Stability of Phase Information.
This paper concerns the robustness of local phase information for measuring image velocity and binocular disparity. It addresses the dependence of phase behavior on the initial filters as well as the image variations that exist between different views of a 3D scene. We are particularly interested in the stability of phase with respect to geometric deformations, and its linearity as a function of spatial position. These properties are important to the use of phase information, and are shown to depend on the form of the filters as well as their frequency bandwidths. Phase instabilities are also discussed using the model of phase singularities described by Jepson and Fleet. In addition to phase-based methods, these results are directly relevant to differential optical flow methods and zero-crossing tracking.
Introduction
An important class of image matching techniques has emerged based on phase information; that is,
the phase behaviour in band-pass filtered versions of different views of a 3-d scene [3, 7, 8, 11, 13, 15,
17, 18, 25, 27, 30, 32, 33]. These include phase-difference and phase-correlation techniques for discrete
two-view matching, and the use of the phase gradient for the measurement of image orientation and
optical flow.
Numerous desirable properties of these techniques have been reported: For two-view matching, the
disparity estimates are obtained with sub-pixel accuracy, without requiring explicit sub-pixel signal
reconstruction or sub-pixel feature detection and localization. Matching can exploit all phase values
(not just zeros), and therefore extensive use is made of the available signal so that a dense set of
estimates is often extracted. Furthermore, because phase is amplitude invariant, the measurements
are robust with respect to smooth shading and lighting variations. With temporal sequences of images,
the phase difference can be replaced by a temporal phase derivative, thereby producing more accurate
1 Published in IEEE Trans. PAMI, 15(12): 1253-1268, 1993
measurements. These computations are straightforward and local in space-time, yielding efficient
implementations on both serial and parallel machines. There is the additional advantage in space-time
that different filters can be used initially to decompose the local image structure according to velocity
and scale. Then, the phase behaviour in each velocity-tuned channel can be used independently to
make multiple measurements of speed and orientation within a single image neighbourhood. This
is useful in the case of a single, densely textured surface where there exist several oriented image
structures with different contrasts, or multiple velocities due to transparency, specular reflection,
shadows, or occlusion [6, 7, 19]. In a recent comparison of several different optical flow techniques,
phase-based approaches often produced the most accurate results [1].
But despite these advantages, we lack a satisfying understanding of phase-based techniques and the
reasons for their success. The usual justification for phase-based approaches consists of the Fourier
shift theorem, an assumption that the phase output of band-pass filters is linear as a function of
spatial position, and a model of image translation between different views. But because of the local
spatiotemporal support of the filters used in practice, the Fourier shift theorem does not strictly apply.
For example, when viewed through a window, a signal and a translated version of it , W (x)s(x) and
are not simply phase-shifted versions of one another with identical amplitude spectra.
For similar reasons, phase is not a linear function of spatial position (and time) for almost all inputs,
even in the case of pure translation. The quasi-linearity of phase often reported in the literature
depends on the form of the input and the filters used, and has not been addressed in detail. Also
unaddressed is the extent to which these techniques produce accurate measurements when there are
deviations from image translation. Fleet et al. [7, 8] suggested that phase has the important property
of being stable with respect to small geometric deformations of the input that occur with perspective
projections of 3-d scenes. They showed that amplitude is sensitive to geometric deformation, but they
provided no concrete justification for the stability of phase.
This paper addresses several issues concerning phase-based matching, the behaviour of phase in-
formation, and its dependence on the band-pass filters. It presents justification for the claims of phase
stability with respect to geometric deformations, and of phase linearity. By the stability of some image
property, we mean that a small deformation of the input signal causes a similar deformation in the
image property, so that the behaviour of that property reflects the structure of the input that we wish
to measure. Here, we concentrate on small affine deformations like those that occur between left and
right stereo views of a slanted surface. For example, scale variations between left and right binocular
views of a smooth surface are often as large as 20% [24]. Although phase deformations do not exactly
match input deformations, they are usually close enough to provide reasonable measurements for many
vision applications.
We address these issues in the restricted case of 1-d signals, where the relevant deformations are
translations and dilations. Using a scale-space framework, we simulate changes in the scale of the
input by changing the tuning of a band-pass filter. In this context our concerns include the extent to
which phase is stable under small scale perturbations of the input, and the extent to which phase is
generally linear through space. These properties are shown to depend on the form of the filters used
and their frequency bandwidths. Situations in which phase is clearly unstable and leads to inaccurate
matching are also discussed, as are methods for their detection. Although we deal here with one
dimension, the stability analysis extends to affine deformations in multiple dimensions.
We begin in Section 2 with a brief review of phase-based matching methods and several comments
on the dependence of phase on the initial filters. Section 3 outlines the scale-space framework used in
the theoretical development that follows in Sections 4 and 5, and Section 6 discusses phase instabilities
in the neighbourhoods of phase singularities. Although the majority of the theoretical results assume
white noise as input, Section 7 briefly discusses the expected differences that arise with natural images.
Finally, Sections 8 and 9 draw conclusions and outline some topics that require further research.
Phase Information from Band-Pass Filters
Phase, as a function of space and/or time, is defined here as the complex argument of a complex-valued
band-pass signal. The band-pass signal is typically generated by a linear filter with a complex-valued
impulse response (or kernel), the real and imaginery components of which are usually even and odd
symmetric. Gabor functions (sinusoidally-modulated Gaussian windows [10]) are perhaps the most
commonly used filter [7, 8, 15, 18, 27, 30, 33]. Other choices of filter include: sinusoidally-modulated
square-wave (constant) windows [30]; filters with nonlinear phase that cycles between \Gamma- and -
only once within the support width of the kernel [31, 32, 33]; quadrature-pair steerable pyramids
constructed with symmetric kernels and approximations to their Hilbert tranforms [9, 28]; log-normal
filters [5, 31]; and derivative of Gaussian filters, and real and imaginary parts of which are the first
and second directional derivatives of a Gaussian envelope [31]. Phase can also be defined from a single
real-valued band-pass signal by creating a complex signal, the real and imaginary parts of which are
the original band-pass signal and its Hilbert transform. In all these cases, the expected behaviour of
phase depends on the filter.
Given the initial filters, phase-based optical flow techniques define image velocity in terms of the
instantaneous velocity of level (constant) phase contours (given by the spatiotemporal phase gradient).
Phase-based matching methods, based on two images, define disparity as the shift necessary to align
the phase values of band-pass filtered versions of the two signals [13, 27, 30]. In this case, a disparity
predictor can be constructed using phase differences between the two views: For example, let R l (x)
and R r (x) denote the output of band-pass filters applied to left and right binocular signals (along
lines). Their respective phase components are then written as OE l
arg[R r (x)], and binocular disparity is measured (predicted) as
(1)
where [OE] 2- denotes the principal part of OE, that is, [OE] 2- 2 (\Gamma-], and k(x) denotes some measure
of the underlying frequency of the signal in the neighbourhood of x. The frequency k(x) may be
approximated by the frequency to which the filter is tuned, or measured using the average phase
derivative (i.e., (OE 0
l
r
(x))=2) from the left and right filter outputs [8, 18]. The phase derivative
is often referred to as the instantaneous frequency of a band-pass signal [2, 26]. Note that linear phase,
and therefore constant instantaneous frequency, implies a sinusoidal signal.
One reason for the success of the predictor in (1) is the fact that phase is often nearly linear over
relatively large spatial extents, with instantaneous frequencies close to the filter tuning. In other words,
the filter output is nearly sinusoidal, so that phase and instantaneous frequency at a single location
provide a good model for the local phase structure of the filter response, in terms of a complex signal
winding sinusoidally (with some amplitude variation) about the origin of the complex plane. When the
left and right signals are shifted versions of one another, and phase is precisely linear over a distance
larger than the shift, then the predictor produces an exact measurement. That is, displacements of a
linear function f(x) are given by differences in function value divided by the derivative of the function:
In more general cases the predictor produces estimates of disparity, and may be used iteratively
to converge to accurate matches of the two local phase signals [8]. The size of the neighbourhood
within which phase is monotonic (and therefore unique) determines the range of disparities that may
be handled correctly by the predictor, and therefore its domain of convergence. Of course, for a
sinusoidal signal the predictor can handle disparities up to half the wavelength of the signal. Because
the domain of convergence depends on the wavelength of the band-pass signal, and hence on the
scale of the filter, small kernels tuned to high frequencies have small domains of convergence. This
necessitates some form of control strategy, such as coarse-to-fine propagation of disparity estimates.
A detailed analysis of the domain of convergence, and control strategies are beyond the scope of this
paper.
Within the domain of convergence, the accuracy of the predictor, and hence the speed with which it
converges, depends on the linearity of phase. As discussed in [8], predictor errors are a function of the
magnitude of disparity and higher-order derivatives of phase. The accuracy of the final measurement,
to which the predictor has converged, will also depend critically on phase stability (discussed below
in detail).
Phase linearity and monotonicity are functions of both the input signal and the form of the filters.
With respect to the filters, it is generally accepted that the measurement of binocular disparity and
optical flow should require only local support, so that the filters should be local in space-time as well as
band-pass. For example, although filter kernels of the form exp[ik 0 x] (i.e. Fourier transforms) produce
signals with linear phase, they are not local. It is also important to consider the correlation between
real and imaginary components of the kernel, as well as differences in their amplitude spectra. To the
extent that they are correlated the phase signal will be nonlinear because the filter response will form
an elliptical path, elongated along orientations of \Sigma-=4 in the complex plane. If they have different
amplitude spectra, then the real and imaginary parts will typically contain different amounts of power,
causing elliptical paths in the complex plane elongated along the real or imaginary axes. Finally, it
is natural to ensure that the filters have no dc sensitivity, for this will often mean that the complex
filter response will not wind about the origin, thereby also introducing nonlinearities, and smaller
domains of convergence [30, 31]. Fortunately, we can deal with these general concerns relatively easily
by employing complex-valued filters with local support in space-time, the imaginary parts of which
are Hilbert transforms of the real parts (i.e. quadrature-pair filters). The real and imaginary parts
are then uncorrelated with similar amplitude spectra and no dc sensitivity.
Gabors functions have Gaussian amplitude spectra with infinite extent in the frequency domain,
and therefore some amount of dc sensitivity. Because of the substantial power in natural signals at
low frequencies this dc sensitivity often introduces a positive bias in the real part of the response.
This problem can be avoided for the most part with the use of small bandwidths (usually less than
one octave), for which the real and imaginary parts of Gabor functions are close approximations to
quadrature pairs. It is also possible to modify the real (symmetric) part of the Gabor kernel by
subtracting its dc sensitivity from it (e.g., [7]). Log-normal kernels have Gaussian amplitude spectra
in log-frequency space, and are therefore quadrature-pair filters. Modulated square-wave filters [30]
and derivative-based real and imaginary parts may have problems because their real and imaginary
parts do not have the same amplitude spectra. In both cases, the odd-symmetric components of the
are more sensitive to low frequencies than the even-symmetric component. This introduces a
bias towards phase values of \Sigma-=2.
3 Scale-Space Framework
To address questions of phase stability and phase linearity, we restrict our attention to band-pass
filters with complex-valued kernels K(x; -), the real and imaginary parts of which form quadrature
pairs (i.e. they are Hilbert transforms of one another [26]). Let - ? 0 denote a scale parameter that
determines the frequency pass-band to which the filter is tuned. Also, let the kernels be normalized
such that
which is defined by
where f   denotes the complex conjugate of f . For convenience, we also assume translational invariance
and self-similarity across scale (i.e., wavelets [20]), so that K(x; -) satisfies
Wavelets are convenient since, because of their self-similarity, their octave bandwidth remains constant,
independent of the scale to which they are tuned. Our results also extend to filters other than wavelets
such as windowed Fourier transforms for which the spatial extent of the effective kernels is independent
of -.
The convolution of K(x; -) with an input signal I(x) is typically written as
Because K(x; -) is complex-valued, the response S(x;
complex-valued, and can be expressed using amplitude and phase as in S(x;
where
The first main concern of this paper is the expected stability of phase under small scale perturbations
of the input. If phase is not stable under scale variations between different views, then the
phase-based measurements of velocity or binocular disparity will not be reliable. To examine this, we
simulate changes in the scale of the input by changing the scale tuning of the filter; if one signal is a
translation and dilation of another, as in
I
because of the filters' self-similarity, the responses S 0
will satisfy
a
That is, if filters tuned to - 0 and - 1 were applied to I 0 (x) and I 1 (x), then the structure extracted
from each view would be similar (up to a scalar multiple) and the filter outputs would be related by
precisely the same deformation a(x). Of course, in practice we apply the same filters to I 0 (x) and
I 1 (x) because the scale factor a 1 that relates the two views is unknown. For phase-matching to yield
accurate estimates of a(x), the phase of the filter output should be insensitive to small scale variations
of the input.
The second concern of this paper is the extent to which phase is linear with respect to spatial
position. Linearity affects the ease with which the phase signal can be differentiated in order to estimate
the instantaneous frequency of the filter response. Linearity also affects the speed and accuracy of
disparity measurement based on phase-difference disparity predictors, as well as the typical extent of
the domain of convergence; if the phase signal is exactly linear, then the disparity can be computed
in just one step, without requiring iterative refinement [8, 18, 27, 30], and the domain of convergence
is \Sigma-. In practice, because of deviations from linearity and monotonicity, the reliable domain of
convergence is usually less than \Sigma-.
For illustration, let K(x; -) be a Gabor kernel 2 [10], Gabor(x; oe(-); k(-)), where
The peak tuning frequency of the Gabor filter is given by
Let the extent of the Gaussian envelope be measured at one standard deviation, and let the bandwidth
fi be close to one octave. Then, the standard deviation of the amplitude spectra oe
From this, it is straightforward to show that the radius of spatial
support is

Figure

1A shows a signal composed of a sample of white Gaussian noise concatenated with a sample
of red Gaussian noise. 3 The two middle images show the amplitude and phase components of the scale-space
Gabor response, ae(x; -) and OE(x; -) ; spatial position is shown on the horizontal axis, and log
scale is shown on the vertical axis (over two octaves). The bottom images show their level contours.
In the context of the scale-space expansion, an image property is said to be stable for image matching
where its level contours are vertical. Figure 1 shows that ae(x; -) depends significantly on scale as its
level contours are not generally vertical. By contrast, the phase structure is generally stable, except
for several isolated regions to be discussed below. Gradient-based techniques applied to low-pass
or band-pass filtered images produce inaccurate velocity estimates, in part, because they implicitly
require that both amplitude and phase behaviour be stable with respect to scale perturbations.
The response S(x; -) defined in (5) is referred to as a scale-space expansion of I(x). It is similar
to band-pass expansions defined by the Laplacian of a Gaussian (r 2 G), but it is expressed in terms
of complex-valued filters (cf. [16, 20, 34, 35]). Interestingly, zero-crossings of filters derived from
directional derivatives of Gaussian envelopes output are equivalent to crossings of constant phase of
complex-valued band-pass filters, the imaginary parts of which are Hilbert transforms of the corresponding
Gaussian derivatives. Here we use the scale-space framework to investigate the effects of
small perturbations of input scale on image properties that might be used for matching. We are
not proposing a new multi-scale representation, nor are we interpreting phase behaviour in terms of
specific image features such as edges.
As mentioned above, although the real and imaginary parts of Gabor kernels do not have identical amplitude spectra,
they are a good appoximation to a quadrature pair for small bandwidths (e.g. less than one octave, measured at one
standard deviation of the Gaussian spectrum).
3 That is, a sample of white noise smoothed with an exponential kernel exp(\Gammajxj=5).
-200200Pixel Location
Input
Intensity
x
l
log
(D) (E)

Figure

1. Gabor Scale-Space Expansion: The input signal (A) consists of a sample of white
Gaussian noise (left side), and a sample of red Gaussian noise (right side). The remaining panels
show the amplitude and phase components of S(x; -) generated by a Gabor filter with a bandwidth of
pixels. The horizontal and vertical axes represent spatial position
and log scale. (B) Amplitude response ae(x; t); (C) Phase response OE(x; t); (D) Level contours of
ae(x; t); (E) Level contours of OE(x; t).
4 Kernel Decomposition
The stability and linearity of phase can be examined in terms of the differences in phase between
an arbitrary scale-space location and other points in its neighbourhood. Towards this end, let S j j
denote the filter response at scale-space position p j
be expressed using inner products instead of convolution:
Phase differences in the neighbourhood of an arbitrary point p 0 , as a
function of relative scale-space position of neighbouring points p 1 , with
be written as
Phase is perfectly stable when \DeltaOE is constant with respect to changes in scale \Delta-, and it is linear
with respect to spatial position when \DeltaOE is a linear function of \Deltax.
To model the behaviour of S(x; -) and \DeltaOE in the neighbourhood of p 0 we write the scale-space
response at p 1 in terms of S 0 and a residual term R(p that goes to zero as k
is,
Equation (14) is easily derived if the effective kernel at p 1 , that is K 1 (x), is written as the sum of two
orthogonal terms, one which is a scalar multiple of K 0 (x), and the other orthogonal to K 0
where the complex scalar z(p
and the residual kernel H(x;
Equation (14) follows from (15) with . The scalar z reflects the
cross-correlation of the kernels K 0 (x) and K 1 (x). The behaviour of R(p related to the signal
structure to which K 1 (x) responds but K 0 (x) does not.
For notational convenience below, let z
Remember that z 1 , H 1 , and R 1 are functions of scale-space position p 1 in relation to p 0 .
Re
Im
R
z S
Figure

2. Sources of Phase Variation: This shows the formation of S 1 in terms of S 0 , the complex
scalar z 1 j z(p; p 0 ), and the additive residual R 1
Equation (14), depicted in Figure 2, shows that the phase of S 1 can differ from the phase of S 0
because of the additive phase shift due to z 1 , and the phase shift caused by the additive residual term
R 1 . Phase will be stable under small scale perturbations whenever both the phase variation due to z 1
as a function of scale and jR 1 j=jz 1 S 0 j are reasonably small. If jR 1 j=jz 1 S 0 j is large then phase remains
stable only when R 1 is in phase with S 0 , that is, if arg[R 1 phase may vary
wildly as a function of either spatial position or small scale perturbations.
5 Phase Stability Given White Gaussian Noise
To characterize the stability of phase behaviour through scale-space we first examine the response of
K(x; -) and its phase behaviour to stationary, white Gaussian noise. Using the kernel decomposition
(15), we derive approximations to the mean phase difference E[\DeltaOE] , and the variation about the mean
expectation. The mean provides a prediction for
the phase behaviour, and the expected variation about the mean amounts to our confidence in the
prediction. These approximations can be shown to depend only on the cross-correlation z 1 (16), and
are outlined below; they are derived in further detail in Appendix A.
Given white Gaussian noise, the two signals R 1 and z 1 S 0 are independent (because the kernels
are orthogonal), and the phase of S 0 is uniformly distributed over (\Gamma-]. If we
also assume that arg[R 1 are uncorrelated and that arg[R 1
the residual signal R 1 does not affect the mean phase difference. Therefore we approximate the mean
E[\DeltaOE] by
where z 1 is a function of scale-space position. Then, from (13) and (18), the component of \DeltaOE about
the approximate mean is given by (cf. Figure 2)
The expected magnitude of \DeltaOE \Gamma -(z 1 ) measures of the spread of the distribution of \DeltaOE about
the mean; 4 it is a function of the magnitude of z 1 S 0 and the magnitude of the component of R 1
perpendicular to the direction of z 1 S 0 in the complex plane (see Figure 2). With the assumptions 5
that uniformly distributed, it is shown in Appendix A that an
approximate bound, b(z 1 ), on E[ j\DeltaOE \Gamma -(z 1 )j ] is given by
It is tightest for small variations about the mean, that is, small values of \DeltaOE \Gamma -(z 1 ) .
5.1 Gabor Kernels
For illustrative purposes we apply these results to Gabor filters. Although they only approximate
quadrature-pair filters for small bandwidths, they admit simple analytic derivation for z 1 which is the
basis for the stability measures. For many other filters, it is more convenient to derive z 1 numerically
from discrete kernels.
For is given by (see Appendix C.1)
oe
defines the filter tunings (11a), oe defines the support widths (11b),
1 .
From (21), the approximate mean phase difference, that is arg[z 1 ], is given by
\Deltakk
The expected magnitude of \DeltaOE about the mean (20) can also be determined from (21) straightfor-
wardly. In particular, from (20) we expect b(z 1 ) to behave linearly in the neighbourhood of p 0 , because
for sufficiently small k (\Deltax; \Delta-) k it can be shown from (21) that jz 1

Figure

3 illustrates this behaviour in the restricted cases of pure dilation and pure translation
between points p 1 and p 0 . Figure 3A shows -(z 1 ) with error bars (the expected deviation about the
4 The expected value of the j\DeltaOEj is one possible measure of the spread of the probability density function. Compared
to the standard deviation it is less sensitive to outliers [12], and in this case, it yields analytic results while the second
moment does not (see Appendix A).
5 The assumption that means that p 0
is not in the immediate neighbourhood of a singular point, where
jS0 j is very small. Singularity neighbourhoods are discussed below in Section 6.
mean, b(z 1 )) as a function of scale for a vertical slice through a Gabor scale-space expansion with
bandwidth Figure 3B shows -(z 1 ) and b(z 1 ) as a function of spatial position for a
horizontal slice through the same scale-space expansion. This shows clearly that the mean is constant
with respect to scale changes and linear as a function of spatial position. The behaviour of b(z 1 ), the
variation about the mean, shows that the mean provides a very good model for the expected phase
behaviour for small translations and dilations. For larger translations and dilations we expect a larger
variation about the mean. This has a direct impact on disparity measurement when dilation occurs
between two views, suggesting that errors will increase with the amount of dilation due to spatial drift
of the phase contours.
In the case of translation, the increased variation about the mean is caused by two factors, namely,
phase nonlinearities, and the distribution of the instantaneous frequencies in the filter output, which
is a function of the bandwidth of the filter. 6 When the tuning frequency of the filter is used to
approximate the instantaneous frequency in the denomenator of the disparity predictor (1) as in
[27, 13, 15], then b(z 1 ) provides a direct measure of the expected predictor errors. Otherwise, if the
instantaneous frequency is measured explicitly as in [8, 18], then b(z 1 ) can be used for an upper bound
on expected predictor errors. The variation about the mean b(z 1 ) can also be used indirectly as an
indication of the range of disparities that a band-pass channel will reliably measure before aliasing
occurs (when phase wraps between - to \Gamma-). This is important to consider when developing a control
strategy to handle relatively large disparities with predictors at several scales and spatial positions.
A more complete illustration of the expected phase behaviour in the scale-space neighbourhood of
p 0 is given in Figure 4. Figure 4A shows level contours of -(z 1 ) for a Gabor filter. The point
lies at the centre, and represents a generic location away from phase singularities (discussed
below). Log scale and spatial position in the neighbourhood of p 0 are shown on vertical and horizontal
axes. These contours illustrate the expected mean phase behaviour, which has the desired properties
of stability through scale and linearity through space. The contours are essentially vertical near p 0 ,
and for fixed \Deltak , the mean phase behaviour -(z 1 ) is a linear function of \Deltax. It is also interesting to
note that the mean phase behaviour does not depend significantly on the bandwidth of the filter; for
Gabor filters it is evident from (21) that -(z 1 ) is independent of fi.

Figures

4B and 4C show level contours of b(z 1 ) for Gabor kernels with bandwidths fi of 0.8 and
1.0 octaves. In both cases, b(z 1 ) is monotonically decreasing as one approaches p 0 in the centre.
As decreases so does the relative magnitude of R 1 , and therefore so does the expected
fluctuation in the phase of S 1 about the phase of z 1 S 0 . By design, b(z 1 ) is a measure of the distribution
of phase differences about the mean. The contours in Figure 4 show that the expected deviation from
small in the vicinity of p 0 . Since -(z 1 ) has the properties we desire (stability through scale
and linearity through space near p 0 ), we can also view b(z 1 ) as a direct measure of phase stability
and phase linearity.
6 The expected distribution of instantaneous frequencies for a given filter is discussed in [6]. A significant proportion
of phase variability is due to the variation in instantaneous frequency in addition to phase nonlinearities. However, the
separation of these two causes is beyond the scope of this paper, as it is not the principal issue.
Relative Scale (in octaves)
Phase
Difference
(mod 2p)
Spatial Location (in l 0 )
Phase
Difference
(mod 2p)

Figure

3. Approximate Mean and Variation About the Mean of \DeltaOE for 1-D Slices of Scale-
Space: The expected behaviour of \DeltaOE is shown for vertical and horizontal slices through the middle
of the scale-space in Figure 4 for Gabor filters with plots show the approximate mean
and the variation about the mean b(z 1 ) (as error bars). (A) Phase behaviour as a function of
\Delta- while Phase behaviour as a function of \Deltax for

Figure

4. Phase Stability with Gabor Kernels: Scale-space phase behaviour near
shown with log scale on the vertical axis over two octaves with
position
on the horizontal axis, x is in the centres of the figures. (A) Level
contours 4, as a function of scale-space position. These contours are
independent of the bandwidth. (B) and (C) Level contours of b(z 1 ) are shown for
Each case shows the contours b(z 1 5; the innermost contours correspond to
It is also evident from Figures 4B and 4C that b(z 1 ) depends on the bandwidth of the filter. This
can be explained from the dependence of oe 0 and oe 1 in (21) on bandwidth as given by (11b). As the
bandwidth increases the amplitude spectra of filters tuned to nearby scales will overlap to a greater
extent, and phase is therefore stable for larger scale perturbations of the input. On the other hand, an
increase in bandwidth implies a decrease in the spatial extent of the kernels, and therefore a decrease
in the spatial extent over which phase is generally linear. The smallest contours in Figures 4B and
correspond to b(z 1 which amounts to a phase difference of about
\Sigma 5% of a wavelength. For contour encloses approximately \Sigma 20% of an octave vertically
and \Sigma 20% of a wavelength spatially. In this case, relative scale changes of 10% and 20% are typically
accompanied by phase shifts of less than 3.5% and 6.6% of a wavelength respectively.
5.2 Verification of Approximations
There are other ways to illustrate the behaviour of phase differences as a function of scale-space posi-
tion. Although they do not yield as much explanatory insight they help to validate the approximations
discussed above.
First, following Davenport and Root [4] it can be shown 7 that the probability density function for
at scale-space positions p 0 and p 1 , for a quadrature-pair kernel and white Gaussian noise
7 See [6] for details.

Figure

5. E[\DeltaOE] and E[ Inputs: Scale-space phase behaviour
based on (23) for Gabor kernels. As above, is the centre, with \Gamma- 0 - x . The same level contours as in Figure 4 are superimposed for comparison. (A) E[\DeltaOE]
for
input is
for \Gamma- \DeltaOE ! -, where A and B are given by
. Given the density function,
we can use numerical integration to find its mean behaviour and the expected variation about the
mean.

Figures

5A and 5B show the behaviour of the mean, and the absolute variation about the
mean, of \DeltaOE as functions of scale-space position for Gabor kernels with a bandwidth of 0.8 octaves.

Figure

5C shows the expected variation of \DeltaOE about the mean for Gabor filters of 1.0 octave. The
mean behaviour in this case is not shown as it is almost identical to Figure 5A. In all three cases level
contours have been superimposed to better illustrate the behaviour. Intensity in Figure 5A reflects
values between \Gamma- and -. Values in the other two range from 0 in the centre to -=2 at the edges
where the distribution of \DeltaOE becomes close to uniform. Comparing Figures 4 and 5, notice that the
bound b(z 1 ) in (20) is tightest for smaller values of \DeltaOE (see Appendix A); the two smallest contours
in

Figures

4 and 5 are extremely close.
It is also instructive to compare the phase behaviour predicted by -(z 1 ) and b(z 1 ) with actual
statistics of phase differences gathered from scale-space expansions of different input signals. Figure
6A shows behaviour of \DeltaOE predicted by -(z 1 ) and b(z 1 ) for one octave Gabor filters as in Figure
3.

Figures

6B and 6C show statistical estimates of E[\DeltaOE] and E[j\DeltaOE \Gamma E[\DeltaOE]j] measured from scale-
Scale Variation (in octaves)
Predicted
Phase Behaviour
Scale Variation (in octaves)
Experimental
Phase Behaviour
Scale Variation (in octaves)
Experimental
Phase Behaviour

Figure

6. Predicted Versus Actual Phase Behaviour: (A) Predicted behaviour of \DeltaOE based
on -(z 1 ) and b(z 1 ) for a Gabor filter with Statistical estimates of E[\DeltaOE] and
that were extracted from Gabor scale-space expansions of white noise (middle) and
scanlines from natural images (bottom).
space expansions of white noise and of scanlines of natural images. In both cases the observed phase
behaviour is in very close agreement with the predicted behaviour. The statistical estimates of E[j\DeltaOE \Gamma
E[\DeltaOE]j] in these and other cases were typically only 1-5% below the bound b(z 1 ) over the scales shown.
5.3 Dependence on Filter
These quantitative measures of expected phase behaviour can be used to predict the performance of
phase matching as a function of the deformation between the input signals. The same measures can
also be used to compare the performance of different filters; for although Gabor filters (8) have been
popular [27, 7, 8, 15, 18], several alternatives have been suggested in the context of phase information.
Weng [30] used a self-similar family of kernels derived from a square-wave (or constant) window:
e ik(-)x if jxj -=2
x
l
log

Figure

7. Modulated Square-Wave Windows: Scale-space phase behaviour is shown for modulated
square-wave windows (24) in comparison to Gabor filters. (A) Level contours b(z 1
for the square-wave kernel. (B) Similar level contours of b(z 1 ) for a Gabor kernel with
The superposition of level contours from square-wave and Gabor kernels.
Other alternatives, common to phase-correlation techniques, are families of filters with fixed spatial
extents but tuned to different scales, also called windowed Fourier transforms [11, 17, 20]. This section
several issues in addition to those in Section 2 that are related to the choice of filter.
The effect of bandwidth on phase stability was illustrated in Figure 4. This dependence can be
explained from the form of z 1 (16) and Parseval's theorem:
K(k) denotes the Fourier transform of K(x). As the bandwidth increases the extent of the
amplitude spectrum increases, and so does the range of scales \Delta- over which -
may
remain highly correlated. Similar arguments hold in space with respect to the extent of spatial support
and phase linearity. For wavelets, the stability of phase with respect to input scale changes is constant
across filters tuned to different scales, but the spatial extent over which phase is expected to be nearly
linear decreases for higher scales as a function of the support width (and therefore the wavelength)
of the filters. For windowed Fourier transforms, the support width is constant for different scales,
and hence so is the expected extent of linearity, but the stability of phase with respect to input scale
perturbations decreases at higher scales as the octave bandwidth decreases.
But does a simultaneous increase in the extent of the kernels' support in space and its amplitude
spectrum produce better scale stability or more extensive linearity through space? For example, compared
to the Gaussian which minimizes a measure of simultaneous extent (the uncertainty relation),
the modulated square-wave kernel (24) has relatively large simultaneous extents in space and frequency
domain. Its amplitude spectrum is particularly broad, and therefore we might expect better stability
than with comparable Gabor filters (e.g., with fi - 1:5).
The expected phase behaviour for the modulated square-wave filter (24) is shown in Figure 7. Its
mean phase behaviour, given by arg[z 1 ] , is very much like that exhibited by Gabor filters in Figure 4A
and is not shown. Figure 7A shows the scale-space behaviour of b(z 1 ) for the modulated square-wave
kernel (24), for which z 1 is given by (see Appendix C.2)
e i\Deltak a
(\Deltak
where \Delta- and \Deltak are defined above, 8 and
\Delta-
\Delta-
For comparison, Figure 7B shows level contours of b(z 1 ) for a Gabor kernel with a bandwidth of

Figure

7C shows the superposition of the level contours from Figures 7A and 7B. These
figures show that the distribution of \DeltaOE about the mean for the modulated suare-wave kernel is two
to three times larger near p 0 , which suggests poorer stability and poorer linearity. Note that the
innermost contour of the Gabor filter clearly encloses the innermost contour of the square-wave filter.
This Gabor filter handles scale perturbations of 10% with an expected phase drift of up to \Sigma 3.6% of
a wavelength, while a perturbation of 10% for the modulated square-wave kernels gives b(z 1
which amounts to a phase difference of about \Sigma7:5%.
The poorer phase stability exhibited by the modulated square-wave kernel implies a wider distribution
of measurement errors. Because of the phase drift due to scale changes, even with perfect phase
matching, the measurements of velocity and disparity will not reflect the projected motion field and
the projected disparity field as reliably. The poorer phase linearity affects the accuracy and speed of
the disparity predictor, requiring more iterations to match the phase values between views. Moreover,
we find that the larger variance also causes a reduction in the range of disparities that can be measured
reliably from the predictor. The poorer linearity exhibited in Figure 7 also contradicts a claim in [30]
that modulated square-wave filters produce more nearly linear phase behaviour.
Wider amplitude spectra do not necessarily ensure greater phase stability. Phase stability is the
result of correlation between kernels at different scales. The shapes of both the amplitude and phase
spectra will therefore play significant roles. The square-wave amplitude spectra is wide, but with
considerable ringing so that jz 1 j falls off quickly with small scale changes.
Another issue concerning the choice of filter is the ease with which phase behaviour can be accurately
extracted from a subsampled encoding of the filter output. It is natural that the outputs
of different band-pass filters be quantized and subsampled to avoid an explosion in the number of
bits needed to represent filtered versions of the input. However, because of the aliasing inherent in
subsampled encodings, care must be taken in subsequent numerical interpolation/differentiation. For
example, we found that, because of the broad amplitude spectrum of the modulated square-wave and
its sensitivity to low frequencies, sampling rates had to be at least twice as high as those with Gabors
(with comparable bandwidths, to obtain reasonable numerical differentiation. If these issues
8 As \Deltak ! 0, this expression for z1 converges to exp[i\Deltaxk 0
are not considered carefully, they can easily cause greater problems in phase-based matching than
differences in stability or linearity between kernels. To alleviate some of these problems Weng [30]
presmoothed the input signal with a Gaussian.
5.4 Amplitude Stability
Although our main concern is phase behaviour, it is also of interest to consider the expected scale-space
behaviour of amplitude. Towards this end, using the same arguments as above for white noise inputs,
it is shown in Appendix B that the expected (mean) amplitude variation as a function of scale-space
position is constant, independent of the direction of . The expected absolute magnitude of
amplitude differences, like phase variations about the mean, will depend on the relative magnitudes
of z 1 S 0 and R 1 . We expect the size of amplitude variations to increase for greater differences in
scale-space distance.
This implies that amplitude often varies slowly through scale-space. However, while level phase contours
exhibit predominantly vertical structure, level amplitude contours will occur at all orientations,
and are therefore not consistently stable with respect to dilations between inputs. This variability is
evident in Figure 1D compared to Figure 1E.
5.5 Multiple Dimensions
Finally, although beyond the scope of the current work, it is important to note that this basic frame-work
can be extended to consider the stability of multidimensional filters with respect to other types
of geometric deformation. In particular, we are interested in the phase behaviour of 2-d oriented filters
with respect to small amounts of shear and rotation as well as scale changes. This analysis can be
done, as above, using the cross-correlation between a generic kernel and a series of deformations of it.
In this way, quantitative approximations can be found to predict the expected degree of phase drift
under different geometric deformations of the input.
6 Singularity Neighbourhoods
The above analysis gives quantitative bounds on the expected stability of phase through scale and its
linearity through space. But from Figure 1 it is clear that phase stability is not uniform throughout
some regions exhibit much greater instability in that the phase contours are nearly horizontal
and not vertical as desired. Jepson and Fleet [14] explained that this phase instability occurs
in the neighbourhoods of phase singularities (locations in space-time where the filter output passes
through the origin in the complex plane). In terms of (14), S 0 is zero at a singularity, and the response
singularity neighbourhood is dominated by the residual term R 1 . Zeros of S(x; -) appear as
black spots in Figure 1B.
From the analysis described in [6, 14] the singularity neighbourhoods and the nature of phase
instability can be characterized in terms of properties of the complex logarithm of the filter response

Figure

8. Detection of Singularity Neighbourhoods: (A) Phase contours that survive the
constraints. (B) Phase contours in regions removed by the constraints. The stability constraint in
was used with
log its x-derivative:
@
@x
log
ae x (x; -)
ae(x; -)
The imaginary part OE x (x; -) gives the local (instantaneous) frequency of the response [2, 26], that
is, the instantaneous rate of modulation of the complex signal. The real part ae x (x;
the relative amplitude derivative. It can be shown that the instantaneous frequency of the filter
output OE x (x; -) is expected to be within the passband of the filter, and the amplitude derivative
ae x (x; is expected to be near zero [6]. The behaviour exhibited in singularity neighbourhoods
is, however, quite different. As jS(x; -)j decreases to zero, j log S(x; -)j increases without bound,
as do the magnitudes of OE x (x; -) and/or ae x (x; This leads to a simple method for
detecting singularity neighbourhoods so that unreliable measurements of binocular disparity and image
velocity can be detected. Jepson and Fleet [14] used two constraints, one on the local frequency
of response, and one on the magnitude of the amplitude derivative. Here we combine them into one:oe k (-)
@
@x
log
where k(-) and oe k are given in (11a) and (11b). As - decreases, this constraint detects
larger neighbourhoods about the singular points. In effect, the left-hand side of (28) reflects the inverse
scale-space distance to a singular point.

Figure

8 shows the application of (28) to the scale-space in Figure 1. Figure 8A shows the phase
contours that survive the constraint, which are predominantly stable; the large black regions are the
singularity neighbourhoods detected by (28). Figure 8B shows those removed by the constraint, which
amounts to about 20% of the entire scale-space area. With respect to the quantitative approximations
to phase behaviour presented in Section 4, we reported that statistics of mean phase differences and
the absolute variation about the mean agreed closely with the bounds. When the phase behaviour in
singularity neighbourhoods is ignored, so that the statistics are gathered only from outside of such
singularity neighbourhoods, we find that the magnitude of the variation of \DeltaOE about the mean is
generally less than half of that predicted by the bound in (20). This detection of unstable regions
is essential to the reliable performance of phase-based matching techniques, and it can be used to
improve the performance of zero-crossing and phase-correlation techniques [11, 17, 21, 22, 29].
7 Natural Images
Unlike white noise, the Fourier harmonics of natural images are often correlated across scales, and
their amplitude spectra typically decay something like 1=k [5]. Both of these facts affect our results
concerning phase stability, the accuracy of phase matching, and instability due to singularity
neighbourhoods.
First, because of amplitude spectra decay with spatial frequency, the filter responses will be biased
to lower frequencies (as compared to white noise). As a result, care is required to ensure that the
filter outputs do not contain too much power at low frequencies. Otherwise, there may be a) more
distortion due to aliasing in a subsampled representation of the response, and b) larger singularity
neighbourhoods, and hence a sparser set of reliable measurements. These problems are evident when
comparing the modulated square-wave filters with Gabor filters (or similar bandwidths) because the
former have greater sensitivity to low frequencies. Second, without the assumption of white noise we
should expect z 1 S 0 and R 1 in (14) to be correlated. This will lead to improved phase stability when
R 1 and S 0 remain in phase, and poorer stability when they become systematically out of phase.
Although we lack a sufficient model of natural images, in terms of local structure, to provide a
detailed treatment of phase stability on general images, several observations are readily available: For
example, with many textured image regions the phase structure appears much like that in Figure
1. We find that the noise-based analysis provides a good model of the expected phase behaviour for
complex structures that regularly occur in natural images.
Moreover, it appears that phase is even more stable in the neighbourhoods of salient image features,
such as those that occur in man-made environments. To see this, note that the output of a filter in a
small region can be viewed as a weighted sum of harmonics. In the vicinity of localized image features
such as edges, bars and ramps, we expect greater phase stability because the phases of the input
harmonics (unlike the white noise) are already coincident. This is clear from their Fourier transforms.
Therefore changing scales slightly, or adding new harmonics at the high or low ends of the passband
will not change the phase of the output significantly.
It is also worth noting that this phase coincidence at the feature locations coincides with local
maxima of the amplitude response [23]. When different harmonics are in phase their amplitudes
combine additively. When out of phase they cancel. Therefore it can be argued that neighbourhoods
of local amplitude maxima correspond to regions in which phase is maximally stable (as long as
the signal-to-noise-ratio is sufficiently high). This is independent of the absolute phase at which the
different harmonics coincide, 9 and is significant for stable phase-based matching.
As one moves away from salient features, such as edges, the different harmonics may become
increasingly out of phase, the responses from different features may interfere, and the amplitude of
response decreases. This yields two main types of instability: 1) where interference from nearby
features causes the total response to disappear at isolated points; and 2) where large regions have
very small amplitude and are dominated by noise. The first case amounts to a phase singularity and
is detectable using the stability constraint (28). In the second case, the phase behaviour in different
views may be dominated by uncorrelated noise, but will not necessarily violate the stability constraint.
For these situations a signal-to-noise constraint is necessary.
To illustrate these points, Figure 9 shows the Gabor scale-space expansion of a signal containing
several step edges (Figure 9A). There are two bright bars, apart.
The scale-space plots were generated by Gabor filters with

Figures

9A and 9B show the amplitude response and its level contours superimposed on the input
signal (replicated through scale) to show the relationship between amplitude variation and the edges.

Figures

9C and 9D show the scale-space phase response and its level contours, and Figures 9E and
9F show the contours that remain after the detection of singularity neighbourhoods using (28) with
1:25, and the phase contours in the neighbourhoods detected by (28). As expected, phase is stable
near the edge locations where the local Fourier harmonics are in phase, and arg[R 1
a wide range of scales. The similarity of the interference patterns between the different edges to the
singularity neighbourhoods shown in Figures 1 and 8 is clear, and these regions are detected using the
stability constraint. Also detected are the regions relatively far from the edges where the amplitude
and phase responses of the filter both go to zero.
However, as explained above, regions in which the filter response decreases close to zero are also
very sensitive to noise. These regions can become difficult to match since uncorrelated noise between
two views can dominate the response. To illustrate this, we generated a different version of the scale-space
in which uncorrelated noise was added to the input independently before computing each scale
(to simulate uncorrelated noise added to different views). The response to the independent noise
patterns satisfied the stability constraint much of the time, but the phase structure was unstable
(uncorrelated) between scales. Figure 9H shows the regions detected by the stability constraint in this
case. As discussed, the regions of low amplitude in the original are now dominated by the response to
the noise and are no longer detected by (28). Another constraint on the signal-to-noise ratio of the
filter output appears necessary. For example, Figure 9I shows the regions in which the amplitude of
the filter output is 5% or less of the maximum amplitude at that scale. This constraint in conjunction
9 Morrone and Burr [23] argue that psychophysical salience (of spatial features) correlates well with phase coincidence
only for certain absolute values of phase, namely, integer multiples of -=2, which are perceived as edges and bars of
different polarities.
with those discussed earlier are sufficient to obtain phase stability comparable to that in the noiseless
case (

Figure

9J).
This also demonstrates the fact that an amplitude constraint alone does not serve our purposes, for
the constraints on instantaneous frequency and amplitude derivative detect different regions. Although
the regions detected by a single threshold applied directly to ae(x; -) will eventually include the regions
detected by (28) if the threshold is large enough, they will also enclose regions of stable phase behaviour.
A single amplitude constraint will remove more of the signal than necessary if relied on to detect all
the instabilities.
8 Further Research
Finally, it is clear that not all variations between two views of a scene are accounted for by affine
geometric deformation and uncorrelated Gaussian noise. Other factors include contrast variations due
to shading and shadows, specular anomalies, effects of anisotropic reflectance, and occlusion boundaries
[6]. Although the behaviour of phase in these cases is not examined here, below we outline some of
the main problems that require investigation.
Many contrast variations are smooth compared to surface texture. Thus, we expect the phase
information from filters tuned to higher frequency to be insensitive to the contrast variations. In other
cases, the contrast becomes a source of significant amplitude modulation in the input, which manifests
itself in the phase behaviour of the filter output. Thus, some illumination changes will perturbation
the disparity measurements in a non-trivial way.
Another difficulty for phase-based techniques, as well as most other matching techniques, is that
the local intensity structure in one view of a 3d scene may be very different from that in another view
of the same scene. Two obvious examples are occlusions, in which a surface may be visible in only
one of the two views, and specular phenomena, in which case highlights may be visible from only one
of two views. In both of these cases we should not expect the two views to be highly correlated in
the usual sense; the difference between views is not well modelled by uncorrelated Gaussian noise.
Moreover, we do not expect these cases to be detected by our stability constraint.
Another topic that is left for further research concerns measures of confidence for phase-based
disparity measures. In our work we have found several ways of detecting unreliable measurements.
One is the stability measure, and another is some form of SNR constraint. A further constraint that
is useful in detecting poor matches, such as those due to occlusion is a correlation measure between
matched regions. The use of these measures, possibly combining them into one measure has not been
addressed in sufficient detail.
9 Discussion and Summary
This paper examines the robustness of phase-based techniques for measuring image velocity and binocular
disparity [7, 8, 11, 13, 15, 17, 18, 25, 27, 30, 32, 33]. Our primary concerns are the effects of
200100Pixel Location
Input
Intensity

Figure

9. Gabor Scale-Space Expansion With Step-Edge Input: The input signal (A) consists
of two bars. Vertical and horizontal axes of the Gabor scale-space represent log scale and spatial
position. Level contour have been superimposed on the input to show the relative location of the edges.
its level contours; (D) and (E) OE(x; -) and its level contours; (F) and (G)
Level phase contours that survive the stability constraint (28), and those detected by it; (H) Regions
of the "noisy-scale-space" that were detected by (28); (I) Regions detected by a simple amplitude
constraint; (J) The level phase contours that survive the union of constraints in (H) and (I).
the filters and the stability of phase with respect to typical image deformations that occur between
different views of 3-d scenes. Using a scale-space framework it was shown that phase is generally
stable with respect to small scale perturbations of the input, and quasi-linear as a function of spatial
position. Quantitative measures of the expected phase stability and phase linearity were derived for
this purpose. From this it was shown that both phase stability and linearity depend on the form of
the filters and their frequency bandwidths. For a given filter type, as the bandwidth increases, the
extent of the phase stability increases, while the spatial extent over which phase is expected to be
linear decreases. In the context of disparity measurement, the bandwidth of the filters should therefore
depend, in part, on the expected magnitude of deformation between left and right views, since the
potential accuracy of phase-based matching depends directly on phase stability.
One of the main causes of instability is the occurrence of phase singularities, the neighbourhoods of
which exhibit phase behaviour that is extremely sensitive to input scale perturbations, small changes
in spatial position, and small amounts of noise. Phase behaviour in these neighbourhoods is a source of
significant measurement error for phase-difference and phase-gradient techniques, as well as gradient-based
techniques, zero-crossing techniques, and phase-correlation techniques. Fortunately, singularity
neighbourhoods can be detected automatically using a simple constraint (28) on the filter output.
This stability constraint is an essential component of phase-based methods. A second constraint is
also needed to ensure a reasonable signal-to-noise ratio.
This basic approach can also be used to examine the stability of multi-dimensional filters to other
types of geometric deformation, such as the stability of 2-d oriented filters with respect to local affine
deformation (rotation, shear, and dilation). As explained here, we may consider the behaviour of
phase information using the cross-correlation of deformed filter kernels z 1 as a function of rotation
and shear in addition to the case of dilation on which we concentrated in this paper. In this way,
quantitative approximations can be found to predict the expected degree of phase drift under different
geometric deformations of the input.

Acknowledgements

We are grateful to Michael Langer for useful comments on earlier drafts of this work. This research
has been supported in part by the Natural Sciences and Engineering Research Council of Canada, and
the Ontario Government under the ITRC centres.



--R

Performance of optical flow techniques.
Estimating and interpreting the instantaneous frequency of a signal.
Object tracking with a moving camera.
Introduction to the Theory of Random Signals and Noise
Relations between the statistics of natural images and the response properties of cortical cells.
Measurement of Image Velocity
Computation of component image velocity from local phase information.

The design and use of steerable filters.
Theory of communication.
Direct estimation of displacement histograms.
Robust Statistics
The measurement of binocular disparity.
Phase singularities in scale-space
Fast computation of disparity from phase differences.
The structure of images.
The phase correlation image alignment method.
Vertical and horizontal disparities from phase.
Multiple motion from instantaneous frequency.
Multifrequency channel decomposition of images and wavelet models.
A computational theory of human stereo vision.
Computational studies toward a theory of human stereopsis.
Feature detection in human vision: a phase-dependent energy model
Research in Binocular Vision


Stereo disparity computation using Gabor filters.
Shiftable multiscale transforms.
Convected activation profiles: Receptive fields for real-time measurement of short-range visual motion
A theory of image matching.
Preattentive gaze control for robot vision.
Hierarchical phase-based disparity estima- tion
A multiresolution stereopsis algorithm based on the Gabor representation.

Scaling theorems for zero-crossings

--TR
Scaling theorems for zero crossings
Computational processes in human vision
Vertical and horizontal disparities from phase
Computation of component image velocity from local phase information
Phase-based disparity measurement
The Design and Use of Steerable Filters
Performance of optical flow techniques
Measurement of Image Velocity

--CTR
Computational model for neural representation of multiple disparities, Neural Networks, v.16 n.1, p.25-37, January
Jun Zhou , Yi Xu , Xiaokang Yang, Quaternion wavelet phase based stereo matching for uncalibrated images, Pattern Recognition Letters, v.28 n.12, p.1509-1522, September, 2007
Tatiana Jaworska, Amplitude elimination for stereo image matching based on the wavelet approach, Machine Graphics & Vision International Journal, v.14 n.1, p.103-120, January 2005
Phillipe Burlina , Rama Chellappa, Analyzing Looming Motion Components From Their Spatiotemporal Spectral Signature, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.18 n.10, p.1029-1033, October 1996
Sherif El-Etriby , Ayoub K. Al-Hamadi , Bernd Michaelis, Dense depth map reconstruction by phase difference-based algorithm under influence of perspective distortion, Machine Graphics & Vision International Journal, v.15 n.3, p.349-361, January 2006
Wang , Song-Chun Zhu, Analysis and Synthesis of Textured Motion: Particles and Waves, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.26 n.10, p.1348-1363, October 2004
Bernd Porr , Bernd Nrenberg , Florentin Wrgtter, A VLSI-Compatible Computer Vision Algorithm for Stereoscopic Depth Analysis in Real-Time, International Journal of Computer Vision, v.49 n.1, p.39-55, August 2002
Horst W. Haussecker , David J. Fleet, Computing Optical Flow with Physical Models of Brightness Variation, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.23 n.6, p.661-673, June 2001
David J. Fleet , Michael J. Black , Oscar Nestares, Bayesian inference of visual motion boundaries, Exploring artificial intelligence in the new millennium, Morgan Kaufmann Publishers Inc., San Francisco, CA,
Djamal Boukerroui , J. Alison Noble , Michael Brady, On the Choice of Band-Pass Quadrature Filters, Journal of Mathematical Imaging and Vision, v.21 n.1, p.53-80, July 2004
local phase information;geometric deformations;frequency bandwidths;initial filters;image processing;robustness;image recognition;binocular disparity;filtering and prediction theory;3D scene;image velocity;zero-crossing tracking;spatial position;phase singularities;frequency stability;phase stability;optical flow