« « n m n r i ï i i r w A t : -«wr„h,-y»' W3. mmw a«PW:.TJz^s**-
This chapter describes the mathem atical and com putational structure of the M ulti channel Gradient Model (McGM) [Johnston99][Johnston98]. The basic im plementation of the algorithm is discussed and some results presented.
3.1 Overview o f th e McGM
The Multi-channel G radient Model (McGM) computes optic flow using an enhanced gradient technique th a t makes use of many spatio-temporal derivative filters. The model extends the basic motion constraint equation by expressing the image sequence as a Taylor expansion, thus introducing higher order derivatives into the image description. Use of higher order derivatives in a Taylor expansion framework has been shown to add robustness to optical flow calculations [Gtte95] and provides a convenient method for analysing image structure contained within a small region of an image [Bainbridge97]. The ratio of the spatial to temporal derivative of any one of the terms in the Taylor expansion can be used to compute image speed, although each individual ratio will be ill conditioned by itself. The McGM combines multiple term s and pools over
space-time to arrive at a much more robust measurement. In addition, the Taylor
expansion is extracted at a number of spatial orientations th a t correspond to the orientation columns found in the prim ate visual cortex. Speed is then measured as a function of spatial orientation. This speed function allows the translational component to be separated from differential components of optic flow.
3.2 T he Im age as a Taylor Expansion
We express an image sequence I{x,y,t) as a Taylor expansion. This expansion yields a vector of gradient measures up to order l,m,n in the three dimensions:
I m n i j k
I{x + p ,y + q,t + r) = J 2 (3-1)
1 = 0 j = Q k = 0
where is the derivative operator • In order to form the Taylor
expansion and extract the image structure, the image m ust be differentiated in space and time. A derivative of a function can be obtained in a well-posed way by convolving it w ith the derivative of a smooth test function [Shwartz50] [Florack96]. The Multi-channel G radient model uses convolution with the Gaussian and its derivatives to differentiate the
image in space. The Gaussian is a unique function in many ways
[Koenderink87] [Koenderink88] and is of particular importance to biology since the derivatives of a Gaussian are considered to approxim ate the sensitivity profiles of simple cell receptive fields in the visual cortex [Young93][Koenderink87]. The Gaussian function is given by equation (3.2)
Go(a;,(7) — 2^2 (3.2)
where a is the standard deviation of the Gaussian and the scale factor ensures the function integrates to unity. This function is plotted in figure 3.1. The derivative the Gaussian can be expressed as a polynomial multiplied by the original Gaussian:
G n M = = i f k T (3 3)
the multiplying polynomial function H^( x) is the Hermite polynomial which can be calculated through the relation:
Gaussian G(x,sigma) sigma=2.5 0.2 0.15 - 0.05 - Position X
F igure 3.1 T he norm alised G aussian blur filter from equation (3 .2 ) with standard deviation a of 2.5 pixels. T he function is sym m etric about zero.
First Derivative of Gaussian G1(x,sigma) sigma=2.5 0.06 -I 0.04 - 0.02 - X O -0.02- ' -0.04 - -0.06 - Position X
F igu re 3.2 T he first derivative of a Gaussian generated using equation (3 .3 ). T h is first order function has one zero crossing, sp littin g the curve into a positive and a negative region.
Second Derivative of Gaussian G2(x,sigma) sigma=2.5 0.02 0.01 - -0.02 - -0.03 Position X
Figu re 3.3 T he second derivative o f a G aussian generated using equation (3 .3 ). T h is second order function has tw o zero crossings, sp littin g the curve into three
The first two derivatives are plotted in figures 3.2 and 3.3. The functions have a number of zero crossings equal to their order of differentiation, splitting each curve into regions of positive and negative values. Images are 2-dimensional, thus in the McGM, 2- dimensional spatial filters are required. The 2-dimensional Gaussian is unique among all other circularly symmetric (rotation invariant) kernels in th a t it can be generated using two separate one-dimensional convolutions [KoenderinkSS]. These 2-dimensional Gaussian filters are generated through multiplication of two orthogonal 1-dimensional Gaussian derivatives:
(^5 (2/5 (3.4)
The separability of the 2-dimensional Gaussian derivatives will prove invaluable for the im plementation and optimisations detailed in Chapter 4. Gaussian filters can be convolved together such th a t higher order derivatives and/or lower resolution kernels can be generated:
Gn{x,(j{) (8) G^{x,a^) = (3-5)
W here ® indicates the convolution operator. Also, although it is of less im portance to us directly, the Gaussian is a solution of the diffusion equation and is unique in scale-space theory [Florack96][Koenderink87]. The diffusion equation (3.8) follows from the observations th at:
^Gr^{x,a) = G^^^{x,a) (3.6)
and thus:
-^Gn{x,a) = G^+2(z,(7) (3.7)
The utilisation of the diffusion equation for com putation at multiple scales will be explored in Chapter 7.
For the temporal dimension, Gaussian derivatives of log-time are used [Johnston94]. The temporal blur function is given by:
GLTAt,
T,a)= k
exp (3.9)where
k =
V^TCK ex p (^ )
The value of