Key words: Velocity Filtering, Matched Filtering, Optical Flow, Passive Ranging.
Velocity filtering is a track-before-detect algorithm that can be utilized to perform field-based passive ranging. In this paper we expand on the theoretical understanding and the performance analysis of the velocity filtering algorithm as applied to optical-flow-based ranging.
Passive ranging is an area of considerable interest for applications such as rotorcraft nap-of-the-earth navigation and spacecraft landing. A forward-looking imaging sensor, such as a FLIR, typically produces the optical flow information necessary for range computation. The principle of passive ranging using optical flow is akin to triangulation; the optical flow is a sequence of angular measurements collected from different locations along the vehicle's flight path with respect to all points in the field of view (FOV).
There are two basic ranging approaches: feature-based and field-based; the first only deals with some chosen objects, while the second regards the scene as a continuum. Feature-based methods suffer from the need to identify objects between successive frames; this is not required with field-based methods because they track all points in the FOV. The general method of Velocity Filtering has been suggested as a potential field-based method.
The relevance of Velocity Filtering to optical-flow passive ranging stems from the concept of filtering in the three-dimensional Fourier Transform (3D-FT) domain (e.g., see [1]). Reed [2] suggested the use of 3D Matched Filtering (MF) for detecting moving constant-velocity objects. This paper develops the MF equations and the Signal-To-Noise (SNR) expressions for the MF output in the frequency domain.
Porat and Friedlander [3] apply filtering in 3D-FT for the purpose of detecting moving objects in mosaic-sensor imagery. They use a bank of directional filters where each filter is defined as a plane in the 3D-FT domain. In the same context, Fries [4] develops a matched filter in the hybrid 2D-FT/1D-temporal domain. His derivation is very general in including camera jitter noise and non-white clutter background.
Stotts [5, 6] implements the MF suggested by Reed [2] and obtains experimental results for the detection of an aircraft-like object. He derives an expression for the filtering SNR and loss as a result of velocity mismatch.
Application of the space-domain implementation of velocity filtering to optical flow was suggested [7] as a field-based method. This work uses without proof the space-domain equivalent of filtering in 3D-FT.
Smith [8], applies the principles of optical flow directly to obstacle avoidance (feature-based) without the intermediate step of ranging by determining whether an object is in or out of the ``Tunnel of Safe Passage". Error analysis and experimental results with real imagery are presented. The algorithm's performance seems to be very dependent on the scene contents.
Our purpose in this paper is to evaluate the performance of the velocity filtering method as applied to optical-flow passive ranging under real-world constraints. We start by reviewing (in Sections 2 and 3) the theory of 3D-FT as applied to constant-speed moving points. In Section 4 we derive the space-domain shift-and-add algorithm from the general 3D Matched-Filtering formulation. In section 5 we find the 2D velocity pass-band of the filter and develop a variable-velocity (hyperbolic) filter for the particular use in optical-flow. We further calculate in Section 6 the depth (range) pass-band and accuracy of this new filter. In Section 7 we address the problems associated with pixel interpolation and object expansion and present experimental results.