Lec2 for ML-4360¶
2.1 Primitives & Transformations¶
\(Homogeneous\)¶
- homogeneous coordinates & inhomogeneous coordinates
- argmented vector[one element out of the whole equivalent class],homogeneous vectors,homogeneous vectors
- points at infinity
- In homogeneous coordinates, the intersection of two lines is given by:
\(\tilde{x}\)=\(\tilde{l1}\) × \(\tilde{l2}\)
- the line joining two points can be compactly written as: \(\tilde{l}\)=\(\tilde{x1}\) × \(\tilde{x2}\)
\(Transformations\)¶
- \(\tilde{l}'\) = \((\tilde{H}^T)^{-1}\)\(\tilde{l}\)
translation 2DOF¶
\(\begin{bmatrix}I&t\\0^{T}&1\end{bmatrix}\)
Euclidean 3DOF¶
\(RR^T=I \ der(R)=1\)
\(\begin{bmatrix}R&t\\0^{T}&1\end{bmatrix}\)
正交矩阵\(A^TA=I\)
- \(A^{-1}=A^T\)
- \(\left|A\right|\) = \(+-1\)
- A的行（列）向量组为n维单位正交向量组
- 正交变换保持向量的长度与内积不变 \(|\sigma\alpha|=|\alpha|\)
Similarity 4DOF¶
\(\begin{bmatrix}sR&t\\0^{T}&1\end{bmatrix}\)
\(RR^{T}=I\)
Affine: 6DOF¶
\(\begin{bmatrix}A&t\\0^{T}&1\end{bmatrix}\)
-
arbitrary \(2×2\) matrix
-
Parallels Remain!
Projective :8DOF¶
- preserve straight lines
- \(\tilde{H}\in R_{3\times 3}\)is an arbitrary homogeneous \(3 × 3\) matrix (specified up to scale)
- DOF（2D）:n(n-1)/2
Direct Linear Transform for Homography Estimation[algorithm DLT¶
- Please Refer to Structure from Motion
2.2 Geometric Image Formation¶
Orthographic projection¶
An orthographic projection simply drops the z component of the 3D point in camera
coordinates \(x_c\) to obtain the corresponding 2D point on the image plane (= screen) \(x_s\)
- Scaled -- The unit for s is \(px/m\) or \(px/mm\) to convert metric 3D points into pixels.
\(\bar{x_s}=\begin{bmatrix}s&0&0&0\\0&s&0&0\\0&0&0&1\end{bmatrix}\).
Perspective projection¶
Chaining Transformations¶
\(\tilde{x}_s=K[R \ t]\bar{x}_w\)
我的理解：\(\bar{x}_s\)是真实的相机坐标系下的坐标，与\(\tilde{x}_s\)差一个系数，如果我们知道这个系数，就可以恢复出\(\tilde{x}_s\)并根据\(\tilde{x}_s\)的计算方法恢复出世界坐标系下的坐标\(\bar{x}_c\)
2.3 Photometric Image Formation¶
\(Rendering\ Equation\)¶
- intensity :power per solid angle
\(dw=\sin\theta\)\(d\theta\)\(d\phi\)
- Irradiance :power per unit area
E(x)=d\(\Phi(x)\)/dA
- Radiannce
\(L(p,w)=d^2\Phi(p,\omega)\)/\(d\omega\)\(dAcos\theta\)
\(BRDF\) "\(Radiance_{out}/Irradiance_{in}\)"¶
\(The\ Reflection\ Equation\)¶
\(L_r(p,w_r)=\int_{H^2}f_r(p,w_i\rightarrow w_r)L_i(p,w_i)cos\theta_id_{w_i}\)
Back To Cameras¶
创建日期: 2023年11月4日 10:47:52