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OPTICAL FLOW COMPUTATION IN THE LOG-POLAR PLANE 
Kostas Daniilidis and Volker Krüger 
Computer Science Institute, Christian-Albrechts University Kiel 
Preusserstr. 1-9, 24105 Kiel, Germany 
email: {kd,vok}@informatik.uni-kiel.de 
KEYWORDS: space-variant sensing, conformal mapping, optical flow. 
ABSTRACT: 
Vision systems with spatially homogeneous resolution are not able to provide a real time response in a dynamically 
changing environment. A reactive behavior necessitates selective sensing in space. Such a selection can be accomplished 
by the combination of a space-variant resolution scheme and a sensor with controllable degrees of freedom. The field 
of view is split into a homogeneous high resolution area - the fovea - and the periphery with decreasing resolution. 
Both in neurobiology and in robot vision, models of the resolution decrease towards the image boundaries have been 
established. The most convincing model is the theory of logarithmic polar mapping. In this paper we propose two new 
methods for the estimation of the optical flow and its spatial derivatives in the log-polar plane. We study analytically 
and experimentally the effects of the polar deformation and the decimation due to subsampling on the computation of 
optical flow. 
1 Introduction 
This paper is concerned with the computation of optical flow in image sequences obtained with the logarithmic polar 
transformation. The log-polar transformation is a model for space-variant resolution in the periphery of the image. 
Space-variant sensing arises as a necessity in systems which must be able to process simultaneously a central region of 
interest (fovea) in detail for recognition tasks and a wide-angle peripheral view for detecting events and new candidates 
for gaze change. Uniform resolution in the peripheral part would result to a computational burden unacceptable for 
real-time reactive behavior. The prerequisite for space-variant sensing is active sensing that means the capability to 
control the gaze direction. The biologically motivated log-polar transformation has a second significant advantage: It 
is a very rich representation regarding recognition tasks (rotation and scaling invariance) as well as navigational tasks 
(ego-motion and time to collision estimation, motion detection). 
We first shortly describe the basic anatomy of the primate's space-variant sensing. The retina consists of three layers. 
In the first layer we find the photo-receptor-cells (PRC) that code the visual information in terms of nervous impulses. 
The PRC are connected either directly via bipolar-cells or indirectly via amacrin- bipolar- and horizontal-cells in the 
second layer to the retinal ganglion cells (RGC) in the third layer. An area of PRC on the retina that are combined 
by one single RGC is called receptive field (RF). PRC are nonuniformly spaced over the human retina. Their heightest 
density can be found within the fovea centralis, but their density decreases with increasing eccentricity. The spatial 
density of RFs on the human retina is related to the density of the PRC. The highest density of RFs is therefore found 
in the fovea where some RF even consist of only one PRC each. In the periphery the density of RF decreases whereas 
their center-size increases nearly linearly with eccentricity. This allows keeping the amount of visual information as 
received by z 10? cones and rods low enough to be processed by only ~ 10° RGC and optic nerve fibers. [Schwartz, 
1977] proposed the complex logarithmic mapping for the retinotopic mapping of the RGC onto the first area of the visual 
cortex (V1). [Weiman and Chaikin, 1979] studied first the properties of the complex logarithmic transformation as a 
conformal mapping and they proposed a logarithmic spiral grid as a digitization scheme for both image synthesis and 
analysis. 
The goal of this paper is to study what kind of information is still preserved after the log-polar transformation which 
can be used for motion tasks. We use the optical flow as an intermediate step and study analytically and experimentally 
the effects of the polar deformation in sec. 3 and the logarithmic subsampling in sec. 5. In particular, 
e we prove that the polar transformation introduces fictitious gray-values curvature that leads to an erroneous 
elimination of the aperture problem, 
e we propose two new methods for the optical flow estimation in the log-polar domain that are superior to methods 
directly transferred from the cartesian domain and we experimentally study their performance in a real sequence, 
e we propose a basis for an analysis of the logarithmic subsampling that allows spectral techniques for the design of 
the necessary low-pass and gradient filters. 
We use (z,y) for the cartesian coordinates and (p,m) for the polar coordinates in the plane. By denoting with 
z — rt jy — pe?" a point in the complex plane the complex logarithmic (or log-polar) mapping is defined as 
w = In(z) (1) 
for every z # 0 where Re(w) = Inp and Im(w) = n + 2kx. To exclude the periodicity of the imaginary part we constrain 
the range of Im(w) to [0, 27). The complex logarithmic mapping is a well-known conformal mapping preserving the angle 
of the intersection of two curves. It is trivial to show that every scaling and rotation about the origin in the z-plane is 
represented in the w-plane by a shift parallel to the real and imaginary axis, respectively. 
IAPRS, Vol. 30, Part 5W1, ISPRS Intercommission Workshop “From Pixels to Sequences”, Zurich, March 22-24 1995