This article describes an implementation of the optical flow estimation method introduced by Zach, Pock and Bischof in 2007. This method is based on the minimization of a functional containing a data term using the L 1 norm and a regularization term using the total variation of the flow. The main feature of this formulation is that it allows discontinuities in the flow field, while being more robust to noise than the classical approach by Horn and Schunck. The algorithm is an efficient numerical scheme, which solves a relaxed version of the problem by alternate minimization. Source CodeA C implementation of this algorithm is provided. The source code and an online demo are accessible at the web page of this article 1 .

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ABSTRACT:The increasing availability of high resolution stereo images from Earth observation satellites has boosted the development of tools for producing 3D elevation models. The objective of these tools is to produce digital elevation models of very large areas with minimal human intervention. The development of these tools has been shaped by the constraints of the remote sensing acquisition, for example, using ad hoc stereo matching tools to deal with the pushbroom image geometry. However, this specialization has also created a gap with respect to the fields of computer vision and image processing, where these constraints are usually factored out. In this work we propose a fully automatic and modular stereo pipeline to produce digital elevation models from satellite images. The aim of this new pipeline, called Satellite Stereo Pipeline and abbreviated as s2p, is to use (and test) off-the-shelf computer vision tools while abstracting from the complexity associated to satellite imaging. To this aim, images are cut in small tiles for which we proved that the pushbroom geometry is very accurately approximated by the pinhole model. These tiles are then processed with standard stereo image rectification and stereo matching tools. The specifics of satellite imaging such as pointing accuracy refinement, estimation of the initial elevation from SRTM data, and geodetic coordinate systems are handled transparently by s2p. We demonstrate the robustness of our approach on a large database of satellite images and by providing an online demo of s2p.Figure 1: 3D point clouds automatically generated from Pléiades stereo datasets, without any manual intervention, with the s2p stereo pipeline. Its implementation can be tested online through a web browser.

The seminal work of Horn and Schunck is the first variational method for optical flow estimation. It introduced a novel framework where the optical flow is computed as the solution of a minimization problem. From the assumption that pixel intensities do not change over time, the optical flow constraint equation is derived. This equation relates the optical flow with the derivatives of the image. There are infinitely many vector fields that satisfy the optical flow constraint, thus the problem is ill-posed. To overcome this problem, Horn and Schunck introduced an additional regularity condition that restricts the possible solutions. Their method minimizes both the optical flow constraint and the magnitude of the variations of the flow field, producing smooth vector fields. One of the limitations of this method is that, typically, it can only estimate small motions. In the presence of large displacements, this method fails when the gradient of the image is not smooth enough. In this work, we describe an implementation of the original Horn and Schunck method and also introduce a multi-scale strategy in order to deal with larger displacements. For this multi-scale strategy, we create a pyramidal structure of downsampled images and change the optical flow constraint equation with a nonlinear formulation. In order to tackle this nonlinear formula, we linearize it and solve the method iteratively in each scale. In this sense, there are two common approaches: one that computes the motion increment in the iterations; or the one we follow, that computes the full flow during the iterations. The solutions are incrementally refined over the scales. This pyramidal structure is a standard tool in many optical flow methods. Source Code A standalone ANSI C implementation is available 1. This file contains two main programs: horn schunck classic.c, which implements the original Horn and Schunck method; and the implementation of the multi-scale approach, in file horn schunck pyramidal.c. This latter implementation is best suited for general image sequences.