The paper presents a new theory of combined moment invariants to Gaussian blur and spatial affine transformation. The blur kernel may be arbitrary oriented, scaled and elongated. No prior information about the kernel parameters and about the underlaying affine transform is required. The main idea, expressed by the Substitution Theorem, is to substitute pure blur invariants into traditional affine moment invariants. Potential applications of the new descriptors are in blur-invariant image recognition and in robust template matching.

We present a method to reconstruct a dense spatio-temporal depth map of a non-rigidly deformable object directly from a video sequence. The estimation of depth is performed locally on spatio-temporal patches of the video, and then the full depth video of the entire shape is recovered by combining them together. Since the geometric complexity of a local spatio-temporal patch of a deforming non-rigid object is often simple enough to be faithfully represented with a parametric model, we artificially generate a database of small deforming rectangular meshes rendered with different material properties and light conditions, along with their corresponding depth videos, and use such data to train a convolutional neural network. We tested our method on both synthetic and Kinect data and experimentally observed that the reconstruction error is significantly lower than the one obtained using other approaches like conventional non-rigid structure from motion.