the construction of two dimensional hilbert huang …...the construction of two dimensional hilbert...

The construction of two dimensional Hilbert Huang transform and its application in image analysis

1 Lihong Qiao, 2Sisi Chen

*1, Henan univ. of technology,[email protected] 2, XingTai University, [email protected]

Abstract Hilbert Huang Transform is a new developed method for signal processing especially suitable for

non-stationary signal processing. In this paper, we propose a two dimensional Hilbert-Huang Transform based on Bidimensional Empirical Mode Decomposition (BEMD) and quaternionic analytic signal. Bidimensional Empirical Mode Decomposition is adaptive signal decomposition method and its decomposition results are almost monocomponent. Quaternionic analytic signal satisfies most of the two dimensional extension properties and is especially suitable for the monocomponent. In detail, the image is first decomposed to several comoponents by Bidimensional Empirical Mode Decomposition. Then by the Quaternionic analytic method, we get the two dimensional Quaternionic analytic signal. Two dimensional Hilbert spectral characters are got include the instantaneous amplitude, the instantaneous phase, and the instantaneous frequencies. We illustrate the techniques on natural images, and demonstrate the estimated instantaneous frequencies using the needle program. These features inflect the intrinsic characters of image and form the basis for a general theory of image processing.

Keywords: Two dimensional Hilbert Huang Transform, quaternionic analytic signal, two dimensional Hilbert spectral character,

1. Introduction

Image analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques. It has many applications in different areas[1-2].Traditionally, Fourier transform can only give meaningful interpretation to linear and stationary processes. However, the real world is neither linear nor stationary, thus we need some new analysis method.

In 1990s, Huang’s group developed a new adaptive time-frequency data analysis method, namely Hilbert-Huang Transform (HHT) [3] at NASA. The development of the Hilbert-Huang Transform has shed new light on spectral estimation for non-stationary signals. Comparisons with traditional Fourier transform and wavelet analysis method, Hilbert-Huang Transform is an adaptive analysis method and is especially suitable for non-linear signals. Hilbert-Huang Transform consists of two parts: Empirical Mode Decomposition (EMD) and the Hilbert spectral analysis (HSA). Based on EMD, any time series data can be decomposed to a finite number of intrinsic modes of oscillations. EMD identifies the intrinsic undulations at different time scales and sifts the so-called intrinsic mode functions (IMFs) out. The IMFs yield instantaneous frequencies as functions of time and give sharp identications of imbedded structures. Then with the Hilbert transform, several important Hilbert spectrum characters are included, such as: the instantaneous amplitude, the instantaneous phase, and the instantaneous frequency, which reflect the intrinsic and instantaneous characters of the signal. These Hilbert spectral characters analysis method offers much better temporal and frequency resolutions. In the past decade, it has been utilized in more than 3000 published works and has been applied in various research fields [4].

Recently people pay more attention to the two dimensional Hilbert-Huang Transform. This method is important for the analysis of the two dimensional nonlinear signals. A possible 2D extension of EMD is the so-called Bidimensional Empirical Mode Decomposition (BEMD).Details of the method are fully available in essay[5,6].Between them, Bidimensional Empirical Mode Decomposition is a suitable method[5].It can decompose image adaptively

The construction of two dimensional Hilbert Huang transform and its application in image analysis Lihong Qiao, Sisi Chen

International Journal of Digital Content Technology and its Applications(JDCTA) Volume6,Number9,May 2012 doi:10.4156/jdcta.vol6.issue9.30

236

according to the intrinsic mode of the signal rather than the fixed mode of the filter. In this essay, we use this method. As for the second part of HHT: the two dimensional Hilbert spectral analysis is a tough problem. First of all, we have to construct the two dimensional analytic method. The two dimensional analytic method are based on the extension of the notion of 1D analytic signal to 2D, or simply a Hilbert-based extension of the 1D Hilbert-based demodulation approach. Many two dimensional Hilbert transform have been proposed, such as the total Hilbert transform [7], the partial Hilbert transform, and the single-orthant Hilbert transform and the quaternionic Hilbert transform[8,9]. However, there are some two dimensional Hilbert transform properties that the first three Hilbert transform method don’t satisfy. Directional EMD and the partial Hilbert transform considering directions [10] were used to obtain the corresponding complex signals. The instantaneous frequencies were computed along the fixed direction. This method is suitable for the texture which has main directions and it is hardly perfectly reconstructed from the modulation results. The partial Hilbert transform make Hilbert transform in x and y direction separately. P.C. Toy once used the partial Hilbert transform and two dimensional EMD [11], but decomposition method has some problems. Bulow once used quaternionic Hilbert transform, but he represented the signal by three phases, which lead to six frequencies, hard to represent. [12-15] Bi-orthant analytic signal was proposed for two dimensional signals, which kept many good properties of two dimensional analytic signals. [16]Besides, Felsberg M. and Sommer G. introduced the monogenic signal [17-19], which is a three-dimensional representation method and leads to two phases. The monogenic signal is an efficient extension of the analytic signal to images, which is not suitable for frequency representing.

In this paper, we illustrate the problem of two dimensional Hilbert Huang transform method, and make some experiment in image analysis. First of all, Hilbert-Huang transform is introduced. This is an adaptive analysis method, especially useful for nonlinear signal. Besides, we introduce the Bidimensional Empirical Mode Decomposition, which is an adaptive decomposition method showing the visually significant structure of the image. Then by the quaternionic analytic method, suitable analytic signal is acquired. Quaternionic analytic method is a new proposed method, which keeps many good properties of 2D analytic signal. In section 3, based on quaternionic analytic method, two dimensional Hilbert spectral characters analysis method are proposed. The experimental results are listed in section 4. Section 5 is the conclusion. 2. Hilbert-Huang transform

Hilbert-Huang transform (HHT) provides a intrinsic analysis method for non-linear and non-stationary signals. Empirical Mode Decomposition (EMD) decomposes a signal into intrinsic mode functions (IMFs). IMFs are mono-components of the signal. By summing all the IMFs, the original signal can be perfectly reconstructed.

Secondly, then Hilbert transform is applied to the IMFs and get the Hilbert spectral characters of each IMFs’. For any function ( )x t of pL class, its Hilbert transform ( )y t is

1 ( )( )

xy t P d

t

, (1)

where P is the Cauchy principal value of the singular integral.

Let ( )x t is the Hilbert transform ( )y t ,we obtain the analytic function,

( )( ) ( ) ( ) ( ) ,i tz t x t y t i a t e (2)

where 1i ，12 2 2( ) ( ( ) ( ) )a t x t y t , 1 ( )

( ) tan( )

y tt

x t .Here ( )a t is the instantaneous amplitude,

and ( )t is the instantaneous phase function. The instantaneous frequency is:


237

( )( )

d tt

dt

. (3)

The final presentation is an energy-frequency-time distribution, which is designated as the Hilbert spectrum. This method provides many intrinsic Hilbert spectrum characters of signal such as: the instantaneous amplitude, the instantaneous phase, and the instantaneous frequencies. This method is especially suitable for the analysis of 1D-non-linear and non-stationary data. This new time-frequency distribution is shown in essay [20, 21]

2. Two dimensional Hilbert-Huang transform 2.1 Bidimensional empirical mode decomposition

Many decomposition methods are proposed for two dimensional signals. Bidimensional empirical mode decomposition (BEMD) [5] extracts IMFs by computing envelopes using radial basis functions interpolation. While Directional Empirical Mode Decomposition (DEMD) defines directional frequency and envelopes as features by 2D, which is combine of two 1D decomposition method.

In this essay, we use Bidimensional Empirical Mode Decomposition in essay [5]. This process is a 2D-sifting process. The modified 2D-sifting process is performed in two steps: extremas detection by neighboring window or morphological operators and surface interpolation by fast radial basis functions The image “lena” is decomposed to several components demonstrated in Figure 1. There are seven IMFs and one residual. This Empirical Mode Decomposition (EMD) is a fully data-driven method and using no predetermined filter.


238

Figure 1. Decomposition results of Lena using BEMD

2.2. Two dimensional Hilbert Transform based on quaternionic analytic method

We can further analyze each IMF by two dimensional Hilbert spectral analyses and get the intrinsic characters of the image. Definitely, it depends on suitable two dimensional Hilbert transform method. We’ll further introduce some two dimensional Hilbert transforms and introduce the properties of 2D analytic signals. Between them, the Quaternionic analytic method is presented.

Firstly, we have to define two dimensional analytic signals so as to get the two dimensional Hilbert spectral characters using analytic method. Some guidelines are needed for the extension of two dimensional analytic signals. Following lists the main properties of the analytic signal in 1D. Any new two dimensional analytic signals should be measured according to the degree to which it extends these properties to higher dimensions. [14]

• The spectrum of an analytic signal is right-sided ( ( , ) 0AF u v , for 0u )

• Hilbert pairs are orthogonal. • The real part of the analytic signal Af is equal to the original signal f .

• The analytic signal is compatible with the associated harmonic transform. In case of the one-dimensional Fourier transform, the property of the analytic signal is called

compatible with the associated harmonic transform, since the real part of the Fourier kernel, i.e. (exp( 2 )) cos( 2 )i ux i ux is the Hilbert transform of sin( 2 )i ux . Two dimensional analytic

signal should satisfy this property, that is the analytic signal is compatible with the associated harmonic transform with transformation kernel K if K and IK are a Hilbert pair.

Two dimensional analytic signals depend on the development of two dimensional Hilbert transform. Different nD Hilbert transform have been proposed in the past, which include: the the partial Hilbert transform, total Hilbert transform and the total Hilbert transform and the quaternionic Hilbert transform. There are some properties that the first three methods don’t satisfy. However, the quaternionic Hilbert transform overcomes the remaining problems. It is compatible with the associated harmonic transform, but it always leads to several useless elements when using it in two dimensional Hilbert spectral character research.

The quaternionic analytic signal can overcome the remaining problems. It is compatible with the associated harmonic transform. [9]Before using it, the basic knowledge of quatenion is introduced.

A quaternion may be represented in hypercomplex form as q a bi cj dk , (4)

where a , b , c and d are real ; i , j , and k are complex operators, which obey the following rules

2 2 1,i j ij ji k (5)

This implies that 2 1, , , , .k jk i kj i ki j ik j (6)

A quaternion has a real part a and an imaginary part.


239

The latter has three components and can be used as a vector quantity, often denoted by. ( )V q bi cj dk .

For this reason, the real part is sometimes referred to as the scalar part of the quaternion ( )S q and the

whole quaternion may be represented by the sum of its scalar and vector parts as ( ) ( )q S q V q . A quaternion with a zero real or scalar part is called a pure quaternion.

The modulus and conjugate of a quaternion follow the definitions for complex numbers

2 2 2 2q a b c d (7)

.q a bi cj dk

(8) A quaternion with a unit modulus is called a unit quaternion. The conjugate of a quaternion is obtained, like the complex conjugate, by negating the imaginary or vector part.

Euler’s formula for the complex exponential generalizes to hypercomplex form

cos sinue u (9)

where

( )

( )

V qu

V q

is a unit pure quaternion.

Any quaternion may be represented in polar form asuq q e

, where u and are referred to as the eigenaxis and eigenangle of the quaternion, respectively. u identifies the direction in 3-space of the

vector part and may be regarded as a true generalization of the complex operator i ,

since2 1u .

( )arctan

( )

V q

S q

is analogous to the argument of a complex number, but is unique

only in the range[0, ] because a value greater than the range can be reduced to this range by negating or reversing the eigenaxis (its sense in 3-space). The approximate representation form in quaternion is quaternionic analytic signal.

Let f is a real two-dimensional signal andqF its quaternionic Fourier transform. In the

quaternionic frequency domain we can define the quaternionic analytic signal of a real signal as

( ) (1 ( ))(1 ( )) ( ),q qAF sign u sign v F

where ( , )x y and ( , )u v .This can be expressed in the spatial domain as follows:

( ) ( ) ( ),q TA H if f n f

(10)

where ( , , )T Tn i j k and H ifis a vector which consists of the total and the partial Hilbert transforms

of f :

1

2

( )

( ) ( )

( )

H i

H i H i

H i

f

f f

f

(11)

Quaternionic analytic signal fulfills most of the desired properties and allows generalizing the instantaneous phase. In essay [14] instantaneous phase was defined as the triple of phase angles of the quaternionic value of the quaternionic analytic signal at each position. Then there would be six instantaneous frequencies, which is not easy to present. In this part, we choose different represent form. To present the amplitude and the phase of the image, the polar form is used. The polar form is used like

thisue

, where

( )

( )

V qu

V q

,

( )arctan

( )

V q

S q

.


240

Similarly, the analytic signal ( )q

Af can be written as

( )qAf Ae

(12) where the amplitude is represented as

1 2

2 2 2 2 ( ) ( ) ( ) ( )H i H i H iA f f f f

(13)

1 2

1 2

( ) ( ) ( )

( ) ( ) ( )

H i H i H i

H i H i H i

f i f j f ku

f i f j f k

(14)

1 2 ( ) ( ) ( )arctan

( )

H i H i H if i f j f k

f

(15)

where u identifies the direction in 3-space of the vector part and is the phase. Then instantaneous frequencies are

1 ( 1, ) ( , )diff x y x y 2 ( , 1) ( , )diff x y x y (16) which represent the frequency in horizontal and vertical directions separately. We can combine them together to present the instantaneous frequency. Then there are seven inherent characters: amplitude, phase, two instantaneous frequency and three components of u , which give more characters of image. 4. Experiments

In this section, we apply the image modulation method to both synthetic and natural images. In detail, we first decompose the image to several IMFs using BEMD. The IMFs are the mono-components of the image. Then we apply the Quaternionic analytic signal to modulate the image and get the two dimensional Hilbert spectral characters of image, such as: instantaneous amplitude, instantaneous frequencies. In our experiment, we just analyze the first IMF of the image by the new method; the other components can be analyzed similarly.

First of all, this new modulation method was applied to image, as well as real-world texture image. In detail, we first decompose the image by BEMD to several IMFs, which are the monocomponents of the image. Then we apply the quaternionic analytic method to modulate the image. In our experiment, we just analyze the first IMF of the image by this new method , the other components can be analyzed similarly.

From Figure 2(a), we can see the original image “lena”, the first IMF of this image is shown in Figure 2(b). Besides, we calculate the reconstruction image Figure 2(j), and it is equal to the original image Figure 2(a). Using the two dimensional Hilbert spectral analysis method, we get an instantaneous amplitude, an instantaneous phase, two instantaneous frequencies and three components of u shown in Figures 2(c)-2(f). The instantaneous amplitude provide a dense local characterization of the local texture contrast as is shown in Figure 2(c).Besides, the local phase provides a dense characterization of the local texture orientation and pattern spacing, as is depicted in Figure 2(d). We put three component of u in RGB band to compose a color image, given in Figure 2(a).


241

(a) Lena (b) IMF1of Lena

(c) amplitude (d) phase

(e) diff1 (f) diff2

(g) needle diagram of instantaneous frequency (h) 1u

(i) 2u (j) 3u


242

(k) u (l) Reconstruction image

Figure 2. Two dimensional Hilbert spectral characters of the IMF1 of image “Lena”

Figure 2(g) shows the needle program of the instantaneous frequency (IF) vector of Figure 2 (b). In the picture, the arrows point in the direction of the instantaneous frequency vector and the length of each needle are proportional to the instantaneous frequency vector's modulus. For clarity, only one needle is shown for each block of 5 5 pixels. This needle diagram reflects the intrinsic characters of image. First, needles are clearly visible in all four quadrants. However, the direction instantaneous frequency estimations using the partial Hilbert transform point to certain directions. Second, as shown in Figure 2(g), much of the salient structure of the image have been interpreted by the analytic image. We can make meaningful associations between individual needles diagram of instantaneous frequencies in Figure 2 (b) and specific features of the image. The needles are longer in the texture part of the image, corresponding to frequency estimations with larger magnitudes, just consistent with the understanding. What’s more, the vector is smaller in the smoother part of the image, As is shown in Figure 2 (g)’s left part,the instantaneous frequency vector also describe the complexity part of Figure 2 (b). These characters reflect the special visual meaning of the image and it is a fully data-driven method and shows the intrinsic characters of image.

5. Conclusions

This paper examines the issue of two dimensional Hilbert-Huang Transform.Two dimensional Hilbert-Huang Transform based on Bidimensional Empirical Mode Decomposition (BEMD) and quaternionic analytic signa are proposed in the essayl. Bidimensional Empirical Mode Decomposition is adaptive signal decomposition method and its suitable for non-stationary signals. Quaternionic analytic signal satisfies most of the two dimensional analytic method. Two dimensional Hilbert spectral characters are got include the instantaneous amplitude, the instantaneous phase, and the instantaneous frequencies. We illustrate the techniques on natural images, and demonstrate the estimated instantaneous frequencies using the needle program. These features offer a new and promising way to analyze images and may use in character extracting, image searching and related fields.

6. References [1] Liu Weihua, "Remote On-Line Supervisory System of Micro Particle Using Image Analysis",

JCIT, Vol. 6, No. 5, pp. 44-50, 2011. [2] Yan Liping, Song Kai, Wang Chao, "Design and Realization of Forest Fire Monitoring System

Based on Image Analysis", JDCTA, Vol. 5, No. 12, pp. 452-458, 2011. [3] N. E. Huang et al., “The empirical mode decomposition and the Hilbert spectrum for non-linear

and non stationary time series analysis”, Proc Royal Soc. London A., vol. 454A: 903-995, 1998. [4] E. Huang, C.C. Chern, et al., “A New Spectral Representation of Earthquake Data: Hilbert

Spectral Analysis of Station TCU129, Chi-Chi, Taiwan, 21 September 1999," Bulletin of The Seismological Society of America, vol. 91, pp. 1310-1338, 2001.

[5] Nunes J C ,et al. “Image analysis by bidimensional empirical mode decomposition”, Image and Vision Computing, vol. 21, no. 12, pp. 1019-1026, 2003.

[6] Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD” , IEEE Signal


243

Processing Letters, vol. 12, no. 10, pp. 701-704, 2005. [7] H. Stark, “An extension of the Hilbert transform product theorem,” Proc. IEEE, vol. 59, pp. 1359-

1360, 1971. [8] Hahn S L .The Analytic, Quaternionic and Monogenic 2D Complex Delta Distributions. Technical

Report 3, Inst. Of Radio electronics, Nowowiejska, Warsaw Univ. of Technology, 2002 [9] T. A. Ell, S. J. Sangwine. “Hypercomplex Fourier transforms of color images”, IEEE Trans Image

Process. vol.16, no.1, pp.22-35, 2007. [10] Zhongxuan Liu, Silong Peng. “Texture segmentation using directional empirical mode

decomposition, (in Chinese) Proceedings of the Pattern Recognition, vol.4, no.3, pp.803-806,2004. [11] P. C. Toy. “AM-FM Image Analysis Using the Hilbert Huang Transform”, IEEE Southwest

Symposium on Image Analysis and Interpretation, pp.13-16, 2008. [12] T. Bulow. “Hypercomplex Spectral Signal Representations for the Processing and Analysis of

Images”, Ph.D. thesis, Christian Albrechts University, 1999. [13] T. Bulow, G. Sommer,“Hypercomplex Signals-A Novel Extension of the Analytic Signal to the

Multidimensional Case”, IEEE Trans. Signal Processing, vol.49, no.11, pp. 2844-2852, 2001. [14] T. Bulow, G. Sommer, “A novel approach to the 2nd analytic signal”, Proc.CAIP’99, pp.25-32,

1999. [15] T. Bulow, G. Sommer. “Multi-dimensional signal processing using an algebraically extended

signal representation”, Proc. Int. Workshop on Algebraic Frames for the Perception-Action Cycle, Kiel, LNCS, pp.148-163,1997.

[16] Guanlei Xu, Xiaotong Wang, Xiaogang Xu, “Bi-orthant Hilbert transform”, Progress in Natural Science,vol.17, no.8, pp. 1120-1129,2007.

[17] J. P. Havlicek, “AM-FM image models”, Ph.D. thesis, The University of Texas at Austin, 1996. [18] M. Felsberg , G. Sommer, “The Monogenic signal”, IEEE Trans Signal Processing, vol.49, no.17,

pp. 3136-3144, 2001. [19] S. L. Hahn, “The Analytic, Quaternionic and monogenic 2D Complex delta distributions”,

Technical Report 3, Inst. Of Radio electronics, Nowowiejska, Warsaw Univ. of Technology, 2002. [20] X. Chen, Z. Wu, and N.E. Huang, “The Time-Dependent Intrinsic Correlation Based on the

Empirical Mode Decomposition", Advances in Adaptive Data Analysis, Vol. 2, No. 2, pp.233-265, 2010.

[21] Stevenson, N.J., Mesbah, M., Boashash, B., “Multiple-view time-frequency distribution based on the empirical mode decomposition, Signal Processing, IET ,Vol. 4, No. 4, pp.447- 456, 2010.


244

the construction of two dimensional hilbert huang …...the construction of two dimensional hilbert...

Documents