A new image flux conduction model and its application to selective

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A New Image Flux Conduction Model and Its Application to Selective Image Smoothing Chwen-Jye Sze, Hong-Yuan Mark Liao, Member, IEEE, and Kuo-Chin Fan, Member, IEEE

Abstract—In this paper, a discrete image flux conduction equation which is completely new in this field is proposed. The new approach starts with formulating a discrete image flux conduction equation based on the concept of heat conduction theory. Based on this discrete equation, the status change at a time point can be directly computed from its spatial neighborhood. To more accurately estimate an image flux, we have used an orthogonal wavelet basis to approximate the gradient of the intensity at each point. Since the proposed approach is discrete by nature, it is not necessary to formulate a continuous PDE to fit the discrete image data set. Furthermore, introduction of different numerical methods to solve the PDE can also be avoided. Since the proposed approach does not require that a PDE be solved, it is therefore more efficient and accurate than the conventional methods. Experimental results obtained using both synthetic signals and real images have demonstrated that the proposed model could effectively handle the selective image smoothing problem. Index Terms—Nonlinear filtering, selective image smoothing, wavelet.

I. INTRODUCTION

I

MAGE smoothing is an important step in low-level computer vision especially when the input images are noisy. An ideal smoothing algorithm is expected to simultaneously remove noises and enhance edges in an image. It is well-known that both noises and edges are high frequency signals. Therefore, it is very difficult to remove one while enhancing (or retaining) the other. As a result, many researchers have devoted themselves to solving the problem, and some satisfactory results have been reported [1], [2]. Among the existing smoothing algorithms, the approach that bases the smoothing procedure on a diffusion/conduction equation has attracted a lot of attention in recent years [3], [4], [9]–[20]. In the literature, signals convoluted with Gaussians of varying width are considered to be the oldest and best-studied representatives of linear diffusion-based smoothing. In 1962, Taizo Iijima [3] presented an axiomatic derivation of the one-dimensional (1-D) Gaussian scale-space filtering technique. In the early 1980’s, Witkin [4] and Koenderink [5] rigorously introduced the notion that scale-

Manuscript received March 24, 1999; revised July 24, 2000. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Sridhar Lakshmanan. C.-J. Sze and H.-Y. M. Liao are with the Institute of Information Science, Academia Sinica, Taipei, Taiwan R.O.C. (e-mail: [email protected]; [email protected]). K.-C. Fan is with the Department of Computer Science and Information Engineering, National Central University, Chung-Li, Taiwan, R.O.C. (e-mail: [email protected]). Publisher Item Identifier S 1057-7149(01)00107-5.

space could be used to represent images at multiple scales. Extensive research works have been done in recent years with regard to the Gaussian scale-space theory. Up to now, more than ten different axioms for the scale-space theory have been derived and summarized [6]. The first nonlinear diffusion filter was proposed by Perona and Malik in 1987 [7], [8]. The basic concept behind their work is to replace Gaussian smoothing, which is equivalent to isotropic diffusion, with directional diffusion that preserves edges. Their work led to number of theoretical and practical research issues which have attracted much attention in the past decade. These research issues have included understanding the mathematical properties of an isotropic diffusion and related variational formulations [9], [8], [10], developing well-posed and stable equations [11], [12], [9], [13], [10], modifying an isotropic diffusion for fast and accurate implementation, applying the diffusion equations in specific applications [14], and studying the relations between an isotropic diffusion and other image processing related problems [15], [16]. An excellent survey of nonlinear diffusion filtering can be found in [17]. A general procedure for conventional nonlinear conduction filtering starts with formulating a conduction equation and determining its corresponding conductivity function. In most cases [18], [19], [8], the selected conductivity function is nonlinear and is dependent on the intensity gradients of an image. Since the data of an image is discrete by nature, its gradients are normally estimated using a numerical method. Once the gradient values of all the pixels are determined, one can directly compute the corresponding conductivity values. Finally, an appropriate numerical method should be introduced to solve the conduction equation, which is a PDE by nature. A previous report [17] mentioned that if numerical methods are introduced to solve a PDE and the data happen to be an image (i.e., discrete data), then the results will turn out to be worse than those obtained by applying Perona–Malik’s method [8]. Surprisingly, Perona and Malik [8] did not explain why their method could outerperform other numerical methods. In [17], Weickert wrote: “Interestingly, practical implementation of the Perona–Malik process works often better than one would expect from the theorem.” The above mentioned phenomenon is the so-called Perona–Malik Paradox [20]. Later, in 1997, Weickert and Benhamonda [18] explained this phenomenon by introducing a discrete nonlinear scale-space framework. In this framework, a spatial discretization on a pixel grid is actually equivalent to providing a well-posed scale-space with many image-simplifying properties. In addition, they explained that the introduction of an explicit time discretization scheme leads to a scheme which does not introduce additional oscillations.

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Recent researches [20], [18] in the area have tried to either explain the Perona–Malik paradox [20], [18] or find a numerical solution which could outerperform the Perona–Malik’s approach [20], [18]. In this paper, we intend to solve the problem using a completely different approach. In the first place, a discrete conduction process which can totally fit a discrete data set will be proposed. Under the circumstances, we don’t have to formulate a continuous PDE to fit the discrete data set. Furthermore, introduction of different numerical methods to solve the continuous PDE can also be avoided. The advantage of the proposed approach is that the error generated due to the approximation process will be significantly reduced. Our approach starts with formulating a discrete image flux conduction equation based on the concept of heat conduction theory. Basically, this equation describes the conduction process in a discrete manner. The conductivity value of each point in the image can be computed based on its corresponding intensity. Therefore, the status of each point in the next state can be directly computed using the image flux equation. Since the proposed approach does not require that a PDE be solved, it is therefore more efficient and accurate than the conventional methods. The rest of this paper is organized as follows. In the next section, some basic concepts of heat conduction will be introduced. In Section III, a wavelet-based isotropic image flux conduction equation and an anisotropic conduction equation will respectively be proposed. We will prove that the isotropic formula satisfies the maximum and minimum principle when using the Haar wavelet basis. Experimental results obtained using both 1-D synthetic signals and two-dimensional (2-D) real images will be reported in Section IV. Section V will include some discussion issues and a concluding remark.

Fig. 1.

Element volume for 1-D heat-conduction analysis.

substitute an appropriate value to solve the problem. However, people usually consider the general case, i.e., where the temperature may be changing with time and heat sources may be present within the body. For an element of thickness , the following energy conservation equation can be established (Fig. 1): Energy conducted in the left plane heat generated within the element change of internal energy energy conducted out from the right plane. The above energy quantities can be represented in mathe(energy conducted matical forms as follows: (energy generated within elein the left plane), (change of internal energy), and ment), (energy conducted out from the right is the energy generated per unit volume, plane), where is the specific heat of the material and is the density of the material. Combining the above relations gives

II. BASIC CONCEPTS OF HEAT CONDUCTION Conduction, convection and radiation are three different modes of heat transfer. Since the conduction theory will be applied in the study, we shall introduce it briefly in the section. It is known that when a temperature gradient exists in a body, there will be energy transfer in the body from the high-temperature regions to the low-temperature regions [21]. Basically, the energy transfer is carried out by means of conduction, and the heat-transfer rate per unit area is proportional to the temperature gradient

(2) or (3) A derivation process which is similar to the 1-D case can be applied to obtain the three-dimensional (3-D) heat-conduction equation (4)

When the proportion constant is introduced, we have

Equation (4) can be further simplified as (1)

is the where represents the heat-transfer rate and temperature gradient in the direction of heat flow. The positive constant is called the thermal conductivity of a material, and the minus sign enforces satisfaction of the second principle of thermodynamics. Equation (1) is known as the Fourier law of heat conduction. Considering the 1-D heat conduction system shown in Fig. 1, if the system is in a steady state, then one can integrate (1) and

(5) is called the thermal diffusivity where the quantity of the material. Equation (1) defines an equation for thermal conductivity. The numerical value of the thermal conductivity indicates how fast heat will flow in a given material. Based on this definition, experimental measurements can be made to determine the thermal conductivity of different materials. In general, the thermal conductivity is strongly temperature-depen-

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Fig. 2. Discrete image flux-based model.

dent. If a solid happen to have different conductivities , in three mutually perpendicular directions, then , and should be defined as and

, ,

and (6)

III. IMAGE FLUX CONDUCTION In this section, we shall first propose a discrete image flux conduction equation which can easily solve the problem mentioned in the introduction. The numerical implementation of the isotropic image flux conduction process will be described in Section III-B. Finally, an anisotropic image flux equation will be proposed in Section III-C.

replace the original continuous conduction model. In what follows, we shall describe the new model in detail. Let be the intensity of a discrete spatial point at time . Let be the image flux computed at point , time . Here, the mathematical form of image flux is the same as that of heat flow in (1). Assume there is no self-generated energy; then, at a specific point , the internal energy change between and is simply the difference between the image flux conducted in and conducted out. Therefore, the image flux of a spatial point at time , i.e., , can be computed based on the internal energy change (see Fig. 2). Under the circumstances, a discrete image flux balance equation can be written as follows: (7) and is a constant that can where be used to control the energy transfer rate. Thus, we have (8)

A. Image Flux-Based Model In Section II, we mentioned that the thickness of the element at point is considered to be . Therefore, the energy conducted into the left plane can be computed using the gradient of point itself, and the energy conducted out from the right plane (Fig. 1) can be calculated using the gradient of the point . The above computation is based on an assumption that at the system is continuous. However, the image data set is discrete by nature, and the above assumption is thus not practical. Therefore, we do believe that a new model which can better characterize the reality is needed. Our idea is as follows: when the thickness of an element is not zero, then the energy conducted into and conducted out from this element cannot be measured by simply calculating the status of a representative point. That is, the status of the element should be decided by the energy change of the representative point’s neighborhood. Based on this concept, we propose a discrete image flux-based analysis model to

From Fig. 2, it is obvious that lows:

can be calculated as fol-

(9) That is, (10) From (10), one can see that the change of internal energy at point is dependent on the image flux of its two neighbors ( and ). If the total image flux change is equivalent to zero at time , then there is no status change of point from to . This means that, when the values of the image flux conducted in and conducted out can be computed at time , the status of the can also be computed. point at time

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When this case is extended to the 2-D one, the internal energy changes in the and directions, respectively, are (11) and (12) The total change of internal energy at

from to

is (13)

conduction equation. Then, an anisotropic image flux conduction equation which is a generalization of an isotropic case will be proposed. B. Numerical Implementation of the Isotropic Image Flux Conduction Equation In Section III-A, we mentioned that the mathematical formulation of image flux is similar to that of heat flow described in (1). In what follows, we shall explain how to calculate the image flux when the input data are discrete. Assume that the status of point at time is known; then, the status of its nearest neighbors at time should be known in advance. Similar to the at time heat flow described in (1), the image flux function is

That is (18) is a conductivity function. In most applications, is nonlinear. Since (18) involves a partial differentiation and the image data are discrete by nature, the design of a good approximation that can best characterize the “differentiation” from a set of discrete data becomes an important issue. In [22], [23], we successfully applied Daubechies scaling functions (continuous and differentiable) to fit a set of discrete data. Because this type of scaling function is similar to a delta function [22], the fitting error is almost negligible. Therefore, the derivatives of the discrete data can be estimated accurately. In [22], [23], an orthogonal wavelet basis was applied to approximate a continuous surface based on a set of discrete surface points. We also proposed an efficient way to deal with the differentiation problem. In this paper, we will use an orthogonal wavelet basis to approximate the intensity function and then calculate the conductivity values and their corresponding image flux. In order to simplify the computation and proofs, we will use the orthogonal Haar wavelet basis to accomplish the task. 1) One-Dimensional Isotropic Image Flux Conduction: Let be a signal at time . Assume that the conductivities at point and at point are two functions . The derivatives of and with respect to can be estimated as follows:

where (14) In sum,

can be written as follows:

(15) from to From (15), we can find that the change of is dependent on the total image flux of its neighborhood along two orthogonal directions. Therefore, if the image flux of every can point at time can be accurately computed, then also be calculated. Based on (15), one can calculate the intensity distribution of an arbitrary image at all instances. Theorem 1 asserts that the image flux conduction equation satisfies the so-called average gray level invariance property [18], [19]. This property is very important because once it is satisfied, one can guarantee that the corresponding dynamic system will not generate extra energy after each iteration. Under these circumstances, one can say that the system is always stable. Theorem 1 (1-D Conservation of Average Grey Level): The after the iteration average grey level of

(19) and

(16) always maintains the average grey level of the initial input image: (17) is the total number of input data points. where The proof of this theorem can be found in Appendix A. The 2-D case can be easily extended; hence, we omit the proof here. In what follows, we shall first introduce how to derive a numerical implementation scheme to solve an isotropic image flux

(20) is a connection coefficient of a wavelet-based difwhere is an integer representing the doferential operator [24], and main of a compact support. The isotropic image flux functions and can be, respectively, approximated as follows:

(21)

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and

(26) and (22)

Therefore, equation:

of (10) can be calculated using the following

(27)

, , The isotropic image flux functions , and can be, respectively, approximated as follows:

where

(28)

(29) (23) Theorem 2 asserts that (23) satisfies the so-called maximum and minimum principle [8] when the Haar wavelet is chosen as a basis [24]. Once (23) satisfies the above mentioned principle, no extra local maxima or minima will be generated when it is applied. It should be noted that the discrete maximum–minimum principle is a very restrictive stability criterion (more restrictive than the von Neumann stability) [19]. Theorem 2 (1-D Maximum–Minimum Principle): Let and be the maximum and minimum , respectively, where of the neighborhood of point and . If , then is always less than or and larger than or equal to when the equal to Haar wavelet is chosen as a basis [24]. The proof of Theorem 2 is given in Appendix B. In most applications, the input data are 2-D by nature; therefore, it is necessary to extend (23) to two dimensions. 2) Two-Dimensional Isotropic Image Flux Conduction: Let be a 2-D image at time . Assume that the conductivities at point , at point , at point , and at point are four functions . Following (19) and (20), , , , and the derivatives of can be, respectively, estimated as follows:

(30) and

(31) of (15) can be calculated directly using Therefore, the following equation:

(24)

(25) (32)

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Theorem 3 asserts that (32) satisfies the maximum and minimum principle [8] when the Haar wavelet is chosen as a basis [24]. Theorem 3 (2-D Maximum–Minimum Principle): Let and be the maximum and minimum of the , respectively, where neighborhood of point

Let

, ,

, and . Then, (35) can be rewritten as follows:

and . If and , then

in (32) is always less than or equal to and larger than or equal to when the Haar wavelet is chosen as a basis [24]. The proof of Theorem 3 is similar to that of Theorem 2; it is thus omitted. In real implementations, other non-Haar bases, such as Daubechies’ bases [25]–[27], also work.

C. Anisotropic Image Flux Conduction In the discussion of the 2-D isotropic case, we only considered the image flux along the and directions independently; that is, no coupling effect was considered. In Section II, we mentioned that in the anisotropic heat conduction case, the magnitude and orientation of heat flow could be the linear combination of the gradients along the two orthogonal directions. Here, we shall define the anisotropic image flux in a similar format. Then, the anisotropic image flux will be plugged into the proposed image flux conduction equation [(15)]. Finally, we shall solve the anisotropic conduction equation by using a wavelet-based approximation. In what follows, we shall proceed with the complete derivation. It is known that in a 2-D anisotropic case, each component can be written as a linear of the image flux at a point combination of the components of the intensity gradient at that point [28], that is,

(36) Assume that, and and

. Then, can be represented in the following form: (37)

and (38) Based on (15), we have

(33) and (39)

(34) When the Haar basis is applied, (39) can be written as and are conwhere at time . Equations (33) and ductivity functions of point (34) can be written into the following matrix form:

(35)

(40)

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(a)

(b)

(c)

Fig. 5. Results obtained by processing the signals in Fig. 3(c) using Perona-Malik’s approach (noise level  = 20).

Fig. 3. Set of 1-D test signals: (a) original signal, (b) signal in (a) with added white noise with variance  10, and (c) signal in (a) with added white noise with variance  = 20.

=

Fig. 6.

Test image: brain.

If we can find some functions of , , and that can satisfy the conditions

,

,

(a)

and

then, the maximum of lowing equation:

can be bounded by the fol-

(b) Fig. 4. (a) Results obtained by processing the signals in Fig. 3(b) using our approach (noise level  = 10). (b) Results obtained by processing the signals in Fig. 3(c) using our approach (noise level  = 20).

(41)

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Fig. 7.

303

Results obtained by applying our approach. (a) Iteration no.

= 100 and (b) iteration no. = 400.

Fig. 8. (a) Test image: house and (b) results obtained by applying our approach for 100 iterations.

IV. EXPERIMENTAL RESULTS

where . Similarly

A. Experiments on Isotropic Conduction in Real Images In this section, we used 1-D ideal signals (synthetic data) and 2-D real images to test the effectiveness of our approach. Two chosen conductivity functions [8], [19] used to conduct a series of experiments were as follows: where

(42)

where where . Therefore, the proposed anisotropic conduction equation can also satisfy the maximum and minimum principle based on some special constraints.

(43) Fig. 3(a) shows a 1-D ideal signal with some step edges. Fig. 3(b) and (c) show two noisy signals (Gaussian white

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noises) with variances 10 and 20, respectively. In this 1-D case, we used (42) as the conductivity function and the parameter was chosen as 3.0. Fig. 4(a) and (b) show the results obtained by processing the signals in Fig. 3(b) and (c), respectively, using our approach. In each set of experiments, we illustrate the results obtained at different iteration numbers (100, 200, 400, and 800 iterations). From these experiments, it is obvious that no matter the introduced noise level was 10 or 20, the detected step edges stayed stable up to 800 iterations. Fig. 5 shows the results obtained by processing the signal in Fig. 3(c) using Perona–Malik’s approach [8]. We found the denoised results obtained by using our approach and those obtained by Perona–Malik’s approach were comparable. For the experiments conducted on 2-D real images, we used (43) as a conductivity function to test the effectiveness of our 2-D isotropic conduction approach. Fig. 6 shows an MRI image. Fig. 7(a) and (b) show the results obtained by applying our approach for 100 iterations and 400 iterations, respectively. Fig. 8 shows the result obtained by applying our approach to the house image. From these results, it is obvious that the edge information could be retained and the noises could be removed by using our approach. B. Experiments on Anisotropic Conduction in Real Images In the following experiments, we extended our method to handle the discrete anisotropic conduction problem. In general, a 2-D conductivity matrix can be defined as follows: (44) When one considers the structure information of an image, a structure tensor matrix can be defined as follows [17]:

Fig. 9. Results obtained by using the proposed anisotropic conductivity matrix (47) with different r values. (a) r 1:0, (b) r = 0:9, (c) r = 0:8, (d) r = 0:7, (e) r = 0:6, and (f) r = 0:5.

=

a simplified anisotropic formula to replace (46) can be written as follows:

(45)

Since every coefficient of a conductivity matrix should be bounded between 0 and 1, a coherence conductivity matrix can be defined as follows:

(46) where ,

,

. For simplicity, we set , and . Thus,

(47) . In the following experiments, we used where and , to six different values: control the conductivity matrix [ (47)]. Fig. 9(a)–(f) shows the results (1000 iterations) obtained by applying (47) with and , respectively. From these results, it is obvious that the edge information could be retained, and that noise could be removed by using our proposed . The poor quality anisotropic approach when result shown in Fig. 9(f) was obtained by using (47) with . The reason for the poor result is that when (47) is close , it no longer satisfies the maximum and minimum to principle anymore.

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V. CONCLUSIONS

,

, and

for all

or

)

In this paper, we have derived a discrete image flux conduction equation from our proposed discrete image flux model using an image flux balance equation. Based on this discrete equation, the status change at a time point can be directly computed based on the image flux of its neighborhood. To more accurately estimate an image flux, we have used an orthogonal wavelet basis to approximate the gradient of the intensity at each point. We have also proved that our discrete equation, which is based on the orthogonal Haar wavelet basis, satisfies the so-called maximum and minimum principle. Experimental results obtained using both synthetic signals and real images have demonstrated that the proposed model could effectively handle the selective image smoothing problem.

(49) and , then the value If is always larger than or of equal to 0. Then, (49) can be written as follows:

APPENDIX A PROOF OF THEOREM 1 This appendix presents the main steps in the proof of The[(10)] can be written orem 1. The average grey level of as

(48) be a periodical function on (1, and can be written as follows:

Let

where

. Similarly,

where we have

. Therefore,

) [29]. Therefore, . Thus, (48) REFERENCES

That is

Therefore, (10) satisfies the average grey level invariance property. APPENDIX B PROOF OF THEOREM 2 This appendix presents the main steps in the proof of Theorem 2. Before proving Theorem 2, (23) can be rewritten in , a new form if the Haar wavelet is chosen (

[1] R. C. Gonzalez and R. E. Woods, Digital Image Processing. Reading, MA: Addison-Wesley, June 1992. [2] R. M. Haralick and L. G. Shapiro, Computer and Robot Vision. Reading, MA: Addison-Wesley, 1992, vol. 1. [3] T. Iijima, “Basic theory of pattern normalization (for the case of a typical one-dimensional pattern)” (in Japanese), Bull. Electrotech. Lab., vol. 26, pp. 368–388, 1962. [4] A. P. Witkin, “Scale-space filtering,” in Proc. 8th Int. Joint Conf. Artifical Intelligence, Karlsruhe, Germany, Aug. 1983, pp. 1019–1022. [5] J. J. Koenderink, “The structure of images,” Biol. Cybern., vol. 50, pp. 363–370, 1984. [6] T. Lindeberg, Scale-Space Theory in Computer Vision. Norwell, MA: Kluwer, 1994. [7] P. Perona and J. Malik, “Scale space and edge detection using anisotropic diffusion,” in IEEE Workshop Computer Vision, 1987, pp. 16–22. [8] , “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell., vol. 12, pp. 629–639, July 1990. [9] F. Catté, P. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Anal., vol. 29, pp. 182–193, Feb. 1992. [10] Y. L. You, W. Xu, A. Tannenbaum, and M. Kaveh, “Behavioral analysis of anisotropic diffusion in image processing,” IEEE Trans. Image Processing, vol. 5, pp. 1539–1553, 1996. [11] L. Alvarez, P. L. Lions, and J. K. Morel, “Image selective smoothing and edge detection by nonlinear diffusion II,” SIAM J. Numer. Anal., vol. 29, pp. 845–866, 1992.

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[12] L. Alvarez, F. Guichard, P. L. Lions, and J. K. Morel, “Axioms and fundamental equation of image processing,” Arch. Ration. Mech. Anal., vol. 123, pp. 200–257, 1993. [13] M. Nitzberg and T. Shiota, “Nonlinear image filtering with edge and corner enhance,” IEEE Trans. Pattern Anal. Machine Intell., vol. 14, pp. 826–833, Aug. 1992. [14] G. Gerig, O. Kübler, R. Kikinis, and F. A. Jolesz, “Nonlinear anisotropic filtering of MRI data,” IEEE Trans. Med. Imag., vol. 11, pp. 221–232, 1992. [15] G. Sapiro, “From active contours to anisotropic diffusion: Relations between basic pde’s in image processing,” in IEEE Int. Conf. Image Processing, Sept. 1996. [16] J. Shah, “A common framework for curve evolution, segmentation, and anisotropic diffusion,” in IEEE Int. Conf. Computer Vision Pattern Recognition, 1996. [17] J. Weickert, “A review of nonlinear diffusion filtering,” in Scale-Space Theory for Computer Vision, Lecture Notes in Computer Science, B. ter Haar Romeny, Ed. New York: Springer, 1997, vol. 1252, pp. 3–28. [18] J. Weickert and B. Benhamouda, “Why the Perona–Malik Filter Works,” Dept. Comput. Sci., Univ. Copenhagen, Denmark, Tech. Rep. DIKU-TR-97/22, August 1997. [19] J. Weickert, B. M. ter Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE Trans. Image Processing, vol. 7, pp. 398–410, Mar. 1998. [20] S. Kichenassamy, “The Perona–Malik paradox,” SIAM J. Appl. Math., vol. 57, pp. 1328–1342, Oct. 1997. [21] J. P. Holman, Heat Transfer, 8th ed. New York: McGraw-Hill, 1997. [22] J. W. Hsieh, H. Y. Mark Liao, M. T. Ko, K. C. Fan, and Y. P. Hung, “Image registration using an edge-based approach,” Comput. Vis. Image Understand., vol. 67, pp. 112–130, Aug. 1997. [23] C. J. Sze, H. Y. Mark Liao, H. L. Hung, K. C. Fan, and J. W. Hsieh, “Multiscale edge detection on range images via normal changes,” IEEE Trans. Circuits Syst. II, vol. 45, pp. 1087–1092, Aug. 1998. [24] G. Beylkin, “On the representation of operators in bases of compactly supported wavelets,” SIAM J. Numer. Anal., vol. 29, pp. 1716–1740, Dec. 1992. [25] I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Commun. Pure Appl. Math., vol. 41, pp. 909–996, 1988. , “The wavelet transform, time-frequency localization and signal [26] analysis,” IEEE Trans. Inform. Theory, vol. 36, pp. 961–1005, Sept. 1990. [27] , Ten Lectures on Wavelets Philadelphia, PA, 1992. [28] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. Oxford, U.K.: Clarendon, 1959. [29] S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell., vol. 11, pp. 674–693, July 1989.

Chwen-Jye Sze was born in Taipei, Taiwan, R.O.C. in 1966. He received the B.S. degree in electrical engineering from the Chinese Culture University, Taipei, in 1992, the M.S. degree in electrical engineering from National Chung-Cheng University, Chiayi, Taiwan, in 1994, and the Ph.D. degree in computer science and information engineering from National Central University, Chung-Li, Taiwan, in 1998. Since October 1998, he has been a Postdoctoral Fellow with the Institute of Information Science, Academia Sinica, Taipei, Taiwan. His research interests are in multimedia processing.

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Hong-Yuan Mark Liao (M’88) received the B.S. degree in physics from National Tsing-Hua University, Hsinchu, Taiwan, in 1981, and the M.S. and Ph.D. degrees in electrical engineering from Northwestern University, Evanston, IL, in 1985 and 1990, respectively. He was a Research Associate with the Computer Vision and Image Processing Laboratory, Northwestern University, during 1990–1991. In July 1991, he joined the Institute of Information Science, Academia Sinica, Taipei, Taiwan, as an Assistant Research Fellow. He was promoted to Associate Research Fellow and then Research Fellow in 1995 and 1998, respectively. From August 1997 to July 2000, he served as the Deputy Director of the Institute. His current research interests are in multimedia signal processing, wavelet-based image analysis, content-based multimedia retrieval, and multimedia protection. He served as the Program Chair of the International Symposium on Multimedia Information Processing (ISMIP) in 1997. He also served on the program committees of several international and local conferences. He is on the editorial boards of the International Journal of Visual Communication and Image Representation, Acta Automatica Sinica, and the Journal of Information Science and Engineering. Dr. Liao was the recipient of the Young Investigators’ Award of Academia Sinica in 1998; the Best Paper Award of the Image Processing and Pattern Recognition Society of Taiwan in 1998; and the paper award of the Image Processing and Pattern Recognition Society of Taiwan in 1996 and 1999. He is on the editorial board of the IEEE TRANSACTIONS ON MULTIMEDIA. He is a member of the IEEE Computer Society.

Kuo-Chin Fan (M’88) was born in Hsinchu, Taiwan, R.O.C., in 1959. He received the B.S. degree in electrical engineering from National Tsing-Hua University, Hsinchu, in 1981. He received the the M.S. and Ph.D. degrees from the University of Florida, Gainesville, in 1985 and 1989, respectively. In 1983, he was a Computer Engineer with the Electronic Research and Service Organization (ERSO). From 1984 to 1989, he was a Research Assistant with the Center for Information Research, University of Florida. In 1989, he joined the Institute of Computer Science and Information Engineering, National Central University, Chung-Li, Taiwan, where he became Professor in 1994. He was the Chairman of the department from 1994 to 1997. Currently, he is the Director of Software Research Center and Computer Center at National Central University. His current research interests include image analysis, pattern recognition, and computer vision. Dr. Fan is a member of SPIE.