The construction o two dimensional Hilbert Huang transorm and its application in image analysis 1 Lihong Qiao, Sisi Chen *1, Henan univ. o technology,iaolihong@16.com, XingTai University, chssxt@163.com Abstract Hilbert Huang Transorm is a new developed method or signal processing especially suitable or non-stationary signal processing. In this paper, we propose a two dimensional Hilbert-Huang Transorm based on Bidimensional Empirical Mode Decomposition (BEMD) and uaternionic analytic signal. Bidimensional Empirical Mode Decomposition is adaptive signal decomposition method and its decomposition results are almost monocomponent. Quaternionic analytic signal satisies most o the two dimensional extension properties and is especially suitable or the monocomponent. In detail, the image is irst 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 reuencies. We illustrate the techniues on natural images, and demonstrate the estimated instantaneous reuencies using the needle program. These eatures inlect the intrinsic characters o image and orm the basis or a general theory o image processing. Keywords: Two dimensional Hilbert Huang Transorm, uaternionic analytic signal, two dimensional Hilbert spectral character, 1. Introduction Image analysis is the extraction o meaningul inormation rom images; mainly rom digital images by means o digital image processing techniues. It has many applications in dierent areas[1-].traditionally, Fourier transorm can only give meaningul 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-reuency data analysis method, namely Hilbert-Huang Transorm (HHT) [3] at NASA. The development o the Hilbert-Huang Transorm has shed new light on spectral estimation or non-stationary signals. Comparisons with traditional Fourier transorm and wavelet analysis method, Hilbert-Huang Transorm is an adaptive analysis method and is especially suitable or non-linear signals. Hilbert-Huang Transorm consists o two parts: Empirical Mode Decomposition (EMD) and the Hilbert spectral analysis (HSA). Based on EMD, any time series data can be decomposed to a inite number o intrinsic modes o oscillations. EMD identiies the intrinsic undulations at dierent time scales and sits the so-called intrinsic mode unctions (IMFs) out. The IMFs yield instantaneous reuencies as unctions o time and give sharp identications o imbedded structures. Then with the Hilbert transorm, several important Hilbert spectrum characters are included, such as: the instantaneous amplitude, the instantaneous phase, and the instantaneous reuency, which relect the intrinsic and instantaneous characters o the signal. These Hilbert spectral characters analysis method oers much better temporal and reuency resolutions. In the past decade, it has been utilized in more than 3000 published works and has been applied in various research ields [4]. Recently people pay more attention to the two dimensional Hilbert-Huang Transorm. This method is important or the analysis o the two dimensional nonlinear signals. A possible D extension o EMD is the so-called Bidimensional Empirical Mode Decomposition (BEMD).Details o the method are ully available in essay[5,6].between them, Bidimensional Empirical Mode Decomposition is a suitable method[5].it can decompose image adaptively International Journal o Digital Content Technology and its Applications(JDCTA) Volume6,Number9,May 01 doi:10.4156/jdcta.vol6.issue9.30 36
according to the intrinsic mode o the signal rather than the ixed mode o the ilter. In this essay, we use this method. As or the second part o HHT: the two dimensional Hilbert spectral analysis is a tough problem. First o all, we have to construct the two dimensional analytic method. The two dimensional analytic method are based on the extension o the notion o 1D analytic signal to D, or simply a Hilbert-based extension o the 1D Hilbert-based demodulation approach. Many two dimensional Hilbert transorm have been proposed, such as the total Hilbert transorm [7], the partial Hilbert transorm, and the single-orthant Hilbert transorm and the uaternionic Hilbert transorm[8,9]. However, there are some two dimensional Hilbert transorm properties that the irst three Hilbert transorm method don t satisy. Directional EMD and the partial Hilbert transorm considering directions [10] were used to obtain the corresponding complex signals. The instantaneous reuencies were computed along the ixed direction. This method is suitable or the texture which has main directions and it is hardly perectly reconstructed rom the modulation results. The partial Hilbert transorm make Hilbert transorm in x and y direction separately. P.C. Toy once used the partial Hilbert transorm and two dimensional EMD [11], but decomposition method has some problems. Bulow once used uaternionic Hilbert transorm, but he represented the signal by three phases, which lead to six reuencies, hard to represent. [1-15] Bi-orthant analytic signal was proposed or two dimensional signals, which kept many good properties o 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 eicient extension o the analytic signal to images, which is not suitable or reuency representing. In this paper, we illustrate the problem o two dimensional Hilbert Huang transorm method, and make some experiment in image analysis. First o all, Hilbert-Huang transorm is introduced. This is an adaptive analysis method, especially useul or nonlinear signal. Besides, we introduce the Bidimensional Empirical Mode Decomposition, which is an adaptive decomposition method showing the visually signiicant structure o the image. Then by the uaternionic analytic method, suitable analytic signal is acuired. Quaternionic analytic method is a new proposed method, which keeps many good properties o D analytic signal. In section 3, based on uaternionic analytic method, two dimensional Hilbert spectral characters analysis method are proposed. The experimental results are listed in section 4. Section 5 is the conclusion.. Hilbert-Huang transorm Hilbert-Huang transorm (HHT) provides a intrinsic analysis method or non-linear and nonstationary signals. Empirical Mode Decomposition (EMD) decomposes a signal into intrinsic mode unctions (IMFs). IMFs are mono-components o the signal. By summing all the IMFs, the original signal can be perectly reconstructed. Secondly, then Hilbert transorm is applied to the IMFs and get the Hilbert spectral characters o p each IMFs. For any unction x() t o L class, its Hilbert transorm y() t is where P is the Cauchy principal value o the singular integral. 1 x( ) yt () P d, (1) t Let x() t is the Hilbert transorm y() t,we obtain the analytic unction, where i 1, and () t i () t zt () xt () yti () ate (), () 1 1 yt () at () ( xt () yt ()), () t tan.here at () is the instantaneous amplitude, x() t is the instantaneous phase unction. The instantaneous reuency is: 37
d () t () t. (3) dt The inal presentation is an energy-reuency-time distribution, which is designated as the Hilbert spectrum. This method provides many intrinsic Hilbert spectrum characters o signal such as: the instantaneous amplitude, the instantaneous phase, and the instantaneous reuencies. This method is especially suitable or the analysis o 1D-non-linear and non-stationary data. This new time-reuency distribution is shown in essay [0, 1]. Two dimensional Hilbert-Huang transorm.1 Bidimensional empirical mode decomposition Many decomposition methods are proposed or two dimensional signals. Bidimensional empirical mode decomposition (BEMD) [5] extracts IMFs by computing envelopes using radial basis unctions interpolation. While Directional Empirical Mode Decomposition (DEMD) deines directional reuency and envelopes as eatures by D, which is combine o two 1D decomposition method. In this essay, we use Bidimensional Empirical Mode Decomposition in essay [5]. This process is a D-siting process. The modiied D-siting process is perormed in two steps: extremas detection by neighboring window or morphological operators and surace interpolation by ast radial basis unctions 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 ully data-driven method and using no predetermined ilter. 38
Figure 1. Decomposition results o Lena using BEMD.. Two dimensional Hilbert Transorm based on uaternionic analytic method We can urther analyze each IMF by two dimensional Hilbert spectral analyses and get the intrinsic characters o the image. Deinitely, it depends on suitable two dimensional Hilbert transorm method. We ll urther introduce some two dimensional Hilbert transorms and introduce the properties o D analytic signals. Between them, the Quaternionic analytic method is presented. Firstly, we have to deine two dimensional analytic signals so as to get the two dimensional Hilbert spectral characters using analytic method. Some guidelines are needed or the extension o two dimensional analytic signals. Following lists the main properties o 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 o an analytic signal is right-sided ( FA( u, v) 0, or u 0 ) Hilbert pairs are orthogonal. The real part o the analytic signal A is eual to the original signal. The analytic signal is compatible with the associated harmonic transorm. In case o the one-dimensional Fourier transorm, the property o the analytic signal is called compatible with the associated harmonic transorm, since the real part o the Fourier kernel, i.e. (exp( i ux)) cos( i ux) is the Hilbert transorm o sin( i ux). Two dimensional analytic signal should satisy this property, that is the analytic signal is compatible with the associated harmonic transorm with transormation kernel K i K and IK are a Hilbert pair. Two dimensional analytic signals depend on the development o two dimensional Hilbert transorm. Dierent nd Hilbert transorm have been proposed in the past, which include: the the partial Hilbert transorm, total Hilbert transorm and the total Hilbert transorm and the uaternionic Hilbert transorm. There are some properties that the irst three methods don t satisy. However, the uaternionic Hilbert transorm overcomes the remaining problems. It is compatible with the associated harmonic transorm, but it always leads to several useless elements when using it in two dimensional Hilbert spectral character research. The uaternionic analytic signal can overcome the remaining problems. It is compatible with the associated harmonic transorm. [9]Beore using it, the basic knowledge o uatenion is introduced. A uaternion may be represented in hypercomplex orm as abicj dk, (4) where a, b, c and d are real ; i, j, and k are complex operators, which obey the ollowing rules i j 1, ij ji k (5) This implies that k 1, jk i, kj iki, j, ik j. (6) A uaternion has a real part a and an imaginary part. 39
The latter has three components and can be used as a vector uantity, oten denoted by. V( ) bicj dk. For this reason, the real part is sometimes reerred to as the scalar part o the uaternion S ( ) and the whole uaternion may be represented by the sum o its scalar and vector parts as S( ) V( ). A uaternion with a zero real or scalar part is called a pure uaternion. The modulus and conjugate o a uaternion ollow the deinitions or complex numbers a b c d (7) a bi cj dk. (8) A uaternion with a unit modulus is called a unit uaternion. The conjugate o a uaternion is obtained, like the complex conjugate, by negating the imaginary or vector part. Euler s ormula or the complex exponential generalizes to hypercomplex orm u e cos usin (9) V( ) u V( ) where is a unit pure uaternion. u e Any uaternion may be represented in polar orm as, whereu and are reerred to as the eigenaxis and eigenangle o the uaternion, respectively.u identiies the direction in 3-space o the vector part and may be regarded as a true generalization o the complex operator i, V( ) arctan sinceu 1. S ( ) is analogous to the argument o a complex number, but is uniue 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 orm in uaternion is uaternionic analytic signal. Let is a real two-dimensional signal and F its uaternionic Fourier transorm. In the uaternionic reuency domain we can deine the uaternionic analytic signal o a real signal as FA ( ) (1 sign( u))(1 sign( v)) F ( ), where ( xy, ) and ( uv, ).This can be expressed in the spatial domain as ollows: T A( ) ( ) n H i( ), (10) T T where n (, i j, k) and Hi is a vector which consists o the total and the partial Hilbert transorms o : Hi( ) 1 Hi( ) Hi( ) Hi( ) (11) Quaternionic analytic signal ulills most o the desired properties and allows generalizing the instantaneous phase. In essay [14] instantaneous phase was deined as the triple o phase angles o the uaternionic value o the uaternionic analytic signal at each position. Then there would be six instantaneous reuencies, which is not easy to present. In this part, we choose dierent represent orm. To present the amplitude and the phase o the image, the polar orm is used. The polar orm is used like V( ) V( ) u arctan this e u V( ), where, S ( ). 40
Similarly, the analytic signal A where the amplitude is represented as ( ) can be written as ( ) A Ae (1) A ( ) Hi( ) ( ) ( ) 1 Hi Hi (13) Hi( ) i ( ) ( ) 1 Hi j Hi k u ( ) i ( ) j ( ) k Hi1 Hi Hi Hi( ) i ( ) ( ) 1 Hi j Hi k arctan (14) ( ) (15) whereu identiies the direction in 3-space o the vector part and is the phase. Then instantaneous reuencies are di 1 ( x 1, y) ( x, y) di ( xy, 1) ( xy, ) (16) which represent the reuency in horizontal and vertical directions separately. We can combine them together to present the instantaneous reuency. Then there are seven inherent characters: amplitude, phase, two instantaneous reuency and three components o u, which give more characters o image. 4. Experiments In this section, we apply the image modulation method to both synthetic and natural images. In detail, we irst decompose the image to several IMFs using BEMD. The IMFs are the monocomponents o the image. Then we apply the Quaternionic analytic signal to modulate the image and get the two dimensional Hilbert spectral characters o image, such as: instantaneous amplitude, instantaneous reuencies. In our experiment, we just analyze the irst IMF o the image by the new method; the other components can be analyzed similarly. First o all, this new modulation method was applied to image, as well as real-world texture image. In detail, we irst decompose the image by BEMD to several IMFs, which are the monocomponents o the image. Then we apply the uaternionic analytic method to modulate the image. In our experiment, we just analyze the irst IMF o the image by this new method, the other components can be analyzed similarly. From Figure (a), we can see the original image lena, the irst IMF o this image is shown in Figure (b). Besides, we calculate the reconstruction image Figure (j), and it is eual to the original image Figure (a). Using the two dimensional Hilbert spectral analysis method, we get an instantaneous amplitude, an instantaneous phase, two instantaneous reuencies and three components o u shown in Figures (c)-(). The instantaneous amplitude provide a dense local characterization o the local texture contrast as is shown in Figure (c).besides, the local phase provides a dense characterization o the local texture orientation and pattern spacing, as is depicted in Figure (d). We put three component o u in RGB band to compose a color image, given in Figure (a). 41
(a) Lena (b) IMF1o Lena (c) amplitude (d) phase (e) di1 () di (g) needle diagram o instantaneous reuency (h) u 1 (i) u (j) u 3 4
(k) u (l) Reconstruction image Figure. Two dimensional Hilbert spectral characters o the IMF1 o image Lena Figure (g) shows the needle program o the instantaneous reuency (IF) vector o Figure (b). In the picture, the arrows point in the direction o the instantaneous reuency vector and the length o each needle are proportional to the instantaneous reuency vector's modulus. For clarity, only one needle is shown or each block o 5 5 pixels. This needle diagram relects the intrinsic characters o image. First, needles are clearly visible in all our uadrants. However, the direction instantaneous reuency estimations using the partial Hilbert transorm point to certain directions. Second, as shown in Figure (g), much o the salient structure o the image have been interpreted by the analytic image. We can make meaningul associations between individual needles diagram o instantaneous reuencies in Figure (b) and speciic eatures o the image. The needles are longer in the texture part o the image, corresponding to reuency estimations with larger magnitudes, just consistent with the understanding. What s more, the vector is smaller in the smoother part o the image, As is shown in Figure (g) s let part,the instantaneous reuency vector also describe the complexity part o Figure (b). These characters relect the special visual meaning o the image and it is a ully data-driven method and shows the intrinsic characters o image. 5. Conclusions This paper examines the issue o two dimensional Hilbert-Huang Transorm.Two dimensional Hilbert-Huang Transorm based on Bidimensional Empirical Mode Decomposition (BEMD) and uaternionic analytic signa are proposed in the essayl. Bidimensional Empirical Mode Decomposition is adaptive signal decomposition method and its suitable or nonstationary signals. Quaternionic analytic signal satisies most o the two dimensional analytic method. Two dimensional Hilbert spectral characters are got include the instantaneous amplitude, the instantaneous phase, and the instantaneous reuencies. We illustrate the techniues on natural images, and demonstrate the estimated instantaneous reuencies using the needle program. These eatures oer a new and promising way to analyze images and may use in character extracting, image searching and related ields. 6. Reerences [1] Liu Weihua, "Remote On-Line Supervisory System o Micro Particle Using Image Analysis", JCIT, Vol. 6, No. 5, pp. 44-50, 011. [] Yan Liping, Song Kai, Wang Chao, "Design and Realization o Forest Fire Monitoring System Based on Image Analysis", JDCTA, Vol. 5, No. 1, pp. 45-458, 011. [3] N. E. Huang et al., The empirical mode decomposition and the Hilbert spectrum or 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 o Earthuake Data: Hilbert Spectral Analysis o Station TCU19, Chi-Chi, Taiwan, 1 September 1999," Bulletin o The Seismological Society o America, vol. 91, pp. 1310-1338, 001. [5] Nunes J C,et al. Image analysis by bidimensional empirical mode decomposition, Image and Vision Computing, vol. 1, no. 1, pp. 1019-106, 003. [6] Damerval, S. Meignen, and V. Perrier, A ast algorithm or bidimensional EMD, IEEE Signal 43
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