Determining multiscale image feature angles from complex wavelet phases
ABSTRACT In this paper, we introduce a new multiscale representation for 2-D images named the inter-coefficient product (ICP). The ICP is a decimated pyramid of complex values based on the dual-tree complex wavelet transform (DT-CWT). The complex phases of its coefficients correspond to the angles of dominant directional features in their support regions. As a sparse representation of this information, the ICP is relatively simple to calculate and is a computationally efficient representation for subsequent analysis in computer vision activities or large data set analysis. Examples of ICP decomposition show its ability to provide an intuitive representation of multiscale features (such as edges and ridges). Its potential uses are then discussed
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Conference Proceeding: Coarse-level object recognition using interlevel products of complex wavelets[show abstract] [hide abstract]
ABSTRACT: This paper introduces the interlevel product (ILP) which is a transform based upon the dual-tree complex wavelet. Coefficients of the ILP have complex values whose magnitudes indicate the amplitude of multilevel features, and whose phases indicate the nature of these features (e.g. ridges vs. edges). In particular, the phases of ILP coefficients are approximately invariant to small shifts in the original images. We accordingly introduce this transform as a solution to coarse scale template matching, where alignment concerns between decimation of a target and decimation of a larger search image can be mitigated, and computational efficiency can be maintained. Furthermore, template matching with ILP coefficients can provide several intuitive "near-matches" that may be of interest in image retrieval applications.Image Processing, 2005. ICIP 2005. IEEE International Conference on; 10/2005
- [show abstract] [hide abstract]
ABSTRACT: This paper describes a form of discrete wavelet transform, which generates complex coefficients by using a dual tree of wavelet filters to obtain their real and imaginary parts. This introduces limited redundancy (2m:1 for m-dimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good well-balanced frequency responses. Here we analyze why the new transform can be designed to be shift invariant and describe how to estimate the accuracy of this approximation and design suitable filters to achieve this. We discuss two different variants of the new transform, based on odd/even and quarter-sample shift (Q-shift) filters, respectively. We then describe briefly how the dual tree may be extended for images and other multi-dimensional signals, and finally summarize a range of applications of the transform that take advantage of its unique properties.Applied and Computational Harmonic Analysis. 01/2001;
Determining Multiscale Image Feature Angles
from Complex Wavelet Phases
Ryan Anderson, Nick Kingsbury, Julien Fauqueur?
Signal Processing Group, Dept. of Engineering, University of Cambridge, UK
Abstract. In this paper, we introduce a new multiscale representation for 2-D images named
the Inter-Coefficient Product (ICP). The ICP is a decimated pyramid of complex values based
on the Dual-Tree Complex Wavelet Transform (DT-CWT). The complex phases of its coeffi-
cients correspond to the angles of dominant directional features in their support regions. As
a sparse representation of this information, the ICP is relatively simple to calculate and is a
computationally efficient representation for subsequent analysis in computer vision activities
or large data set analysis. Examples of ICP decomposition show its ability to provide an in-
tuitive representation of multiscale features (such as edges and ridges). Its potential uses are
Wavelets, once used primarily for compression, have found new uses for image content analysis.
The ability of the wavelet transform to isolate image energy concisely into spatial, directional, and
scalar components have allowed it to characterize the multiscale profile of non-stationary signals,
including 2-D images, very effectively. In particular, complex wavelets have shown a strong ability
to consistently represent object structures in 2-D images for object recognition and computer vision
In this paper, we explore methods of building upon the phase information of complex wavelets
to yield intuitive image representations. To date, complex wavelet magnitudes have typically been
used in place of real wavelets to improve the consistency of segmentation, denoising, etc. However,
phase information, which indicates the offset of directional features within the support region of a
wavelet coefficient, has found less application to date in analysis and coding applications (although
stereo matching and motion estimation are two examples of its use). Recently, in , Romberg et
al. have described a probabilistic model, the Geometric Hidden Markov Tree (GHMT), which uses
phase as well as magnitude information to infer the angle and offset of contour segments in the
vicinity of a complex wavelet coefficient. In this paper, we introduce a faster method to calculate the
angle of directional energy in the vicinity of a coefficient. This method, which we have named the
Inter-Coefficient Product (ICP), may find use in large-scale image analysis or real-time computer
vision, where computational complexity must be minimized. We introduce the ICP as a complement
to another phase-based transform, the Interlevel Product (ILP) . Upon describing the background
of complex wavelets (section 2), we will develop the ICP transform in section 3 and show example
ICP decompositions in section 4. We conclude in section 5 with a discussion of potential uses of the
ICP and its relationship to the DT-CWT and the ILP.
?This work has been carried out with the support of the UK Data & Information Fusion Defence Technology
2The Dual-Tree Complex Wavelet Transform
Standard real wavelets, such as the Haar and Daubechies wavelets, suffer from shift dependence.
Shift dependence implies that the decomposition of image energy between levels of a multiscalar
decomposition can vary significantly, if the original image is shifted prior to decomposition. This
variation limits the effectiveness of the real wavelet transform to consistently represent an image
object at multiple scales.
Complex wavelets, including the linearly-separable Dual-Tree Complex Wavelet  have been
created to address the problems of shift dependence. A complex wavelet is a set of two real wavelets
with a 90◦phase difference. For 2-D image analysis, the DT-CWT produces d = 1...6 directional
subbands at approximately
labelled with equally-spaced angles of 15◦, 45◦, 75◦, 105◦, 135◦, and 165◦respectively). The impulse
responses in each of these subbands are shown in figure 1. We note that the magnitude responses
of each these subbands can be used to infer feature orientations. However, the lack of precision in
these methods is the primary motivation for us to seek a superior representations through the use
of complex phase information.
10(for convenience, these subbands are often
DT CWT real part
DT CWT imaginary part
Fig.1. The real and imaginary impulse responses of the DT-CWT for each of the six subbands.
Figure 2 shows both the phase (fig. 2a) and magnitude (fig. 2b) responses of a DT-CWT coefficient
to a shifting step response in 1-D. In particular, we observe that this phase response is consistently
linear with respect to the feature offset, in the vicinity of the wavelet coefficient. If we define DW
as the distance between adjacent coefficients, as indicated by the vertical lines in figure 2a, then we
have experimentally determined the relationship between coefficient phase and feature offset to be
−4.49/DW radians per unit length. With this ratio, we can convert DT-CWT phase to a spatial
offset of an edge or impulse, or vice-versa.
In the 2-D case, the phase and magnitude relationships described above apply to edges and ridges
oriented in the direction of the subband. The ratios for 2-D subbands differ from the 1-D example;
for subbands 1, 3, 4, and 6, the ratio is −4.49/?DWcosπ
or −6.35/DW radians per unit length. for
?or −4.72/DW radians per unit length.
For subbands 2 and 5, the ratio is −4.49/?DWcosπ
a) The complex phase of a decimated level 3
DT-CWT coefficient (located at the central
dotted line), in the presence of a step edge
at all possible offsets (the x axis). Note that
when an edge or ridge occurs anywhere be-
tween the coefficient and its immediate neigh-
bours (shown as the vertical dashed lines),
the phase response is linear. This linearity
will be used to infer the offset of the edge, rel-
ative to the coefficient location, at this scale.
−25−20 −15−10−5 0510152025
b) The magnitude response of the same DT-
CWT coefficient under the same conditions.
The overall magnitude is calculated from real
and imaginary components, as shown.
Fig.2. Illustration in the 1-D case of the behaviour of the phase and magnitude of a DT-CWT coefficient
(level 3, in this case) in the presence of a step edge, at the indicated x coordinate, relative to the coefficient.
these 2-D cases, the subband offset of a feature is defined in the direction normal to the subband, as
shown in figure 3.
dA= Subband Offset
Fig.3. Definition of the subband offset dA in the 2-D case between a coefficient location A and a step edge.
3The Inter-Coefficient Product
In this section, we introduce the Inter-Coefficient Product (ICP). We begin in sections 3.1 and 3.2
by showing how the orientation of a 2-D feature (such as an edge or ridge) can be determined from
the phase difference of two adjacent DT-CWT coefficients. This derivation leads us naturally to the
definition of the ICP in section 3.3.
3.1Determination of feature orientation from neighbour coefficients
Consider Figure 4, which shows a feature (a step edge, in this case) that spans the support regions
of two horizontally adjacent DT-CWT coefficients at locations A=(x,y) and B=(x+1,y), at some
arbitrary level. This figure illustrates the trigonometric relationship between the angle of the feature
and its subband offsets with respect to these coefficients.
Dw sin π/10
dA- dB- Dw sin π/10
Dw cos π/10
Fig.4. Trigonometric relationship between the angle θ1 = α +
its subband offsets dA and dB to two horizontally adjacent
A=(x,y) and B=(x + 1,y).
10of a feature (step edge, in this case) and
10subband DT-CWT coefficients located at
As this step edge is closest to the d = 1,
Wl(x + 1,y,1) will correspondingly have large magnitudes in this subband only. From figure 4, we
can see that the angle of the feature, θ1, can be calculated as
10subband, the DT-CWT coefficients Wl(x,y,1) and
10+ α =
10+ tan−1dA− dB− Dwsinπ
where dA and dB are the subband offsets of the edge to the two coefficients as defined in figure
3, using subband 1. These two offset lengths are equal to
respectively1, according to our phase/offset relationships described at the end of section 2. Thus, we
4.72? Wl(x,y,1) and
4.72? Wl(x + 1,y,1)
1We denote the phase argument of a complex number c as? c = arg(c).
can rewrite Equation (1) as
4.72? Wl(x,y,1) −DW
?? Wl(x,y,1) −? Wl(x + 1,y,1)
4.72? Wl(x+1,y,1) − Dwsinπ
We note that for −π
to an individual subband, we can assume that α ≈ tanα. Applying this approximation twice to
Equation (2), we can simplify this expression to
5< α <π
5, which is the approximate range of feature angles that will contribute
4.49[? Wl(x,y,1) −? Wl(x + 1,y,1)]
? Wl(x,y,1) −? Wl(x + 1,y,1)
Thus, for subband 1, we merely divide the phase difference between two horizontally adjacent DT-
CWT coefficients by 4.49 to obtain the angle of the dominant feature in their vicinity.
3.2Feature orientation calculations for all subbands
In the previous section example, we considered the
particular example of step edge. In the general case, to detect a feature at any orientation, the same
type of calculation is achieved for all six subbands based on phase difference between appropriate
neighbour DT-CWT coefficients: we compare horizontal neighbours for the
(d = 1,6), vertical neighbours for the2π
subband. We have already determined θ1(see Equation (4)).
By symmetry, the geometric relationship in the
subbands d = 3,4, and 6, where feature angles in subbands 3 and 4 are measured relative to the
vertical axis. For these two subbands, therefore, we would therefore modify equation (4) to addπ
to the angle. However, in theπ
with the angles of dominant features with two diagonally adjacent coefficients. Using subband 2, the
angle of the feature is related to coefficient offsets by the following simpler equation:
10subband (d = 1) given the orientation of our
5subbands (d = 3,4), and diagonal neighbours for
4subbands (d = 2,5). Thus six orientation angles are calculated θ1,...,θ6, one for each
10subband d = 1 can be equally applied to
4subbands 2 and 5, the Wlvalues possess a different relationship
4+ tan−1dA− dB
Performing the same substitutions, assumptions, and simplifications as with our previous exam-
ple, we establish the linear relationship between θ2and the Wlcoefficient phases to be
8.98[? Wl(x,y + 1,2) −? Wl(x + 1,y,2)](6)
As with the previous example, this relationship is identical for subband 5, except that3π
be added to the phase of θ5in the operation above.