THE MONOGENIC CURVELET TRANSFORM Martin Storath â â M6 ...
THE MONOGENIC CURVELET TRANSFORM Martin Storath † †
M6 - Mathematische Modellbildung, Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, D-85748 Garching bei München, Germany ABSTRACT
In this article, we reconsider the continuous curvelet transform from a signal processing point of view. We show that the analyzing elements of the curvelet transform, the curvelets, can be understood as analytic signals in the sense of the partial Hilbert transform. We then replace the usual curvelets by the monogenic curvelets, which are analytic signals in the sense of the Riesz transform. They yield a new transform, called the monogenic curvelet transform, which has the interesting property that it behaves at the fine scales like the usual curvelet transform and at the coarse scales like the monogenic wavelet transform. In particular, the new transform is highly anisotropic at the fine scales and yields a well-interpretable amplitude/phase decomposition of the transform coefficients over all scales. Index Terms— Curvelet transform, Analytic signal, Monogenic signal, Hilbert transform, Riesz transform 1. INTRODUCTION The continuous curvelet transform is a multiscale transform which allows to resolve the singularities of an image together with their orientations, e.g., the positions and the directions of edges [1]. To this end, the curvelet transform increases the anisotropy of its analyzing elements – the curvelets – as the scale decreases, thus the curvelets have higher directional selectivity at the fine scales. However, from a signal processing point of view, the curvelets have another nice property, which has not been exposed so far. That is, the curvelets are analytic signal filters in the sense of the partial Hilbert transform. Analytic signal filters are important in signal processing, because they yield a meaningful decomposition of the filtered signal into amplitude and phase, where the amplitude has the interpretation of the envelope of the signal. The analytic signal of one-dimensional functions is based on the Hilbert transform. The aforementioned two-dimensional generalization of the analytic signal by the partial Hilbert transform has several drawbacks, which we point out in section 2.2. We resolve these problems by switching to another generalization of the analytic signal, the monogenic signal. Thus we replace the usual curvelets by monogenic curvelets, which yield the new monogenic curvelet transform.
Several authors already use the monogenic signal in image processing. Especially the monogenic wavelet transform [2, 3, 4] opens many new applications like AM/FM analysis [3] or descreening [2]. The disadvantage of the monogenic wavelet transform is their poor directional selectivity. To gain higher anisotropy, the authors in [5] propose higher order Riesz transforms, which leads to a similar approach like the steerable pyramid. However, the degree of directional selectivity of that approach does not adapt to the scale, so the resolution of the orientations of the image singularities is still not optimal [1]. The monogenic curvelets, in contrast, adopt the scale-adaptive anisotropy of the usual curvelets, which results in a better resolution of the orientations. The article is organized as follows. First we give an introduction to the analytic signal and its generalizations to two dimensions, namely the Hilbert-analytic signal and the monogenic signal. We then identify the usual curvelets as Hilbert-analytic signal. From there we derive the monogenic curvelets and the new monogenic curvelet transform. Finally, we show that the monogenic curvelet transform behaves like the usual curvelet transform at the fine scales and like the monogenic wavelet transform at the coarse scales. 2. ANALYTIC SIGNAL CONCEPTS Throughout this article we use (r, ω) for polar coordinates in the frequency domain, and x = (x1 , x2 ) and ξ = (ξ1 , ξ2 ) for Cartesian coordinates in the spatial domain and in the frequency domain, respectively. Further, f is always a realvalued and square-integrable function. 2.1. The Analytic Signal in 1D Let H : L2 (R, R) → L2 (R, R) be the Hilbert transform defined in the Fourier domain by d(s) = i sign (s) fb(s). Hf Note that Hf is also real-valued. A complex-valued function g : R → C whose imaginary part is the Hilbert transform of its real part, i.e., Im g = −H(Re g), is called analytic signal. In 1D signal processing the analytic signal is used to decompose a signal into amplitude and phase, so loosely spoken into a signal intensity and a signal structure.
Monogenic wavelets [2]
Curvelets [1]
Monogenic curvelets
Coarse scales
Directionality
Low
Low
Low
(a ≥ α0 )
Analytic signal concept
Monogenic signal
No analytic signal concept
Monogenic signal
Fine scales
Directionality
Low
High (scale-adaptive)
High (scale-adaptive)
(a < α0 )
Analytic signal concept
Monogenic signal
Hilbert-analytic signal
Monogenic signal
Table 1. Comparison between the monogenic wavelets, the usual curvelets, and the monogenic curvelets. 2.2. The Hilbert-Analytic Signal A straightforward extension to 2D is achieved by letting the Hilbert transform operate on parallel lines which point towards a fixed orientation θ. This extension is called partial Hilbert transform with respect to the angle θ [6], defined in the Fourier domain for f ∈ L2 (R2 , R) by d b H θ f (ξ) = i sign (cos(θ)ξ1 + sin(θ)ξ2 ) f (ξ). We call a complex-valued function g : R2 → C Hilbertanalytic signal, if its imaginary part is the partial Hilbert transform with respect to any angle θ of its real part, i.e., Im g = −Hθ (Re g). The Hilbert-analytic signal has two major drawbacks. The first one is that the partial Hilbert transform has a purely one-dimensional nature, thus, the Hilbert-analytic signal does not take into account sufficiently the structure of true twodimensional signals. The second problem is the dependence on the angle θ. Though there are functions f which only allow one sensible choice for θ, in general, Hθ yields a different analytic signal of f for every different θ ∈ [0, 2π) . 2.3. The Monogenic Signal In order to overcome the aforementioned problems, a different generalization of the analytic signal for 2D was introduced in [7], which is called the monogenic signal. It is considered to be the proper generalization of the 1D-analytic signal for image processing [7] and has proved its utility in several image processing applications [2, 3, 5]. The monogenic signal is based on the Riesz transform. For f ∈ L2 R2 , R the Riesz transform with respect to the axis xν , ν = 1, 2, is defined by ξν b d R f (ξ). ν f (ξ) = i |ξ| Note that Rν f is a real-valued function. Unlike the partial Hilbert transform, the Riesz transform does not depend on any orientation θ. Instead we have two transforms R1 f and R2 f with respect to the fixed coordinate axes x1 and x2 . Consequently, a sensible representation of the monogenic signal needs two imaginary parts, one for R1 f and one for R2 f. Hence we have to switch from the complex numbers to a hypercomplex algebra which possesses at least two imaginary
units. Assume for the moment that we have such an algebra with the two imaginary units i an j . Then the monogenic signal Mf is defined by Mf := f − i R1 f − j R2 f. The most obvious choice for the desired hypercomplex algebra are the quaternions H := {h = a + i b + j c + k d : a, b, c, d ∈ R} , which are an extension of the complex numbers with the three imaginary units i , j , and k . In short, the quaternions are a 4-dimensional R−vector space, whose basis, denoted by {1 , i , j , k } , is a non-commutative algebra with the properties i 2 = j 2 = k 2 = −1 and k = i j = −j i . Like for the complex numbers a conjugation is defined by h = a + i b +√j c + k d = a−i b−j c−k d, and √ the absolute value is |h| = hh = |a + i b + j c + k d| = a2 + b2 + c2 + d2 . Thus, the monogenic signal maps the real-valued function f to the quaternion-valued function Mf. For a quaternion-valued function g, the amplitude and the phase are defined in analogy to the complex numbers. The amplitude is the (quaternionic) absolute value |g| and the phase is the angle between the imaginary and the real part g| tan−1 ( |Im Re g ). In particular, if g is a monogenic signal, then the amplitude |g| has the interpretation as the envelope of g. 3. THE MONOGENIC CURVELET TRANSFORM 3.1. The Usual Continuous Curvelet Transform We first recall the definition of the usual curvelet transform [1]. Let W be a compactly supported positive radial window function and V be a positive angular window function satisfying some admissibility conditions [1]. Let a ∈ R+ , b ∈ R2 , and θ ∈ [0, 2π) . Then a curvelet γabθ is defined by γabθ (x) = γa00 (ρθ (x − b)) where ρθ is a planar rotation by the angle θ and γa00 is defined by its Fourier transform ω γ ba00 (r, ω) = a3/4 W (ar)V √ . (1) a
1 0.5 0 −0.5
(a) From left to right: βa0θ = Re γa0θ , Hθ βa0θ = Im γa0θ , R1 βa0θ , and R2 βa0θ .
0.4 0.2 0 −0.2 −0.4
[ [ [ (b) From left to right: βba0θ , i H θ β a0θ , i R1 β a0θ , and i R2 β a0θ .
Fig. 1. Example of the filters of usual curvelets and monogenic curvelets for some anisotropic scale a < α0 and θ = π/8 in the time domain (a) and in the frequency domain (b). The angular windowing is well-defined only for scales a smaller than a fixed scale α0 [1]. Thus for the coarser scales a ≥ α0 the transform is continued by a purely radial window W (ar) γ ba00 (r, ω) = a √ . π
(2)
Note that γabθ is complex-valued for a < α0 and real-valued for a ≥ α0 . The (continuous) curvelet transform Γf of a function f is defined by ( R+ × R2 × [0, 2π) → C Γf : (3) (a, b, θ) 7→ hγabθ , f i . 3.2. Interpretation of Curvelets as Hilbert-Analytic Signals We show that γabθ is a Hilbert-analytic signal for a < α0 . To this end, we define the real-valued curvelet βa00 by symmetrizing γ ba00 with respect to the origin in the Fourier domain 1 βba00 (ξ1 , ξ2 ) := (b γa00 (ξ1 , ξ2 ) + γ ba00 (−ξ1 , −ξ2 )). 2 Now a simple calculation (omitting the subscripts)
3.3. Definition and Properties of the Monogenic Curvelet Transform We define the monogenic curvelet transform Mf by ( R+ × R2 × [0, 2π) → H Mf : (a, b, θ) 7→ hMβabθ , f i ,
(5)
where hMβabθ , f i = hβabθ , f i + i hR1 (βabθ ), f i + j hR2 (βabθ ), f i .
b 1 , ξ2 ) − i (i sign(ξ1 )β(ξ b 1 , ξ2 )) γ b(ξ1 , ξ2 ) = β(ξ b 1 , ξ2 ) − i H d = β(ξ 0 β(ξ1 , ξ2 ) yields γa00 = βa00 − i H0 βa00 . By the translation invariance of Hθ , i.e., (H0 βabθ )(x) = (H0 βa0θ )(x−b), and the rotation covariance of Hθ , i.e., (H0 βa00 )(ρθ x) = (H−θ βa0θ )(x), we get that γabθ is a Hilbert-analytic signal, so γabθ = Re γabθ + i Im γabθ = βabθ − i H−θ βabθ .
We will refer to the usual curvelet transform also as Hilbertanalytic curvelet transform. At this point we bring the monogenic signal into play. In section 2.3 we stated that the proper generalization of the analytic signal is not the Hilbert-analytic signal but the monogenic signal. This motivates to replace the Hilbertanalytic curvelets γabθ by new monogenic curvelets Mβabθ . This yields a new quaternion-valued transform, which we call monogenic curvelet transform.
(4)
The monogenic curvelet transform has a Calderón-like reproducing formula Z ∞ Z 2π Z da Mf (x) = hMβabθ , f i Mβabθ (x) db dθ 3 a 2 0 0 R and a Parseval formula Z ∞ Z 2π Z 2 kf k2 = 0
0
R2
2
|Mf (a, b, θ)| db dθ
da . a3
0.8
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.6 0.4 0.2 0 −0.2 −0.4
(a) From left to right: βa0θ = γa0θ , R1 βa0θ , and R2 βa0θ . 0.5
0
−0.5
[ [ (b) From left to right: βba0θ , i R 1 β a0θ , and i R 2 β a0θ .
Fig. 2. Filters of usual curvelets and monogenic curvelets for some isotropic scale a ≥ α0 in the time domain (a) and in the frequency domain (b). 3.4. Comparison Between the Hilbert-Analytic Curvelet Transform and the Monogenic Curvelet Transform A comparison of the complex Hilbert-analytic curvelet coefficients with the quaternionic monogenic curvelet coefficients requires to represent the complex numbers Γf (a, b, θ) as quaternions. To this end we embed the curvelet coefficients into the quaternions by an isometric and injective mapping ι defined by ι : Γf (a, b, θ) 7→ Re (Γf (a, b, θ)) + i cos(θ) Im (Γf (a, b, θ)) − j sin(θ) Im (Γf (a, b, θ)) . With the help of ι we get for all b and θ the inequality |ι(Γf (a, b, θ)) − Mf (a, b, θ)| ≤ a2 CW,V kf k2 with a constant CW,V depending only on the window functions W and V. Thus as a tends to 0, the Hilbert-analytic curvelet coefficients and the monogenic curvelet coefficients converge to each other uniformly in b and θ. At the coarse scales, in contrast, the transforms differ strongly. The concept of the Hilbert-analytic signal is not applicable to the isotropic scales, thus, γabθ remains a purely real-valued function for a ≥ α0 . Hence the amplitude |γabθ | boils down to the absolute value of the real numbers. This results in an oscillatory behaviour of |Γf (a, ·, θ)| (Fig. 3 (a)). The concept of the monogenic signal on the other hand can be applied to all scales, so Mβabθ is an analytic signal also at the coarse scales. Thus the amplitude |Mf (a, b, θ)| has the interpretation of an envelope of f and consequently does not oscillate (Fig. 3 (b)). The similarity to the monogenic wavelet transform is more obvious. Mβabθ is a monogenic wavelet for every a ≥ a0 . Thus for the choice α0 = 0, Mf simplifies to a monogenic wavelet transform as in [2] or [3]. (See Table 1).
(a) |Γδ (a, ·, 0)|
(b) |Mδ (a, ·, 0)|
Fig. 3. Amplitude responses of the Hilbert-analytic curvelet transform (a) and of the monogenic curvelets transform (b) for a ≥ α0 . δ denotes the Dirac distribution. The Hilbertanalytic amplitude oscillates in radial direction whereas the monogenic amplitude decays monotonously. 4. CONCLUSION We introduced a new transform which unifies the main advantages of the monogenic wavelet transform and of the curvelet transform. In particular, the monogenic curvelet coefficients split into meaningful amplitude and phase components over all scales. Furthermore, the anisotropy of the analyzing elements increases at the fine scales, which results in excellent directional selectivity. 5. REFERENCES [1] E. J. Candès and D. L. Donoho, “Continuous curvelet transform: I. Resolution of the wavefront set,” Appl. Comput. Harmon. Anal., vol. 19, pp. 162–197, 2003. [2] S. Held, M. Storath, P. Massopust, and B. Forster, “Steerable wavelet frames based on the Riesz transform,” IEEE Trans. Im. Proc., vol. 19, no. 3, pp. 653–667, March 2010. [3] M. Unser, D. Sage, and D. Van De Ville, “Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform,” IEEE Trans. Im. Proc., vol. 18, no. 11, pp. 2402–2418, November 2009. [4] S. C. Olhede and G. Metikas, “The monogenic wavelet transform,” IEEE Trans. Sig. Proc., vol. 57, no. 9, pp. 3426–3441, 2009. [5] M. Unser and D. Van De Ville, “Higher-order Riesz transforms and steerable wavelet frames,” in Proc. IEEE Int. Conf. Im. Proc., Cairo, 2009, pp. 3801–3804. [6] T. Bülow and G. Sommer, “Hypercomplex signals — a novel extension of the analytic signal to the multidimensional case,” IEEE Trans. Sig. Proc., vol. 49, pp. 2844– 2852, 2001. [7] M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Sig. Proc., vol. 49, no. 12, pp. 3136–3144, December 2001.
M6 - Mathematische Modellbildung, Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, D-85748 Garching bei München, Germany ABSTRACT
In this article, we reconsider the continuous curvelet transform from a signal processing point of view. We show that the analyzing elements of the curvelet transform, the curvelets, can be understood as analytic signals in the sense of the partial Hilbert transform. We then replace the usual curvelets by the monogenic curvelets, which are analytic signals in the sense of the Riesz transform. They yield a new transform, called the monogenic curvelet transform, which has the interesting property that it behaves at the fine scales like the usual curvelet transform and at the coarse scales like the monogenic wavelet transform. In particular, the new transform is highly anisotropic at the fine scales and yields a well-interpretable amplitude/phase decomposition of the transform coefficients over all scales. Index Terms— Curvelet transform, Analytic signal, Monogenic signal, Hilbert transform, Riesz transform 1. INTRODUCTION The continuous curvelet transform is a multiscale transform which allows to resolve the singularities of an image together with their orientations, e.g., the positions and the directions of edges [1]. To this end, the curvelet transform increases the anisotropy of its analyzing elements – the curvelets – as the scale decreases, thus the curvelets have higher directional selectivity at the fine scales. However, from a signal processing point of view, the curvelets have another nice property, which has not been exposed so far. That is, the curvelets are analytic signal filters in the sense of the partial Hilbert transform. Analytic signal filters are important in signal processing, because they yield a meaningful decomposition of the filtered signal into amplitude and phase, where the amplitude has the interpretation of the envelope of the signal. The analytic signal of one-dimensional functions is based on the Hilbert transform. The aforementioned two-dimensional generalization of the analytic signal by the partial Hilbert transform has several drawbacks, which we point out in section 2.2. We resolve these problems by switching to another generalization of the analytic signal, the monogenic signal. Thus we replace the usual curvelets by monogenic curvelets, which yield the new monogenic curvelet transform.
Several authors already use the monogenic signal in image processing. Especially the monogenic wavelet transform [2, 3, 4] opens many new applications like AM/FM analysis [3] or descreening [2]. The disadvantage of the monogenic wavelet transform is their poor directional selectivity. To gain higher anisotropy, the authors in [5] propose higher order Riesz transforms, which leads to a similar approach like the steerable pyramid. However, the degree of directional selectivity of that approach does not adapt to the scale, so the resolution of the orientations of the image singularities is still not optimal [1]. The monogenic curvelets, in contrast, adopt the scale-adaptive anisotropy of the usual curvelets, which results in a better resolution of the orientations. The article is organized as follows. First we give an introduction to the analytic signal and its generalizations to two dimensions, namely the Hilbert-analytic signal and the monogenic signal. We then identify the usual curvelets as Hilbert-analytic signal. From there we derive the monogenic curvelets and the new monogenic curvelet transform. Finally, we show that the monogenic curvelet transform behaves like the usual curvelet transform at the fine scales and like the monogenic wavelet transform at the coarse scales. 2. ANALYTIC SIGNAL CONCEPTS Throughout this article we use (r, ω) for polar coordinates in the frequency domain, and x = (x1 , x2 ) and ξ = (ξ1 , ξ2 ) for Cartesian coordinates in the spatial domain and in the frequency domain, respectively. Further, f is always a realvalued and square-integrable function. 2.1. The Analytic Signal in 1D Let H : L2 (R, R) → L2 (R, R) be the Hilbert transform defined in the Fourier domain by d(s) = i sign (s) fb(s). Hf Note that Hf is also real-valued. A complex-valued function g : R → C whose imaginary part is the Hilbert transform of its real part, i.e., Im g = −H(Re g), is called analytic signal. In 1D signal processing the analytic signal is used to decompose a signal into amplitude and phase, so loosely spoken into a signal intensity and a signal structure.
Monogenic wavelets [2]
Curvelets [1]
Monogenic curvelets
Coarse scales
Directionality
Low
Low
Low
(a ≥ α0 )
Analytic signal concept
Monogenic signal
No analytic signal concept
Monogenic signal
Fine scales
Directionality
Low
High (scale-adaptive)
High (scale-adaptive)
(a < α0 )
Analytic signal concept
Monogenic signal
Hilbert-analytic signal
Monogenic signal
Table 1. Comparison between the monogenic wavelets, the usual curvelets, and the monogenic curvelets. 2.2. The Hilbert-Analytic Signal A straightforward extension to 2D is achieved by letting the Hilbert transform operate on parallel lines which point towards a fixed orientation θ. This extension is called partial Hilbert transform with respect to the angle θ [6], defined in the Fourier domain for f ∈ L2 (R2 , R) by d b H θ f (ξ) = i sign (cos(θ)ξ1 + sin(θ)ξ2 ) f (ξ). We call a complex-valued function g : R2 → C Hilbertanalytic signal, if its imaginary part is the partial Hilbert transform with respect to any angle θ of its real part, i.e., Im g = −Hθ (Re g). The Hilbert-analytic signal has two major drawbacks. The first one is that the partial Hilbert transform has a purely one-dimensional nature, thus, the Hilbert-analytic signal does not take into account sufficiently the structure of true twodimensional signals. The second problem is the dependence on the angle θ. Though there are functions f which only allow one sensible choice for θ, in general, Hθ yields a different analytic signal of f for every different θ ∈ [0, 2π) . 2.3. The Monogenic Signal In order to overcome the aforementioned problems, a different generalization of the analytic signal for 2D was introduced in [7], which is called the monogenic signal. It is considered to be the proper generalization of the 1D-analytic signal for image processing [7] and has proved its utility in several image processing applications [2, 3, 5]. The monogenic signal is based on the Riesz transform. For f ∈ L2 R2 , R the Riesz transform with respect to the axis xν , ν = 1, 2, is defined by ξν b d R f (ξ). ν f (ξ) = i |ξ| Note that Rν f is a real-valued function. Unlike the partial Hilbert transform, the Riesz transform does not depend on any orientation θ. Instead we have two transforms R1 f and R2 f with respect to the fixed coordinate axes x1 and x2 . Consequently, a sensible representation of the monogenic signal needs two imaginary parts, one for R1 f and one for R2 f. Hence we have to switch from the complex numbers to a hypercomplex algebra which possesses at least two imaginary
units. Assume for the moment that we have such an algebra with the two imaginary units i an j . Then the monogenic signal Mf is defined by Mf := f − i R1 f − j R2 f. The most obvious choice for the desired hypercomplex algebra are the quaternions H := {h = a + i b + j c + k d : a, b, c, d ∈ R} , which are an extension of the complex numbers with the three imaginary units i , j , and k . In short, the quaternions are a 4-dimensional R−vector space, whose basis, denoted by {1 , i , j , k } , is a non-commutative algebra with the properties i 2 = j 2 = k 2 = −1 and k = i j = −j i . Like for the complex numbers a conjugation is defined by h = a + i b +√j c + k d = a−i b−j c−k d, and √ the absolute value is |h| = hh = |a + i b + j c + k d| = a2 + b2 + c2 + d2 . Thus, the monogenic signal maps the real-valued function f to the quaternion-valued function Mf. For a quaternion-valued function g, the amplitude and the phase are defined in analogy to the complex numbers. The amplitude is the (quaternionic) absolute value |g| and the phase is the angle between the imaginary and the real part g| tan−1 ( |Im Re g ). In particular, if g is a monogenic signal, then the amplitude |g| has the interpretation as the envelope of g. 3. THE MONOGENIC CURVELET TRANSFORM 3.1. The Usual Continuous Curvelet Transform We first recall the definition of the usual curvelet transform [1]. Let W be a compactly supported positive radial window function and V be a positive angular window function satisfying some admissibility conditions [1]. Let a ∈ R+ , b ∈ R2 , and θ ∈ [0, 2π) . Then a curvelet γabθ is defined by γabθ (x) = γa00 (ρθ (x − b)) where ρθ is a planar rotation by the angle θ and γa00 is defined by its Fourier transform ω γ ba00 (r, ω) = a3/4 W (ar)V √ . (1) a
1 0.5 0 −0.5
(a) From left to right: βa0θ = Re γa0θ , Hθ βa0θ = Im γa0θ , R1 βa0θ , and R2 βa0θ .
0.4 0.2 0 −0.2 −0.4
[ [ [ (b) From left to right: βba0θ , i H θ β a0θ , i R1 β a0θ , and i R2 β a0θ .
Fig. 1. Example of the filters of usual curvelets and monogenic curvelets for some anisotropic scale a < α0 and θ = π/8 in the time domain (a) and in the frequency domain (b). The angular windowing is well-defined only for scales a smaller than a fixed scale α0 [1]. Thus for the coarser scales a ≥ α0 the transform is continued by a purely radial window W (ar) γ ba00 (r, ω) = a √ . π
(2)
Note that γabθ is complex-valued for a < α0 and real-valued for a ≥ α0 . The (continuous) curvelet transform Γf of a function f is defined by ( R+ × R2 × [0, 2π) → C Γf : (3) (a, b, θ) 7→ hγabθ , f i . 3.2. Interpretation of Curvelets as Hilbert-Analytic Signals We show that γabθ is a Hilbert-analytic signal for a < α0 . To this end, we define the real-valued curvelet βa00 by symmetrizing γ ba00 with respect to the origin in the Fourier domain 1 βba00 (ξ1 , ξ2 ) := (b γa00 (ξ1 , ξ2 ) + γ ba00 (−ξ1 , −ξ2 )). 2 Now a simple calculation (omitting the subscripts)
3.3. Definition and Properties of the Monogenic Curvelet Transform We define the monogenic curvelet transform Mf by ( R+ × R2 × [0, 2π) → H Mf : (a, b, θ) 7→ hMβabθ , f i ,
(5)
where hMβabθ , f i = hβabθ , f i + i hR1 (βabθ ), f i + j hR2 (βabθ ), f i .
b 1 , ξ2 ) − i (i sign(ξ1 )β(ξ b 1 , ξ2 )) γ b(ξ1 , ξ2 ) = β(ξ b 1 , ξ2 ) − i H d = β(ξ 0 β(ξ1 , ξ2 ) yields γa00 = βa00 − i H0 βa00 . By the translation invariance of Hθ , i.e., (H0 βabθ )(x) = (H0 βa0θ )(x−b), and the rotation covariance of Hθ , i.e., (H0 βa00 )(ρθ x) = (H−θ βa0θ )(x), we get that γabθ is a Hilbert-analytic signal, so γabθ = Re γabθ + i Im γabθ = βabθ − i H−θ βabθ .
We will refer to the usual curvelet transform also as Hilbertanalytic curvelet transform. At this point we bring the monogenic signal into play. In section 2.3 we stated that the proper generalization of the analytic signal is not the Hilbert-analytic signal but the monogenic signal. This motivates to replace the Hilbertanalytic curvelets γabθ by new monogenic curvelets Mβabθ . This yields a new quaternion-valued transform, which we call monogenic curvelet transform.
(4)
The monogenic curvelet transform has a Calderón-like reproducing formula Z ∞ Z 2π Z da Mf (x) = hMβabθ , f i Mβabθ (x) db dθ 3 a 2 0 0 R and a Parseval formula Z ∞ Z 2π Z 2 kf k2 = 0
0
R2
2
|Mf (a, b, θ)| db dθ
da . a3
0.8
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.6 0.4 0.2 0 −0.2 −0.4
(a) From left to right: βa0θ = γa0θ , R1 βa0θ , and R2 βa0θ . 0.5
0
−0.5
[ [ (b) From left to right: βba0θ , i R 1 β a0θ , and i R 2 β a0θ .
Fig. 2. Filters of usual curvelets and monogenic curvelets for some isotropic scale a ≥ α0 in the time domain (a) and in the frequency domain (b). 3.4. Comparison Between the Hilbert-Analytic Curvelet Transform and the Monogenic Curvelet Transform A comparison of the complex Hilbert-analytic curvelet coefficients with the quaternionic monogenic curvelet coefficients requires to represent the complex numbers Γf (a, b, θ) as quaternions. To this end we embed the curvelet coefficients into the quaternions by an isometric and injective mapping ι defined by ι : Γf (a, b, θ) 7→ Re (Γf (a, b, θ)) + i cos(θ) Im (Γf (a, b, θ)) − j sin(θ) Im (Γf (a, b, θ)) . With the help of ι we get for all b and θ the inequality |ι(Γf (a, b, θ)) − Mf (a, b, θ)| ≤ a2 CW,V kf k2 with a constant CW,V depending only on the window functions W and V. Thus as a tends to 0, the Hilbert-analytic curvelet coefficients and the monogenic curvelet coefficients converge to each other uniformly in b and θ. At the coarse scales, in contrast, the transforms differ strongly. The concept of the Hilbert-analytic signal is not applicable to the isotropic scales, thus, γabθ remains a purely real-valued function for a ≥ α0 . Hence the amplitude |γabθ | boils down to the absolute value of the real numbers. This results in an oscillatory behaviour of |Γf (a, ·, θ)| (Fig. 3 (a)). The concept of the monogenic signal on the other hand can be applied to all scales, so Mβabθ is an analytic signal also at the coarse scales. Thus the amplitude |Mf (a, b, θ)| has the interpretation of an envelope of f and consequently does not oscillate (Fig. 3 (b)). The similarity to the monogenic wavelet transform is more obvious. Mβabθ is a monogenic wavelet for every a ≥ a0 . Thus for the choice α0 = 0, Mf simplifies to a monogenic wavelet transform as in [2] or [3]. (See Table 1).
(a) |Γδ (a, ·, 0)|
(b) |Mδ (a, ·, 0)|
Fig. 3. Amplitude responses of the Hilbert-analytic curvelet transform (a) and of the monogenic curvelets transform (b) for a ≥ α0 . δ denotes the Dirac distribution. The Hilbertanalytic amplitude oscillates in radial direction whereas the monogenic amplitude decays monotonously. 4. CONCLUSION We introduced a new transform which unifies the main advantages of the monogenic wavelet transform and of the curvelet transform. In particular, the monogenic curvelet coefficients split into meaningful amplitude and phase components over all scales. Furthermore, the anisotropy of the analyzing elements increases at the fine scales, which results in excellent directional selectivity. 5. REFERENCES [1] E. J. Candès and D. L. Donoho, “Continuous curvelet transform: I. Resolution of the wavefront set,” Appl. Comput. Harmon. Anal., vol. 19, pp. 162–197, 2003. [2] S. Held, M. Storath, P. Massopust, and B. Forster, “Steerable wavelet frames based on the Riesz transform,” IEEE Trans. Im. Proc., vol. 19, no. 3, pp. 653–667, March 2010. [3] M. Unser, D. Sage, and D. Van De Ville, “Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform,” IEEE Trans. Im. Proc., vol. 18, no. 11, pp. 2402–2418, November 2009. [4] S. C. Olhede and G. Metikas, “The monogenic wavelet transform,” IEEE Trans. Sig. Proc., vol. 57, no. 9, pp. 3426–3441, 2009. [5] M. Unser and D. Van De Ville, “Higher-order Riesz transforms and steerable wavelet frames,” in Proc. IEEE Int. Conf. Im. Proc., Cairo, 2009, pp. 3801–3804. [6] T. Bülow and G. Sommer, “Hypercomplex signals — a novel extension of the analytic signal to the multidimensional case,” IEEE Trans. Sig. Proc., vol. 49, pp. 2844– 2852, 2001. [7] M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Sig. Proc., vol. 49, no. 12, pp. 3136–3144, December 2001.
