Cube Image Derivatives Dataset

This dataset contains grayscale 41 × 41 images of a cube together with first- and second-order derivatives of the images with respect to six rigid transformations: translations X, Y, Z and rotations RotX, RotY, RotZ about the basis axes through the cube center. For each sample, 28 channels are provided: the base image, 6 first derivatives, and 21 second derivatives. The dataset also includes geometric data: the coordinates of the three vertices adjacent to a common cube vertex, together with the corresponding first- and second-order derivatives of these coordinates with respect to the same transformations. The remaining vertices can be reconstructed from these three vertices, given the fixed cube side length of 0.4. The main HDF5 datasets are /img/data with shape {N, 28, 41, 41} and /geo/data with shape {N, 28, 9}. Small transformations of the image $\mathcal{I}$ of a cube $C$, corresponding to translations by $(x,y,z)$ and rotations by $(\alpha,\beta,\gamma)$ angles about the $x$, $y$ and $z$-axes are given by \[\begin{split}T_x(x)\,T_y(y)\,T_z(z)\,R_z(\gamma)\,R_y(\beta)\,R_x(\alpha)\,\mathcal{I}(C)\;\approx\;\mathcal{I}(C)+\begin{pmatrix}2 & 3 & 4 & 5 & 6 & 7\end{pmatrix}\begin{pmatrix}x\\y\\z\\\alpha\\\beta\\\gamma\end{pmatrix}\\[1ex]+\frac{1}{2}\begin{pmatrix}x & y & z & \alpha & \beta & \gamma\end{pmatrix}\begin{pmatrix}8 & 9 & 10 & 11 & 12 & 13\\9 & 14 & 15 & 16 & 17 & 18\\10 & 15 & 19 & 20 & 21 & 22\\11 & 16 & 20 & 23 & 24 & 25\\12 & 17 & 21 & 24 & 26 & 27\\13 & 18 & 22 & 25 & 27 & 28\end{pmatrix}\begin{pmatrix}x\\y\\z\\\alpha\\\beta\\\gamma\end{pmatrix}.\end{split}\] Here the numbers denote the corresponding dataset components. The corresponding expression for the cube vertices is analogous.