Normalize the Coordinates
Compute a similarity transformation \(T_1\) and \(T_2\) such that the centroid of the points is moved to the origin, and the maximum distance from the origin is scaled to \(\sqrt{2}\).
Formulate the Equation
For each pair of correspondences \((x_1,y_1,1)\) and \((x_2,y_2,1)\), expanding the equation \((x_2,y_2,1)^T\mathbf{F}(x_1,y_1,1) = 0\), we obtain the following expressions:
\[ x1x2F_{11}+x2y1F_{12}+x2F_{13}+y2x1F_{21}+y1y2F_{22}+y2F_{23}+x1F_{31}+y1F_{32}+F_{33}=0 \]
Combining these equations results in a linear equation for \(\mathbf{F}\).
Solve for F
Using Singular Value Decomposition (SVD) to solve for \(\mathbf{F}\), the final result is given by \(\mathbf{T_2}^T \mathbf{F} \mathbf{T_1}\).
| Viewpoint 1 (Points) | Viewpoint 2 (Epipolar Lines) |
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Using the formula \(\mathbf{E} = \mathbf{K}_2^T \mathbf{F} \mathbf{K}_1\), we get the estimated \(\mathbf{E}\) as \[ \begin{bmatrix} -0.41524274 & -8.81748894 & -2.15055808 \\ -1.3803559 & 0.79067134 & 69.08265036 \\ -3.58890876 & -68.47438701 & 1. \end{bmatrix} \]
Formulate the Equation
After formulating the equation using the eight-point algorithm, we
apply Singular Value Decomposition (SVD) to obtain the solution space,
represented as \(<\mathbf{M}_1,
\mathbf{M}_2>\). The general solution of the linear equation
can be expressed as \(\lambda \mathbf{M}_1 +
(1 - \lambda) \mathbf{M}_2\). Next, we use the fact that the
fundamental matrix has rank two, and solve the equation \(\det(\lambda \mathbf{M}_1 + (1 - \lambda)
\mathbf{M}_2) = 0\) using
np.polynomial.polynomial.polyroots. If there are three real
solutions, we must select the correct one. For this assignment, we
assume the epipoles lie outside the image, which helps identify the
correct solution.
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We sample the minimum number of points required to estimate \(\mathbf{F}\) using the RANSAC algorithm. To determine if a correspondence pair is an inlier, we compute the sum of squared distances between the corresponding points and their estimated epipolar lines. If the error falls below a predefined threshold, the pair is classified as an inlier.
| Viewpoint 1 (Points) | Viewpoint 2 (Epipolar Lines) | % of inliers vs. # of iterations |
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For a 3D point \(\mathbf{x} = (x, y, z)\) with its projection on the image plane at ((u, v)), we have the constraint \((\mathbf{P} (x, y, z)^T) \times (u, v)^T = 0\). This allows us to formulate a \(4 \times 3\) matrix \(\mathbf{A}\) such that \(\mathbf{A} \mathbf{x} = 0\). The solution to this linear system can then be obtained using singular value decomposition (SVD).
| Multi-view images | Output |
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We use cv2.ORB_create to extract features and
cv2.BFMatcher to compute potential matches. Then, we use
the RANSAC algorithm implemented in q2 to compute \(\mathbf{F}\)
| Viewpoint 1 (Points) | Viewpoint 2 (Epipolar Lines) |
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