chapter 6 feature-based alignment advanced computer vision
TRANSCRIPT
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Chapter 6Feature-based alignment
Advanced Computer Vision
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Feature-based Alignment
• Match extracted features across different images
• Verify the geometrically consistent of matching features
• Applications:– Image stitching– Augmented reality– …
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Feature-based Alignment
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Feature-based Alignment
• Outline:– 2D and 3D feature-based alignment– Pose estimation– Geometric intrinsic calibration
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2D and 3D Feature-based Alignment
• Estimate the motion between two or more sets of matched 2D or 3D points
• In this section:– Restrict to global parametric transformations– Curved surfaces with higher order transformation– Non-rigid or elastic deformations will not be
discussed here.
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2D and 3D Feature-based Alignment
Basic set of 2D planar transformations
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2D and 3D Feature-based Alignment
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2D Alignment Using Least Squares
• Given a set of matched feature points • A planar parametric transformation:
• are the parameters of the function • How to estimate the motion parameters ?
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2D Alignment Using Least Squares
• Residual:
• : the measured location• : the predicted location
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2D Alignment Using Least Squares
• Least squares:– Minimize the sum of squared residuals
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2D Alignment Using Least Squares
• Many of the motion models have a linear relationship:
• : The Jacobian of the transformation
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2D Alignment Using Least Squares
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2D Alignment Using Least Squares
• Linear least squares:
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2D Alignment Using Least Squares
• Find the minimum by solving:
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Iterative algorithms
• Most problems do not have a simple linear relationship– non-linear least squares– non-linear regression
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Iterative algorithms
• Iteratively find an update to the current parameter estimate by minimizing:
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Iterative algorithms
• Solve the with:
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Iterative algorithms
• : an additional damping parameter– ensure that the system takes a “downhill” step in
energy– can be set to 0 in many applications
• Iterative update the parameter
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Projective 2D Motion
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Projective 2D Motion
• Jacobian:
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Projective 2D Motion
• Multiply both sides by the denominator() to obtain an initial guess for
• Not an optimal form
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Projective 2D Motion
• One way is to reweight each equation by :
• Performs better in practice
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Projective 2D Motion
• The most principled way to do the estimation is using the Gauss–Newton approximation
• Converge to a local minimum with proper checking for downhill steps
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Projective 2D Motion
• An alternative compositional algorithm with simplified formula:
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Robust least squares
• More robust versions of least squares are required when there are outliers among the correspondences
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Robust least squares
• M−estimator:apply a robust penalty function to the residuals
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Robust least squares
• Weight function • Finding the stationary point is equivalent to
minimizing the iteratively reweighted least squares:
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RANSAC and Least Median of Squares
• Sometimes, too many outliers will prevent IRLS (or other gradient descent algorithms) from converging to the global optimum.
• A better approach is find a starting set of inlier correspondences
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RANSAC and Least Median of Squares
• RANSAC (RANdom SAmple Consensus)• Least Median of Squares
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RANSAC and Least Median of Squares
• Start by selecting a random subset of correspondences
• Compute an initial estimate of • RANSAC counts the number of the inliers,
whose • Least median of Squares finds the median of
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RANSAC and Least Median of Squares
• The random selection process is repeated times
• The sample set with the largest number of inliers (or with the smallest median residual) is kept as the final solution
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Preemptive RANSAC
• Only score a subset of the measurements in an initial round
• Select the most plausible hypotheses for additional scoring and selection
• Significantly speed up its performance
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PROSAC
• PROgressive SAmple Consensus• Random samples are initially added from the
most “confident” matches• Speeding up the process of finding a likely
good set of inliers
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RANSAC
• must be large enough to ensure that the random sampling has a good chance of finding a true set of inliers:
• : • :
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RANSAC
• Number of trials to attain a 99% probability of success:
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RANSAC
• The number of trials grows quickly with the number of sample points used
• Use the minimum number of sample points to reduce the number of trials
• Which is also normally used in practice
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3D Alignment
• Many computer vision applications require the alignment of 3D points
• Linear 3D transformations can use regular least squares to estimate parameters
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3D Alignment
• Rigid (Euclidean) motion:
• We can center the point clouds:
• Estimate the rotation between and
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3D Alignment
• Orthogonal Procrustes algorithm• computing the singular value decomposition
(SVD) of the 3 × 3 correlation matrix:
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3D Alignment
• Absolute orientation algorithm• Estimate the unit quaternion corresponding to
the rotation matrix • Form a 4×4 matrix from the entries in • Find the eigenvector associated with its largest
positive eigenvalue
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3D Alignment
• The difference of these two techniques is negligible
• Below the effects of measurement noise• Sometimes these closed-form algorithms are
not applicable• Use incremental rotation update
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Pose Estimation
• Estimate an object’s 3D pose from a set of 2D point projections– Linear algorithms– Iterative algorithms
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Pose Estimation - Linear Algorithms
• Simplest way to recover the pose of the camera
• Form a set of linear equations analogous to those used for 2D motion estimation from the camera matrix form of perspective projection
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Pose Estimation - Linear Algorithms
• : measured 2D feature locations• : known 3D feature locations
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Pose Estimation - Linear Algorithms
• Solve the camera matrix in a linear fashion• multiply the denominator on both sides of the
equation• Denominator():
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Pose Estimation - Linear Algorithms
• Direct Linear Transform (DLT)• At least six correspondences are needed to
compute the 12 (or 11) unknowns in • More accurate estimation of can be obtained
by non-linear least squares with a small number of iterations.
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Pose Estimation - Linear Algorithms
• Recover both the intrinsic calibration matrix and the rigid transformation
• and can be obtained from the front 3 × 3 sub-matrix of using factorization
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Pose Estimation - Linear Algorithms
• In most applications, we have some prior knowledge about the intrinsic calibration matrix
• Constraints can be incorporated into a non-linear minimization of the parameters in and
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Pose Estimation - Linear Algorithms
• In the case where the camera is already calibrated: the matrix is known
• we can perform pose estimation using as few as three points
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Pose Estimation - Linear Algorithms
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Pose Estimation - Linear Algorithms
• : unit directions of • The visual angle between any pair of 2D
points and must be the same as the angle between their corresponding 3D points and
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Pose Estimation - Linear Algorithms
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Pose Estimation - Linear Algorithms
• The cosine law gives:
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Pose Estimation - Linear Algorithms
• We can take any triplet of constraints and using Sylvester resultants to obtain a quartic equation in :
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Pose Estimation - Linear Algorithms
• Given five or more correspondences, we can obtain a linear estimate (using SVD) for the values of
• Computed to estimate and
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Pose Estimation - Linear Algorithms
• We can generate a 3D structure consisting of the scaled point directions
• Use 3D Alignment to obtain the desired pose estimation
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Pose Estimation - Linear Algorithms
• Minimal PnP solutions can be quite noise sensitive
• Also suffer from bas-relief ambiguities • Use the linear algorithm to obtain an initial
pose estimation• Optimize this estimation using the iterative
algorithm
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Pose Estimation - Iterative Algorithms
• The most accurate (and flexible) way to estimate pose is using non-linear least squares
• Projection equations:
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Pose Estimation - Iterative Algorithms
• Iteratively minimize the robustified linearized reprojection errors:
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Pose Estimation - Iterative Algorithms
• An easier to understand (and implement) version of the above non-linear regression problem can be constructed by re-writing the projection equations as a concatenation of simpler steps
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Pose Estimation - Iterative Algorithms
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Geometric Intrinsic Calibration
• Intrinsic and extrinsic calibration can be computed at the same time– Intrinsic calibration:
Internal camera calibration parameters– Extrinsic calibration:
Pose of the camera• The classic approach to camera calibration
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Calibration patterns
• One of the more reliable ways to estimate a camera’s intrinsic parameters
• In photogrammetry, it is common to set up a camera in a large field looking at distant calibration targets whose exact location has been pre-computed using surveying equipment
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Calibration patterns
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Calibration patterns
• Span the calibration object as much of the workspace as possible if a smaller calibration rig needs to be used
• Estimate the covariance in the parameters to determine the quality of calibration
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Vanishing points
• A man-made scene with a lot of rectangular objects such as boxes or room walls
• Intersect the 2D lines corresponding to 3D parallel lines to compute their vanishing points
• Use these to determine the intrinsic and extrinsic calibration parameters
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Vanishing points
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Vanishing points
• Assume a simplified form for the calibration matrix where only the focal length is unknown
• : , , or • : th column of the rotation matrix
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Vanishing points
• Columns of the rotation matrix are orthogonal
• We can obtain an estimate for
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Vanishing points
• It is also possible to estimate the optical center as the orthocenter of the triangle formed by the three vanishing points
• It is more accurate to re-estimate usingnon-linear least squares
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Rotational motion
• When no calibration targets are available but you can rotate the camera around its front nodal point
• We can calibrate the camera can from a set of overlapping images by assuming that it is undergoing pure rotational motion
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Rotational motion
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Radial distortion
• When images are taken with wide-angle lenses, it is often necessary to model lens distortions such as radial distortion– barrel distortion– pincushion distortion
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Radial distortion
• : radial distortion parameters
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Radial distortion
• A lot of different ways to estimate the radial distortion parameters
• One of the simplest and most useful way:– Take an image of a scene with a lot of
straight lines– Adjust the parameters until all of the lines are
straight– Plumb-line method
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Radial distortion
• Another approach is to use several overlapping images and to combine the estimation of the radial distortion parameters with the image alignment process
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Radial distortion
• We can add another stage to the generalnon-linear minimization pipeline when a known calibration target is used
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Radial distortion
• Sometimes we need more general models of lens distortion
• The general approach of either using calibration rigs with known 3D positions or self-calibration through the use of multiple overlapping images of a scene can both be used