A new generalised α scale spaces quadrature filters

Pattern Recognition ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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A new generalised α scale spaces quadrature filters Ahror Belaid a,b,, Djamal Boukerroui b a b

University of Abderrahmane Mira of Bejaia, Laboratoire d'Informatique Médicale (LIMED), 06000, Bejaia, Algeria Université de Technologie de Compiègne – CNRS, Heudiasyc UMR 7253 BP 20529, 60205 Compiègne Cedex, France

art ic l e i nf o

a b s t r a c t

Article history: Received 22 August 2013 Received in revised form 11 March 2014 Accepted 24 March 2014

The α scale spaces is a recent theory that open new possibilities of phase-based image processing. It is a parameterised class ðα A 0; 1Þ of linear scale space representations, which allows a continuous connection beyond the well-known Gaussian scale space (α ¼ 1). In this paper, we make use of this unified representation to derive new families of band pass quadrature filters, built from derivatives and difference of the α scale space generating kernels. This construction leads to generalised α kernel filters including the commonly known families derived from the Gaussian and the Poisson kernels. The properties of each family are first studied and then experiments on one- and two-dimensional signals are shown to exemplify how the suggested filters can be used for edge detection. This work is complemented by an experimental evaluation, which demonstrates that the new proposed filters are a good alternative to the commonly used Log-Gabor filter. & 2014 Elsevier Ltd. All rights reserved.

Keywords: α Scale spaces Quadrature filters Local phase information Monogenic signal

1. Introduction Based on physiological evidence, the computer vision community introduced models inspired by the Human Visual System (HVS) to solve vision problems. Such motivation is driven by the very high performance of the HVS in solving most computer vision problems. Interestingly, several physiological experiences have suggested that image structures like lines, edges, junctions and orientations play an important role in the HVS. Consequently, these features have always been considered as central in the analysis since the early days. Their detection has therefore been a fundamental operation that needed to be processed in a reliable and robust way. Feature detection has been extensively studied in the literature and still remains an active field of research [1–28]. The study of this large literature suggests that the most classical processing tools are amplitude-based techniques, both in the spatial and frequency domains. Consequently, throughout the history of digital image processing, smoothing and differentiation have been subjects of intense study. A variety of optimal differential operators have been proposed to solve different computer vision problems. For instance, edges and lines detection have received a particular attention [3,5,7,8,10,16,20,25]. Differentiation is highly sensitive to noise, but can be reduced or avoided using an appropriate scale selection for the smoothing function. Amplitude-based techniques however are known to be sensitive to smooth shading and lighting variations. Furthermore, an image contains many types of edges (not only step-edges)

 Corresponding author.

and the interesting features in the image often do not have the idealised shape supposed by the feature model. These may considerably reduce the optimality of the detection and lead to unpredictable multiple responses. Of course, the design of a successful vision algorithms requires “A careful analysis of the problem specification and known constraints from image formation and priors must be married with efficient and robust algorithms” [29]. Hence, recent contributions have reached a high degree of sophistication [27]. These may include, for instance, scale selection and blur estimation methods [12,13]; statistical models based detection [18]; linear [30,31] and non-linear [32,33] scale space methods and links to regularisation theory [34]. An alternative approach to amplitude based techniques is the use of phase information. The framework provides a single unified theory for detecting a wide range of features, rather than being specialised for a single feature type such as step edges. Following the publication of the Local Energy (LE) model of feature detection [35], phase-based feature detection has been investigated extensively. This model postulates that features are perceived at points in an image where the Fourier components are maximally in phase. A wide range of feature types gives rise to points of high phase congruency (PC). Also, it has been shown that this model successfully explains a number of psychophysical effects on human feature perception. This observation has led to the development of a number of phase-based feature detection algorithms [14,17,19,36–43]. Numerous additional advantages of the use of phase-based measurements have been reported. Perhaps the most desirable property of phase is that it is theoretically invariant to brightness and contrast. Hence it is, in principle, robust against typical variations in image formation.

http://dx.doi.org/10.1016/j.patcog.2014.03.029 0031-3203/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: A. Belaid, D. Boukerroui, A new generalised α scale spaces quadrature filters, Pattern Recognition (2014), http: //dx.doi.org/10.1016/j.patcog.2014.03.029i

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2

One of the popular methods to estimate local information of real signals is based on the analytic representation of the signal. It is a complex signal that satisfies, in the one-dimensional case, the Cauchy–Riemann equation and is defined as [44,45] f A ðxÞ ¼ f ðxÞ þ if H ðxÞ

2. Context and background

ð1Þ

pffiffiffiffiffiffiffiffi where i ¼  1 and f H ðxÞ is the Hilbert transform of a given signal f(x). In the Fourier domain Eq. (1) corresponds to F A ðωÞ ¼ FðωÞ  ½1 þ signðωÞ:

Asymmetry (FAM ) measures [41,57]. A discussion on the importance of the shape parameter α in this context is finally given.

ð2Þ

Hence, the analytical signal corresponding to f is obtained by suppressing all its negative frequencies and multiplying all its positive frequencies by two. The local amplitude and local phase information are defined in a straightforward manner from the complex representation of f(x) as the Amplitude and the Argument of the complex signal fA(x). A direct calculation of these local quantities, however, cannot be done in a phase-based technique. This is mainly because the Hilbert transform/analytical signal is defined over the whole signal/spectrum of the signal, while localisation in both space and frequency is highly desirable in the context of feature detection. In other words, we have to design an operator to approximate these quantities in a small, spatial span and over a narrow range of frequencies (scales) to enhance spatial localisation and to avoid the effect of noise. In practice, the estimation of the local signal properties uses a pair of band-pass quadrature filters, an even filter fe(x) and an odd filter fo(x) [46,47]. What is quite certain is that the estimation is intrinsically noisy and depends critically on the choice of the quadrature filters pairs [48,49]. Furthermore, as these local properties are scale dependent, their use for feature detection requires also their scale invariance (at least in a certain range) in order to detect only salient features and not noise. Therefore, the only reasonable approach that has proven itself in practice is to combine several scales. As mentioned earlier, this is in favour for the development of scale space representations and theories [23]. It is in this double context of scale space and quadrature filters that Felsberg et al. [50,51] have introduced the monogenic Poisson scale space filter. This new theory has opened new possibilities of phase-based image processing in scale space. Initially, Felsberg and Duits worked independently on the Poisson scale space following different approaches. Later on, they worked together on a generalised form, which is named the α scale spaces [52] that had appeared initially in [53]. It is a parameterised class (α A 0; 1) of linear scale space representations which allows a continuous connection between the Poisson scale space (α ¼ 1=2) and the well known Gaussian scale space (α ¼ 1) [30]. The axioms underlying this unified framework are studied in [54]. A first comparison of a subset of these representations (α A ½1=2; 1) is due to Duits et al. [55]. They approached the problem only from a topological point view. In this paper, we make use of this unified α scale space representations to derive new families of band-pass quadrature filters. These α kernels lead to the commonly known families of filters: Difference of Poissons [56], Difference of Gaussians, Poisson derivative (also known as the Cauchy filter [37,48]) and Gaussian Derivative filters. In the following sections, first the α scale spaces generating kernel are introduced. The motivation and the usefulness of quadrature filter pairs as a tool for the estimation of local properties of signals are recalled. Then new band-pass filter families are derived and their tuning properties are studied. Furthermore, we illustrate the use of the new filters for edge detection using the Monogenic Phase Congruency (PCM ) and the Monogenic Feature

2.1.

α Scale space kernels

Recently, Duits et al. [54] reinvestigated the one-parameter class of scale space filters introduced by Pauwels et al. [53] in the Fourier domain. The set of axioms leading to what is now known by the α scale spaces (α A 0; 1) are identified and studied. This family of smoothing kernels forms a continuous transition between the identity operator (α ¼ 0), the Poisson scale space (α ¼ 1=2) and the classical well known Gaussian scale space (α ¼ 1). Note that only the gaussian case satisfies the non-enhancement of the local spatial extrema axiom (equivalent to the strong causality). A weaker causality principle is, however, satisfied by all α scale spaces [54,58]. Given a 1D real signal f(x), its α scale spaces representation vðαÞ : R  R þ -R is the unique solution for the following pseudo differential evolution system [52]: 8 ∂ < ∂s v ¼  ð  ΔÞ α v ð3Þ vðx; sÞ ¼ f ðxÞ: : lim s-0 Here s 4 0 is a scaling parameter, Δ is the Laplace operator, f represents the initial condition and α A 0; 1. It has been shown in [52] that the unique solution of (3) is given by means of a convolution operation: vðαÞ ðx; sÞ ¼ ðK sðαÞ nf ÞðxÞ:

ð4Þ

ðαÞ

The kernel K is defined in the spatial domain by the following expression [52]: Z 1 K ðαÞ ðx; sÞ ¼ qs;α ðtÞGt ðxÞ dt; ð5Þ 0

where qs;α is the inverse of the Laplace transform of μ⟶expð  sμα Þ and Gt(x) is the usual Gaussian kernel Gt ðxÞ ¼ 1=ð4π tÞ1=2 expð x2 =4tÞ. The corresponding expression in the Fourier domain is given by [52] KðαÞ ðω; sÞ ¼ expð  sjωj2α Þ:

ð6Þ

In practice, it is preferred to manipulate the bandwidth and the tuning frequency (which depends on s), rather than manipulating the parameter s. There is no tuning frequency common to all applications; the tuning frequency depends on the image and the type of structures to detect (fine or coarse structures). Moreover, a multi-scales approach offers a better control on the edge detection quality, for example, Kovesi [14] suggests that six scales are sufficient in practice for edge detection. 2.2. Quadrature filters As we have already mentioned earlier, the analytical signal and the corresponding quadrature filters provide a powerful framework for the extraction of local properties of signals. Such tool is well known to practitioners in signal processing [35,44,46]. Until a decade ago, two (and more generally N-) dimensional signal theory suffered from the absence of an isotropic extension of the analytical signal/Hilbert transform. In 2001, Felsberg and Sommer were the first to propose a 2D isotropic analytical signal, the monogenic signal [56]. This novel generalisation is based on the Riesz transform, which is used instead of the Hilbert transform, and preserves the core properties of the 1D analytic signal that decomposes a signal into information about structure (local phase) and energy (local amplitude). The basic idea is to design an odd isotropic filter, which is vector-valued rather than scalar-valued. Because the extension is rotation invariant, its use has become

Please cite this article as: A. Belaid, D. Boukerroui, A new generalised α scale spaces quadrature filters, Pattern Recognition (2014), http: //dx.doi.org/10.1016/j.patcog.2014.03.029i

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popular to solve image processing problems [59]. Links to scale space theory [50,51,60], color images [26] and extensions to wavelets [61] and to higher orders have been proposed [43,62,63]. In the context of feature detection, although the importance of quadrature filter has been highlighted [47], regrettably, little work has been done on the selection of the radial part of the quadrature filters. Several families of pairs of quadrature filters have been proposed and applied in the literature: Gabor, log-Gabor, Gaussian derivatives, Difference of Gaussians, Cauchy, Loglets [64]. The properties of these classical families have been studied and some insight on their use have been reported in [48,49]. 3. New band-pass quadrature filters In this section, we aim to derive two new quadrature families using the generating kernel of the α scale spaces given by Eq. (6). We will proceed in the same manner as reported by Boukerroui et al. [48]. The new families provide a unified formulation as most of the above mentioned families will be regarded as particular cases. The main results are given in the form of five propositions (see also Table 1). We first recall hereafter the needed common definitions, namely the filter's octave bandwidth and filter's normalisations. Two normalisation conditions have been used most frequently in the literature [48]:

 maximum condition 1 max F ðωÞ ¼ 1 ) nc ¼ ; ω F ð ω0 Þ

 unit energy condition J FðxÞ J ¼

1 J F ð ωÞ J ¼ 2π

ð7Þ

1 1

F ðωÞF ðωÞ dω ¼ 1;

ð8Þ

where F ðωÞ and F(x) are the filter kernels in the frequency and spatial domain, respectively. The half-response frequency bandwidth β (in octaves) is defined as   ω ð9Þ β ¼ log 2 2 ;

ω1

where ω1 and ω2 are the solutions of F ðω and ω1 o ω2 . 3.1.

respectively, for α ¼ 1=2 and α ¼ 1. Using the derivative property of the Fourier transform, we define the following family of 1D α Scale Spaces Derivative (ASSD) quadrature filter in the frequency domain: ( nc ωa expð ðsωÞ2α Þ if ω Z 0 F ASSD ðωÞ ¼ ð10Þ 0 otherwise where the derivative parameter a A R þ , meaning we are using fractional order derivatives. In order for the filters to satisfy the DC condition and to be also invariant to an additive ramp we impose a 4 1. nc is a normalisation constant and s represents the scale parameter. Important properties and tuning parameters of the ASSD filter namely, the tuning frequency, filter's normalisations and the bandwidth, are given in the following propositions. Proposition 3.1. The peak tuning frequency of the α scale spaces derivative filter is given by (see Definition B.1) 1 a 1=2α ω0 ¼ : ð11Þ s 2α Proposition 3.2. The unit normalisation constant nc of the α scale spaces derivative filter defined by (8) is given by (see Appendix B.2) pffiffiffiffiffiffiffi ð2a þ 1Þ=4α a þ ð1=2Þ πα2 s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nc ¼ 2 : ð12Þ  ffi

Γ

Þ ¼ 12

with nc given by (7)

2a þ 1 2α

Proposition 3.3. The octave bandwidth of the α scale spaces derivative filter is given by (see Appendix B.3)

β¼ Z

3

ln



Wð  1;μÞ Wð0;μÞ



2α lnð2Þ

3.2. Difference of

where μ ¼ 

1 e22α=a

:

ð13Þ

α scale spaces filter

A second way to build a band-pass filter given a low pass filter is to use the difference operator. Thus, we study a new generalised band-pass filter built from the difference of two α scale spaces filters (DoSS). The impulse response of this new filter, in the frequency domain, is given by F DoSS ðωÞ ¼ nc expð  12 ðs1 ωÞ2α Þ  nc expð  12 ðs2 ωÞ2α Þ; ω 4 0 and 0 o s1 o s2 :

ð14Þ

Proposition 3.4. The peak tuning frequency of the difference of α scale spaces filter is given by (see Appendix C.1)

α Scale spaces derivative filter

This new family of filters is based on the derivatives of the generating kernel of the α scale spaces and leads to the Poisson derivatives (PoD) and the Gaussian Derivatives (GD) families,

ω0 ¼

 1=2α ð4αÞ1=2α log ðγ Þ 2 α s2 γ 1

with γ ¼

s1 : s2

ð15Þ

Table 1 Summary of the ASSD and DoSS filters' properties. α A 0; 1. The special cases corresponding to the Poisson kernel (α ¼ :5) and the Gaussian kernel (α ¼ 1) are also reported. Filters

Fourier domain

α scale spaces derivative filter (ASSD)

nc ωa expð  ðsωÞ2α Þ

Poisson derivative filter (PoD)

nc ωa expð  sωÞ

Gaussian derivative filter (GD)

nc ωa expð  ðsωÞ2 Þ

Difference of α scale spaces filter (DoSS) Difference of Poisson (DoP) Difference of Gaussian (DoG)

! ! ðs1 ωÞ2α ðs2 ωÞ2α  nc exp  2 2  s ω  s ω 1 2  nc exp  nc exp  2 2 ! ! ðs1 ωÞ2 ðs2 ωÞ2  nc exp  nc exp  2 2 nc exp 

Tuning frequency

Bandwidth

1  a 1=2α s 2α a s rffiffiffiffi 1 a s 2  1=2α ð4αÞ1=2α log ðγÞ s2 γ 2α  1   2 log ðγÞ s2 γ  1  1=2 2 log ðγÞ 2 s2 γ  1

  1 Wð  1; μÞ 1 log 2 ; μ ¼  2α=a 2α Wð0; μÞ e2   Wð 1; μÞ 1 log 2 ; μ ¼  1=a Wð0; μÞ e2   1 Wð  1; μÞ 1 log 2 ; μ ¼  2=a 2 Wð0; μÞ e2 Numerical (Fig. 6) Numerical (Fig. 6) Numerical (Fig. 6)

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Proposition 3.5. The difference of α scale spaces filter's unit normalisation constant is given by (see Appendix C.2) #  1=2 pffiffiffiffiffiffiffiffiffiffiffi " 2 παs2 1 21=2α ffi  2 : 1 þ nc ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi   γ s22α  1 ðγ 2α þ 1Þ Γ 1

ð16Þ

2α

with γ as given in Proposition 3.4. 4. 1D analysis of the new filters 4.1.

α Scale spaces derivatives filters

Fig. 1 shows the behaviour of the bandwidth of the ASSD filter as a function of α and the derivative parameter a. The arrow in the left figure indicates increasing values of α from 0.1 to 1 in steps of 0.1. Observe that there is a common value, β ¼ 2:5, that defines the ASSD filter's bandwidth almost for all values of α. This will be useful to set a common value for β during the experiments over all possible values of α. In the context of feature detection using quadrature filters, a number of authors suggested the use of a bandwidth around 2.5 octaves. The literature also suggests that small values of the derivative parameter a may be preferred, as differentiation increases noise. Consequently, the exponential behaviour of the filter's bandwidth function of a and α suggests the use of α values

8

4.2. Difference of

6

6

α scale spaces filters

In this section, we reiterate the previous experiments in the case of the DoSS filter. The bandwidth as function of γ for some values of α is shown on Fig. 6. As we can see immediately, the values of the bandwidth grow exponentially for alpha close to 0.1.

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close to 1. Furthermore, the frequency domain representation of the ASSD filter, shown in Fig. 2, demonstrates that the ASSD filter has less aliasing when α is close to one. Therefore in our first experimental analysis of the filters we limit the range of α to ½0:5; 1. An example of a local amplitude and a local phase scalograms of a given 1D signal is shown in Figs. 3 and 4, respectively. The signal is a combination of different types of lines and edges. The right part of the signal is a noisy version of its left part, obtained by the addition of a white Gaussian noise (SNR ¼7 dB). The scalograms are obtained with the ASSD filter with a bandwidth β ¼ 2:5 octaves and for three different values of the shape parameter α. It can be observed that α has a little effect on the Amplitude scalogram. Although, the inspection of the local phase scalograms, shown in Fig. 4, indicates that phase information is more persistent along scales when α is close to 0.5, this experiment suggests that the influence of the choice of α on local energy based edge detection methods will probably be low. The level lines of the local phase scalograms, shown on Fig. 5, confirm this.

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Fig. 1. The bandwidth of the ASSD filter. Left: bandwidth as function of the derivative parameter a for α A f0:1; 0:2; …; 1g. The arrow shows increasing values of α. Right: contour map of the bandwidth as function of a and α.

Fig. 2. The ASSD filter in the Fourier domain for different values of α A f0:1; 0:2; …; 1g. β ¼ 2:5 octaves and w0 ¼ 2π=20. Left: linear scale. Right: logarithmic scale.

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Fig. 3. An example of a 1D signal and its amplitude scalogram computed using the ASSD filter. From top to bottom: original signal (from WaveLab); Amplitude scalograms for α A f0:5; 0:75; 1g. The bandwidth β ¼ 2:5. The lower tuning frequency w0 ¼ 2π=30. There are 100 logarithmic scales with a scaling factor of 1.05 between successive scales.

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Fig. 4. Local phase scalograms corresponding to the example shown in Fig. 3.

As mentioned earlier, localisation in both space and frequency is highly desirable. Thus, a filter with a bandwidth higher than 4 octaves is not practical. This implies that interesting values of α are between 0.5 and 1. Also, we observe that one can set a

common bandwidth of 3.53 octaves for α A ½0:5; 1. Fig. 7 shows the DoSS filter in Fourier domain for some values of α. Note that the DoSS family has less aliasing when α approaches 0.5, while the ASSD filter has less aliasing for α values close to 1.

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Fig. 5. Level contours of the local phase scalograms shown in Fig. 4 for α ¼ 0:5 (red line) and α ¼ 1 (blue line). Top: 0 and π level contours; bottom: 7 π=2 level contours. A local phase of π=2,  π=2, 0 and π corresponds to a negative step edge, a positive step edge, a roof (line/bar) and a valley, respectively. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Fig. 6. Bandwidth of the DoSS filter. Left: bandwidth as function of γ for α A f0:1; 0:2; …; 1g. Right: contour map of the bandwidth as function of γ and α.

Fig. 7. The DoSS filter in the Fourier domain for α A f0:5; 0:6…; 1g. β ¼ 3:53 octaves and w0 ¼ 2π=20. Left: linear scale. Right: logarithmic scale.

The inspection of the local amplitude and the local phase scalograms shown, respectively, in Figs. 8 and 9, suggests that the shape parameter α has a real impact on feature detection.

Indeed, the case of α ¼ 1 offers a better localisation than for α ¼ 0:5, and the phase is more persistent along scales (see the isocontours shown in Fig. 10). However, we may expect a better

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Fig. 8. Amplitude scalogram computed using the DoSS filter of the 1D signal shown in Fig. 3. From top to bottom: Amplitude scalograms for α A f0:5; 0:75; 1g. The bandwidth β ¼ 3:53. The lower tuning frequency w0 ¼ 2π=30. There are 100 logarithmic scales with a scaling factor of 1.05 between successive scales.

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Fig. 9. Local phase scalograms corresponding to the example shown in Fig. 8.

SNR for α close to 0.5. This claim is supported by the higher local amplitude response observed on the amplitude scalogram. Notice also the influence of the noise on the phase scalograms. This is probably due to the fact that the DoSS filter has more aliasing for larger values of α. What we can learn from this one-dimensional analysis is that, the two studied filters react differently according to the variation of the shape parameter α. In fact, the variation of the parameter α seems not to affect greatly the quality of feature detection in the case of the ASSD family. However, we may expect a significant difference in terms of detection performance for the DoSS filter between α ¼ 1 and α ¼ 0:5.

5. Application to image edge detection Phase-based processing has attracted a lot of attention in image analysis since the work of Felsberg and Sommer [56]. Indeed, phase information has been used in numerous applications, such as segmentation, registration, stereo matching, optical flow, denoising, texture characterisation and corner and edge detection (see [48,57], and references therein). Thus, the new proposed quadrature families may be useful in a variety of applications. In this work, we choose to illustrate the use of the proposed filters for edge detection. Specifically, we are interested in the influence of the parameter α on the detection performance. To this end, we use two measures

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Fig. 10. Level contours of the local phase scalograms shown in Fig. 9 for α ¼ 0:5 (red line) and α ¼ 1 (blue line). Top: 0 and π level contours; Bottom: 7 π=2 level contours. A local phase of π=2,  π=2, 0 and π corresponds to a negative step edge, a positive step edge, a roof (line/bar) and a valley, respectively. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Fig. 11. 2D DoSS kernels in Fourier domain for certain parameters. From left to right: the isotropic even part and the pair composing the odd part.

of contour detection, namely the Monogenic Phase Congruency (PCM ) and the Monogenic Feature Asymmetry (FAM ) [57,65]. Both measures are phase based and use multiple scales for the analysis of the given image. The PCM and FAM are evaluated on the Berkeley Segmentation DataSet (BSDS500) [66,67]. It consists of 500 natural images and ground-truth human annotations. This is one of the most complete dataset available for our purpose. It has the advantage of providing multiple human-labelled segmentations per image and it has been used by several research groups for evaluation.

5.1. The PCM and FAM measures The monogenic signal at a given scale s of a 2D signal f ðxÞ can be represented by a scalar valued even and a vector valued odd filtered

responses (see Figs. 11 and 12), with the following simple trick: evens ¼ cnf ; odds ¼ ðcnh1 nf ; cnh2 nf Þ; where n is the convolution operator in image space, c is the spatial domain representation of an isotropic band-pass filter at scale s, and h ¼ ðh1 ; h2 Þ is the generalised Hilbert transform kernel, hðxÞ ¼ ðh1 ; h2 Þ ¼

1 x : 2π jxj3

Given the above, the multiple scales local amplitude and local energy responses are defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ¼ ∑ even2s þ jodds j2 ; s

E2 ¼ ∑ðeven2s þ jodds j2 Þ: s

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Fig. 12. 2D DoSS kernels in spatial domain for certain parameters. From left to right: the isotropic even part and the pair composing the odd part.

The monogenic Phase Congruency can be defined as [14,65]  

E ⌊E Tc  ; PCM ¼ W 1  arccos Aþε Eþε

ð17Þ

where ⌊c denotes zeroing of negative values, T is a noise threshold estimate, based on high frequency wavelet responses, and W is a weighting function that penalises filter response spread [14]. Here, ε is used to avoid zero division. We also define the multiple scales monogenic Feature Asymmetry by [14,57,68] FAM ¼

∑s ⌊jodds j jevens j  T s c qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ∑s even2s þ jodds j2 þ ε

ð18Þ

In the experimental study, the wavelength of the finest scale ω0 is set to 10. In order to detect fine as well as coarse structures, we consider three scales with a multiplicative scaling factor of Section 2.1. 5.2. Evaluation The evaluation is carried out by comparing machine generated contours to human ground-truth data using the precision–recall framework introduced in [69]. In our context, the precision– recall curves are obtained by varying the detection threshold. Precision is the fraction of detections which are true positives, while recall is the fraction of positives that are detected. Usually a relative cost between these quantities is defined, which focuses attention at a specific point on the precision–recall curve. The F1 measure, or harmonic mean of precision and recall, defined as F1 ¼ 2

Precision  Recall ; Precision þ Recall

captures this trade-off. Thus, the location of the maximum of the above measure along the curve defines the optimal threshold. The F1 measure obtained at the optimal detector threshold provides a summary score. When a fixed threshold is used for all images in the dataset, the optimal threshold with respect to the F1 cost

was refereed in [67] as the optimal dataset scale (ODS).1 The authors also defined the optimal image scale (OIS) when the optimal threshold is selected on a per-image basis. Therefore, following [67], we use the below quantities as evaluation scores: 1. The best F1 cost when choosing an optimal scale for the entire dataset (ODS). 2. The average F1 cost when choosing an optimal scale per image (OIS). 3. The Average Precision (AP) on the full recall range (equivalently, the area under the precision–recall curve). The main results of our experiments are summarised in Table 2. The optimal parameters for each filter as well as the relative F-measure scores are shown (see also Fig. 13). This experiment was carried out using 200 test images from BSDS500. The details of these results – using a sample of 100 test images – are reported in Tables 3 and 4 for the ASSD filter, and in Tables 5 and 6 for the DoSS filter. We run experiments for the two proposed filters over the full range of the possible values of their parameters α, a and γ. The maximum scores according to the parameters are shown in the right and bottom columns. These scores allow us to compare the effects of the different parameters of each filter. 5.3. Discussion Table 2 reports the most important results of the contour detection experiments. First, we notice that in general the FAM outperforms the PCM and this is regardless of the used band-pass quadrature filter. One reason for this may be the fact that the FAM is a specialised detector to step edges, as opposed to the PCM , which can detect all types of edges. As a consequence of its general behaviour, in comparison to the FAM measure, the PCM measure is more sensitive to noise and performs less on textured images. We may also argue that there is probably more step edges in natural images than other types of edges, which gives an advantage to 1

Here following [67], by scale we mean the detection threshold.

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Table 2 Results for the PCM and the FAM contour detectors on the BSDS benchmark using ASSD, DoSS and Log-Gabor (LG) filters. The BSDS benchmark consists of 200 test images. The best parameters' values for α and β, and the F-measure scores when choosing an Optimal Dataset Scale (ODS) or an Optimal Image Scale (OIS), as well as the Average Precision (AP) are shown. Filter

Algorithm

α

β

ODS

OIS

AP

ASSD

PCM FAM PCM FAM PCM FAM

0.1 0.1 1 1 – –

2.80 3.80 3.36 3.36 3.50 3.50

0.59 0.61 0.59 0.61 0.59 0.61

0.60 0.62 0.61 0.63 0.60 0.62

0.50 0.47 0.52 0.48 0.48 0.45

DoSS LG

Fig. 13. Precision–Recall curves of the PCM and FAM contour detectors on the BSDS500 [66,69] using the ASSD, DoSS and Log-Gabor filters. The highest F1 measure using the optimal parameters (see Table 2) is also reported for PCM (solid lines) and FAM (dotted lines). Left: ASSD (blue color) and DoSS (red color) filters. Right: Log-Gabor filter. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Table 3 Results obtained with the PCM measure, on a sample of 100 test images from the BSDS benchmark, using the ASSD filters. The F-measure scores (ODS) for varying values of α and a, and the maximum values each line and column are shown. a

α 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Max

7 6 5 4 3 2 1

0.579 0.577 0.575 0.571 0.562 0.544 0.517

0.569 0.574 0.577 0.577 0.575 0.566 0.533

0.543 0.555 0.565 0.573 0.575 0.571 0.543

0.517 0.532 0.547 0.562 0.572 0.572 0.549

0.491 0.510 0.529 0.548 0.565 0.571 0.552

0.467 0.488 0.511 0.534 0.556 0.569 0.553

0.445 0.468 0.493 0.519 0.545 0.565 0.554

0.426 0.450 0.477 0.505 0.535 0.561 0.553

0.408 0.433 0.461 0.493 0.525 0.556 0.552

0.393 0.418 0.447 0.480 0.515 0.550 0.548

0.579 0.577 0.577 0.577 0.575 0.572 0.554

Max

0.579

0.577

0.575

0.572

0.571

0.569

0.565

0.561

0.556

0.550

0.579

Table 4 Results of the FAM measure on the BSDS benchmark using the ASSD filters. a

α 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Max

7 6 5 4 3 2 1

0.592 0.594 0.596 0.597 0.594 0.584 0.554

0.572 0.578 0.583 0.589 0.593 0.593 0.574

0.549 0.560 0.569 0.578 0.585 0.590 0.579

0.529 0.541 0.553 0.566 0.577 0.586 0.579

0.508 0.523 0.539 0.554 0.569 0.581 0.578

0.490 0.506 0.524 0.543 0.561 0.576 0.575

0.474 0.491 0.511 0.532 0.553 0.572 0.572

0.460 0.478 0.498 0.521 0.545 0.567 0.570

0.448 0.466 0.486 0.511 0.537 0.562 0.567

0.438 0.456 0.476 0.501 0.530 0.558 0.565

0.592 0.594 0.596 0.597 0.594 0.593 0.579

Max

0.597

0.593

0.590

0.586

0.581

0.576

0.572

0.570

0.567

0.565

0.597

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Table 5 Results of the PCM measure on the BSDS benchmark using the DoSS filters. γ

α 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Max

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492 0.492

0.508 0.508 0.508 0.508 0.508 0.508 0.508 0.507 0.507

0.520 0.520 0.520 0.520 0.520 0.520 0.520 0.521 0.521

0.535 0.535 0.535 0.535 0.536 0.536 0.537 0.538 0.539

0.552 0.552 0.552 0.552 0.553 0.553 0.554 0.555 0.556

0.561 0.562 0.562 0.562 0.563 0.564 0.565 0.566 0.568

0.564 0.565 0.565 0.565 0.566 0.568 0.570 0.573 0.575

0.565 0.565 0.566 0.567 0.568 0.570 0.573 0.576 0.578

0.560 0.560 0.562 0.564 0.567 0.570 0.575 0.578 0.580

0.550 0.552 0.553 0.557 0.561 0.567 0.574 0.579 0.582

0.565 0.565 0.566 0.567 0.568 0.570 0.575 0.579 0.582

Max

0.492

0.508

0.521

0.539

0.556

0.568

0.575

0.578

0.580

0.582

0.582

Table 6 Results for the FAM measure on the BSDS benchmark using the DoSS filters. γ

α 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Max

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.495 0.496 0.496 0.496 0.496 0.495 0.495 0.495 0.495

0.532 0.532 0.532 0.532 0.532 0.532 0.532 0.532 0.532

0.557 0.557 0.557 0.557 0.557 0.557 0.557 0.558 0.559

0.571 0.571 0.571 0.571 0.572 0.573 0.574 0.575 0.578

0.578 0.578 0.578 0.579 0.579 0.581 0.583 0.585 0.590

0.579 0.579 0.580 0.580 0.581 0.583 0.586 0.590 0.596

0.577 0.577 0.578 0.579 0.581 0.583 0.586 0.591 0.598

0.572 0.573 0.574 0.575 0.578 0.581 0.585 0.591 0.599

0.566 0.567 0.568 0.570 0.573 0.577 0.582 0.590 0.599

0.558 0.558 0.560 0.563 0.567 0.573 0.580 0.588 0.599

0.579 0.579 0.580 0.580 0.581 0.583 0.586 0.591 0.599

Max

0.496

0.532

0.559

0.578

0.590

0.596

0.598

0.599

0.599

0.599

0.599

Fig. 14. Level curves of the ODS F1 measure of the ASSD filter, function of a and α. PCM (left); FAM (right). The red line represents the best score. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

the FAM measure on this benchmark. This observation is also supported by the few illustrative natural images shown in Figs. 15 and 16. Of a particular note is the optimum value of α in the case of the ASSD filter, shown in Table 2. Gaussian derivative (α ¼ 1) and Poisson derivative (α ¼ 0.5) filters seem not to be the best choice regardless of the chosen detection algorithm. Indeed, our experiments suggest that the best α value for the derivative filters is α ¼ 0:1. The level curves of the ODS F1 score function of a and α, shown in Fig. 14, demonstrate clearly that the detection performance is inversely proportional to α. Also notice the high correlation between the iso-contours of the F1 score (Fig. 14) and those of

the bandwidth β shown in Fig. 1(b). This experiment highlights the importance of the use of a large bandwidth in the context of the study. From Figs. 1(b) and 14, the high F1 scores correspond to bandwidths in the range [2.5, 4]. Interestingly, this observation also applies to the DoSS family. Indeed, the bandwidth of this family increases rapidly when α decreases (see Fig. 6). The best detection performances are therefore obtained for α values close to one. The DoG filter for γ ¼ 0.1 gives the best performances for both measures. Our experiments highlight the importance of the choice of the shape parameter α. Its influence can be better assessed by taking the maximum values of the scores for a given α value over the

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Fig. 15. Edge detection results on the BSDS500 benchmark. From top to bottom: original images, the corresponding human segmentations and the segmentation results for the ASSD and the DoSS filters, respectively, using the PCM and the FAM measures with the optimal parameters.

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Fig. 16. Additional results. The organisation of the images follows the one in Fig. 15.

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whole range of the second parameter (a for the derivative family and γ for the difference family) as reported in Tables 3–6. The maximum scores for the ASSD filter vary approximatively by 5% while those of the DoSS filters vary by 15%. These variations are higher than those for the derivative parameter a and the difference parameter γ. The performance of the DoSS filter appears more dependent on the values of α compared to the ASSD filter. This was expected, as we have shown that the bandwidth of the DoSS filter depends critically on α. This also comes in support of the previous 1D analysis of the two filters. Indeed, observe the influence of α on the local phase and local amplitude scalograms shown in Figs. 3, 4, 8 and 9. For the ASSD filter, the best choice seems to occur when α is close to 0.1, whereas for the DoSS filter, better results are obtained when α is close to 1. A general remark, all experiments highlight the importance of the bandwidth in the studied context. Therefore, the choice of α is linked with those of a and γ. Although the experimental study presented here is limited, it suggests that the new α scale space derivative filter with α ¼ 0:1 is generally more efficient than the extensively used Gaussian/ Poisson derivative filters. More broadly, the comparison results show that the new derived band-pass quadrature families are a good alternative for the commonly used Log-Gabor filter in the context of phase based feature detection. This has already been underlined for the special cases of the Cauchy and Gaussian families in the work of Boukerroui et al. [48]. 6. Conclusion Two new parameterised classes (α A 0; 1) of band-pass quadrature filters are defined. They are built from derivatives and differences of the α scale space generating kernel. The parametric nature of the proposed families allows to define a wide range of unexplored band-pass filters. On the other hand, the unified framework simplifies the processing of several families of filters, mainly their implementation, comparison and choice of filters. The most interesting and best known in the literature are the families of Derivative/Difference of Gaussian (α ¼ 1) and Poisson (α ¼ 1=2) filters. These families are special cases among the new general proposed filters. First, important properties and filters characteristics are defined and analysed in the 1D case. We then looked more closely at the influence of the shape parameter α for one and twodimensional signals in the context of phase based edge detection. In this context, based on a quantitative evaluation on 200 natural images, we have shown that the new derived band-pass quadrature families offer more flexibility. More importantly, we have shown that the bandwidth of the filters is probably the key parameter. Consequently, the detection performance depends not only on the generating low-pass kernel (Gaussian, Poisson), but also on the family type (Derivative or Difference). The optimal settings of the new proposed filters give a slightly better results than the commonly used Log-Gabor filter. Finally, it is important to highlight that a further analysis of the new filters is still needed and will be the subject of future research.

Conflict of interest The authors do not have any conflict of interest.

presentation of this paper. We also thank Izumi OTANI for English proofreading. Appendix A. Preliminary theoretical equations To perform some calculation, we make use of the following known results.

 Gamma equality equation: Z

1 n

x expð  cxÞ dx ¼ 0

We would like to thank the anonymous reviewers for their suggestions, which helped to improve the content and the

when n A R; n 4  1 and c A R þ ;

: cnn!þ 1

when n A N and c A R þ :

c

ð19Þ

 The Lambert WðxÞ function is defined as the inverse function of [70]: y expðyÞ ¼ x:

ð20Þ

The solution is given by y ¼ WðxÞ or in short we have WðxÞeWðxÞ ¼ x.

Appendix B. Definition of the ASSD filter properties B.1. Tuning frequency The peak tuning frequency is obtained as the solution of the following equation2: ∂ F SSD ðωÞ ¼ 0: ∂ω

ð21Þ

From the definition of the Fourier transform of the ASSD filter given by (10) we get ∂F SSD ðωÞ ¼ nc ωa  1 expð  ðsωÞ2α Þða 2αðsωÞ2α Þ: ∂ω Therefore the positive solution of (22) is given by 1 a 1=2α ω0 ¼ : s 2α

ð22Þ

B.2. Unit normalisation constant In order to compute nc in the case of the unit energy condition, we make use of Parseval's theorem J F SSD ðxÞ J ¼ 1=2π J F SSD ðωÞ J . Thus, the condition (8) leads to the following equation: Z 1 n2c ω2a expð  2ðsωÞ2α Þ dω ¼ 2π : ð23Þ J F SSD ðωÞ J ¼ 0

The above integral is easily computed using the result (19). Indeed, substituting ω ¼ w1=2α and dω ¼ ð1=2αÞw1=2α  1 dw in the above equation, we get Z 1 1 J F SSD ðωÞ J ¼ n2c w2a=2α expð 2s2α wÞ w1=2α  1 dw 2α 0 Z n2 1 ð2a þ 1Þ=2α  1 ¼ c w expð  2s2α wÞ dw 2α 0 2a þ 1  Γ 2α n2 ð24Þ ¼ c ð2a þ 1Þ=2 α 2a þ 1 ¼ 2π : 2α 2 s Finally, we have pffiffiffiffiffiffiffi ð2a þ 1Þ=4α a þ 1=2 πα2 s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : nc ¼ 2  ffi

Γ

Acknowledgments

8 < Γ ðnn þþ11Þ

2a þ 1 2α

ð25Þ

2 Here we omit the condition on the second derivative as we expect a single maximum point.

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B.3. Bandwidth Finding the solutions ω1 and by (7) is equivalent to solving ua expð  ω2α s2α Þ ¼

ω2 of F SSD ðωÞ ¼ 12 with nc given

F0 ; 2

ð26Þ

where F 0 ¼ F SSD ðω0 Þ ¼ ωa0 expð  ω20α s2α Þ. Putting both sides of the above equation to the power of 2α=a and then multiplying by s2α 2α=a gives     s2α 2α 2α s2α 2α s2α 2α F 0 2α=a ¼ ω exp  ω2α : ð27Þ a a a 2 Now, notice that (27) is in the form of zez ¼ c with   s2α 2α 2α s2α 2α F 0 2α=a z¼  ω and c ¼  : a a 2 Thus, making use of (20) leads us to ! !   s2α 2α 2α s2α 2α F 0 2α=a 2  2α=a  u ¼W  ¼W  : a a 2 e

ð28Þ

Appendix C. Definition of the DoSS filters properties C.1. Tuning frequency Solving for the positive solution of ð∂=∂ωÞF DoSS ðωÞ ¼ 0 leads us to ! ! ðs1 ωÞ2α ðs2 ωÞ2α s21α exp  ¼ s22α exp  : ð31Þ 2 2

ð32Þ

Denoting by γ ¼ s1 =s2 , the tuning frequency is finally given by  1=2α ð4αÞ1=2α lnðγ Þ ω0 ¼ : ð33Þ s2 γ 2α  1 C.2. Unit normalisation constant Following the same procedure as for the SSD family, we have: J F DoSS ðωÞ J ¼

1

0

Z ¼ n2c

"

! !#2 ðs1 ωÞ2α ðs2 ωÞ2α  nc exp  nc exp  dω 2 2

1

0

expð  ðs1 ωÞ2α Þ dω þ n2c

Z  2n2c

0

1

Z

1 0

expð  ðs2 ωÞ2α Þ dω

  1 exp  ω2α ðs21α þ s22α Þ dω: 2

The previous integrals can be computed using the results given in (19). After rearranging we get the following: #  " Γ 1 1 21=2α J F DoSS ðωÞ J ¼ n2c 2α 1 þ 2 2α  1 2α : 2αs2 γ s2 ðγ þ 1Þ

ð35Þ

2α

where μ ¼  2  2α=a =e and Wðk; Þ is the kth branch of the Lambert function and k¼ 0 or  1. The octave bandwidth of the SSD filter is then given by    1;μÞ ln Wð Wð0;μÞ 2  2α=a with μ ¼  : ð30Þ β¼ 2α lnð2Þ e

Z

Making the change of variable ω ¼ w1=2α leads us to Z 1 1 ð1=2α  1Þ w expð  ðs1 Þ2α wÞ dw J F DoSS ðwÞ J ¼ n2c 2α 0 Z 1 1 ð1=2α  1Þ w expð  ðs2 Þ2α wÞ dw þ n2c 2α 0  2α  Z 1 s þ s22α 1 ð1=2α  1Þ w dw: w exp  1  2n2c 2α 2 0

Solving J F DoSS ðωÞ J ¼ 2π for nc, we get the final result #  1=2 pffiffiffiffiffiffiffiffiffiffiffi " 2 παs2 1 21=2α ffi 2 : nc ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ   γ s22α  1 ðγ 2α þ 1Þ Γ 1

The last equality is obtained by replacing ω0 by its value in F 0 and then simplifying. Finally, the solution is given by 1 a 1=2α ω¼  Wðk; μÞ1=2α ; ð29Þ s 2α

Taking the logarithm of both sides and rearranging gives   log ss12 ω2α ¼ 4α 2α 2α : s2  s1

15

ð34Þ

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Ahror Belaid received the M.S. degree in Operation Research from University of Abderrahmane Mira, Bejaia, Algeria. He received his Ph.D. degree at the Department of Information Processing Engineering, Heudiasyc laboratory, Compiègne University of Technology, France. He is currently an Associate Professor with the Department of Operation Research, Abderrahmane Mira University. His current research interests include the phase-based images segmentation and edge detection.

Djamal Boukerroui was borne in Bejaïa, Algeria, in 1972. He received the B.S. degree in 1990, and the M.S. degree in electronics in July 1995 form ENP of Algiers (Algeria). He received the Ph.D. degree in Image Processing at CREATIS Laboratory of INSA of Lyon (France) in 2000. From March 2000 to August 2002 he was a Research Assistant at the Medical Vision Laboratory, Department of Engineering Science, University of Oxford where he worked on the analysis of echocardiographic image sequences. Since September 2002, he joined the Department of Information Processing Engineering of Compiègne University of Technology as Maître de Conférences, and he is a member of HEUDIASYC [research unit associated with CNRS (#7253)]. His current main research interests are in low level image processing and specifically its application in ultrasound image analysis.

Please cite this article as: A. Belaid, D. Boukerroui, A new generalised α scale spaces quadrature filters, Pattern Recognition (2014), http: //dx.doi.org/10.1016/j.patcog.2014.03.029i