all other matched points is computed. This builds a distance curve for each keypoint, where order is determined by the closest matching feature vectors. The compiled keypointcurves for each image are then matched. After the curve match with the lowest error is found, points with the highest individual error are removed.
In the right hand frame in
For each matched keypoint the relative distances to other matched keypoints are compiled and ordered by the closestmatching feature descriptors. The curves computed for each image are matched. Iteratively one curve is moved forward and subtracted against the other. The position yielding the least error is found. The points corresponding to sections of the curve with the highest error are omitted from the match model.
The following equation illustrates the structure based iterative outlier elimination algorithm describe above with respect to
where, f is the sum of errors between corresponding points ((p_{i}, q_{j}) in one curve and (p_{k}, q_{l})) in the corresponding curve.
Vanishing Points
Once correspondence between lines is identified and outliers matches are eliminated, vanishing points can be identified from the remaining matching lines in a given frame, and absolute rotation can be computed from the vanishing points.
If v_{1}, v_{2}, and v_{3 }correspond to the matching vanishing points in each direction, the rotation can be computed as:
is the singular value decomposition.
If only two vanishing points are found in the scene the cross product of the normalized points can be computed in order to find the rotation matrix as follows:

 7. Hybrid Motion Model Computation from Matching with Feature Points, Lines, and Vanishing Points.
Once corresponding features are identified between image frames, the motion of the features can be used to determine camera rotation and movement between frames. The following steps describe the computation of camera rotation and movement between image frames.
 a. Compute the absolute rotation based on the identification of principal directions in the scene. These principal directions correspond to vanishing points computed as the intersections of projections of groups of parallel lines in the scene. Two orthogonal groups are sufficient to compute the absolute rotation.
 i. From the absolute position computed at each frame the interframe motion is computed.
 b. For at least a triplet of frames, (where a triplet is defined as the current image frame and the preceding two frames), estimate interframe camera position at every frame based on feature points matched across the triplet.
 i. For a pair of frames three or more points are used to find the essential matrix candidates for feature keypoint matches.
 ii. Triangulate one matching keypoint for each candidate essential matrix. An essential matrix for which points project in front of each camera view is a valid essential matrix. Rotation and translation of the second frame with respect to the previous can be extracted from the essential matrix.
 c. From a triplet of frames and line correspondences in those frames, compute a linebased trifocal tensor and the estimation of interframe camera position at every frame. (A method for identifying line correspondences between frames is described below in the section entitled “Three Frame Constraints for Line Correspondences”. The method is outlined below.
 i. From a triplet of frames and the linebased trifocal tensor the extraction of the two interframe rotations is performed.
 ii. From a triplet of frames and the interframe rotations the computation of the two interframe translations up to one scale factor is performed.
 d. In the instance in which points cannot be identified or a motion model (9) from (bc) yields an improbable motion result, c can be performed and the rotation from the vanishing points is maintained. The average of the previous three frames'"'"' translations is used. If there are not three frames, than the number of previous frames up to three frames is used.
 a. Compute the absolute rotation based on the identification of principal directions in the scene. These principal directions correspond to vanishing points computed as the intersections of projections of groups of parallel lines in the scene. Two orthogonal groups are sufficient to compute the absolute rotation.
The following sections describe line representation and the use of tracked lines in different image frames to determine camera rotation and movement between frames.
Line Representations
In order to determine camera rotation and movement between image frames, line movement and rotation between image frames may be determined. Before movement and rotation can be determined, lines in each image frame must be parameterized.
d=X×1 (1)
The Plücker coordinates (1, d) satisfy the constraints
∥l∥=1 and l^{T}d=0 (2)
The magnitude of the line momentum d is equal to the distance from the line to the origin.
ThreeFrame Constraints for Line Correspondences
Let us assume m lines L_{i}=1 . . . m in 3D space captured by a camera in time points t_{0 }and t_{1 }(
We denote that (1, d) the Plücker coordinates of a line ^{3}. We denote with n_{i }the normal to the viewing plane π at time t_{i}. This normal can be estimated directly from the line equation in the image plane where x and y represent axis in Cartesian coordinates and −a/b is the slope of the line:
ax+by+c=0. (3)
and c is the y axis intercept.
We denote with A_{r }and B_{r }the rotation from time t_{0 }to time t_{1 }and t_{2}, respectively. We use same subscripts for the translations a_{t }and b_{t}.
l_{1}=A_{r}l_{0}d_{1}=A_{r}d_{0}+a_{t}×A_{r}l_{0} (3.1)
l_{2}=B_{r}l_{0}d_{2}=B_{r}d_{0}+b_{t}×B_{r}l_{0}. (3.2)
If we eliminate 1 and ∥d∥ we obtain an equation with the only unknowns being the rotations and translations.
A_{r}^{T}d_{1}=d_{0}+A_{r}^{T}a_{t}×l_{0} (4)
B_{r}^{T}d_{2}=d_{0}+B_{r}^{T}b_{t}×l_{0} (5)
taking the vector product from left hand side and the right hand side.
A_{r}^{T}d_{1}×B_{r}^{T}d_{2}=d_{0}×(B_{r}^{T}b_{t}×l_{0})+(A_{r}^{T}a_{t}×l_{0})×d_{0}+(A_{r}^{T}a_{t}×l_{0})×(B_{r}^{T}b_{t}×l_{0}). (6)
This yields our geometric consistency equation for three frames with line correspondences which contains only rotations
n_{0}^{T}(A_{r}^{T}n_{1}×B_{r}^{T}n_{2})=0. (7)
First we compute the rotation matrices. This can be solved by iterative minimization. Next we are able to find two equations for translations which can be solved as a system of linear equations using the solved rotations from the previous step. To compute the translations, we use the following equations:
d_{0}×A_{r}^{T}d_{1}+d_{1}^{T}a_{t}l_{0}=0 (8)
d_{0}×B_{r}^{T}d_{2}+d_{2}^{T}b_{t}l_{0}=0. (9)
n_{2}^{T}b_{t}(n_{0}×A_{r}^{T}n_{1})=n_{1}^{T}a_{t}(n_{0}×B_{r}^{T}n_{2}). (10)
 8. For every new frame, the computation of the absolute rotation using current estimates of lines and points in space derived from the last step is performed. The following steps illustrate an exemplary method for calculating the absolute rotation of the camera.
 a. Update the absolute rotation of the camera based on the product of the current rotation and the interframe rotation computed from tripletframes based on the product of vanishing keypoint rotations.
 i. Step a only needs to be performed if step b yields a confidence below a threshold.
 b. Update the absolute rotation of the camera based on the product of the current rotation and the interframe rotation computed from tripletframes based on keypoint correspondences.
 i. Step b only needs to be performed if step c yields a confidence below a threshold.
 c. Update the absolute rotation of the camera based in the product of the current rotation and the interframe rotation computed from three frames based on line correspondences.
 d. Estimate the absolute rotation of the camera given estimates of current 3D lines positions in space and corresponding matches of 2D lines in images.
 e. Fuse the rotation estimates in steps 8ad, including a potential estimate read from an Inertial Measurement Unit (IMU) if available. For example, the camera may have an onboard IMU that senses and outputs its own rotation. If an IMU is present, the output from the IMU may be used to verify rotation estimates.
 i. The rotation matrices produced by 8ad are multiplied by a confidence from their relative covariance matrices and summed.
 a. Update the absolute rotation of the camera based on the product of the current rotation and the interframe rotation computed from tripletframes based on the product of vanishing keypoint rotations.
 9. For every new frame, compute the absolute position of the camera using current estimates of lines and points in space. The absolute position of the camera for each new frame may be computed using the following steps:
 a. Estimate absolute position of the camera given estimates of current 3D points in space adjusted with the absolute rotation and corresponding matches of 2D points in images is performed.
 b. Estimate absolute position of the camera given estimates of current 3D lines positions in space adjusted with the absolute rotation and corresponding matches of 2D lines in images is performed.
 c. Fuse the translation estimates in 8ad adjusted with potential acceleration measurements read from an Inertial Measurement Unit if available is performed.
 i. The rotation matrices produced by 8ad are multiplied by a confidence from their relative covariance matrices and summed.
 10. The global scale of all measurements can be adjusted at a specified frequency as error propagates.
 a. Because of the use of the triplet, absolute scale of current triangulation can be immediately obtained at the next frame using the scale between the previous frame and the current frame. Having triangulations in the same scale an update of the keypoint/line absolute positions as well as an update of the absolute translation and rotation can be obtained at each step using ICL (Iterative Closest Line) or ICP (Iterative Closest Point) over the 3D line segments or points in order to increase accuracy of pose and map estimation.
 b. Conventional Bundle Adjustment methods can also be used.
 8. For every new frame, the computation of the absolute rotation using current estimates of lines and points in space derived from the last step is performed. The following steps illustrate an exemplary method for calculating the absolute rotation of the camera.
Referring to
Thus, the subject matter described herein includes improved methods for visual odometry using the antipodal transform. The methods described herein improve the technological field of visual odometry by accurately identify feature correspondence between frames and with less computational resources required by convention methods. The methods described herein improve the functionality of a computer that computes visual odometry correspondences because less computational resources are required than with convention methods. A computer programmed with the visual odometry methods described herein thus constitutes a special purpose processing device with applications to navigation, augmented reality, robotics, and other technological fields where tracking camera motion from video images captured by the camera is desirable.
The documents corresponding to each of the following citations is hereby incorporated herein in its entirety.
The following citations provided additional detail on the terms in parentheses preceding each citation and which appear in the description hereinabove. (ICL)
 Alshawa, Majd. “ICL: Iterative closest line A novel point cloud registration algorithm based on linear features.” Ekscentar 10 (2007): 5359. (ICP)
 Rusinkiewicz, Szymon, and Marc Levoy. “Efficient variants of the ICP algorithm.” 3D Digital Imaging and Modeling, 2001. Proceedings. Third International Conference on. IEEE, 2001. (LevenbergMarquardt)
 Moré, Jorge J. “The LevenbergMarquardt algorithm: implementation and theory.” Numerical analysis. Springer Berlin Heidelberg, 1978. 105116. (Nested Shape Descriptors)
 J. Byrne and J. Shi, “Nested Shape Descriptors”, International Conference on Computer Vision (ICCV'"'"'13), Sydney Australia, 2013. Visual Odometry
 Nistér, David, Oleg Naroditsky, and James Bergen. “Visual odometry.” Computer Vision and Pattern Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE Computer Society Conference on. Vol. 1. IEEE, 2004. (Distance Transform)
 Felzenszwalb, Pedro, and Daniel Huttenlocher. Distance transforms of sampled functions. Cornell University, 2004. (Distance Transform)
 Ragnemalm, Ingemar. “The Euclidean distance transform in arbitrary dimensions.” Pattern Recognition Letters 14.11 (1993): 883888. (Essential Matrix)
 LonguetHiggins, H. Christopher. “A computer algorithm for reconstructing a scene from two projections.” Readings in Computer Vision: Issues, Problems, Principles, and Paradigms, M A Fischler and O. Firschein, eds (Sep. 10, 1981): 133135. (Singular Value Decomposition)
 Golub, Gene H., and Charles F. Van Loan. Matrix computations. Vol. 3. JHU Press, 2012. (Bundle Adjustment)
 Triggs, Bill, et al. “Bundle adjustment—a modern synthesis.” Vision algorithms: theory and practice. Springer Berlin Heidelberg, 2000. 298372.
It will be understood that various details of the presently disclosed subject matter may be changed without departing from the scope of the presently disclosed subject matter. Furthermore, the foregoing description is for the purpose of illustration only, and not for the purpose of limitation.