Tomographic reconstruction using nonseparable ... - Semantic Scholar
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[5] A. Kumar, Y. B.-Shalom, and E. Oron, “Precision tracking based on segmentation with optimal layering for imaging sensors,” IEEE Trans. Pattern Anal. Machine Intell., vol. 17, pp. 182–188, Feb. 1995. [6] A. G. Bors¸ and I. Pitas, “Optical flow estimation and moving object segmentation based on median radial basis function network,” IEEE Trans. Image Processing, vol. 7, pp. 693–702, May 1998. [7] J. Konrad and E. Dubois, “Bayesian estimation of motion vector fields,” IEEE Trans. Pattern Anal. Machine Intell., vol. 14, pp. 910–927, Sept. 1992. [8] A. G. Bors¸ and I. Pitas, “Median radial basis function neural network,” IEEE Trans. Neural Networks, vol. 7, pp. 1351–1364, Nov. 1996. [9] V. Vapnik, Estimation of Dependences Based on Empirical Data. New York: Springer-Verlag, 1982. [10] B. Widrow and S. D. Stearns, Adaptive Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1985.
Tomographic Reconstruction Using Nonseparable Wavelets Fig. 11. PSNR of the predicted frame in the “Hamburg taxi” image sequence. “–” denotes the PSNR of the proposed tracking algorithm. “- -” represents the PSNR prediction considering the initial MRBF model on 4 4 pixel blocks. “-.” denotes the PSNR between the actual frame and that used for prediction.
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Indy Workstation. The trained network, can be used for those successive frames which match the model according to a criterion [6]. In this case, 95 s are required for segmenting the moving objects and the optical flow for 20 frames when using 424 pixel blocks. When employing tracking as described in this study, only 68 s are necessary for the same frames using pixel resolution segmentation. In the first case only 3040 vectors had been processed while in the second case their number was 48 640. The segmentation provided by the tracking algorithm is quite good as it can be observed from the experimental results and provides a good basis for prediction-based frame reconstruction. The prediction PSNR of the tracking algorithm is better than when considering the initial MRBF model for segmenting all the frames and assuming just the previous moving object features for reconstruction, as it can be observed from Fig. 11. VI. CONCLUSION We propose a moving object tracking algorithm derived from the Bayesian theory. The optical flow and the segmentation features are jointly modeled by the MRBF network in the initial stage. The occluding and unlabeled regions are detected and classified appropriately. The proposed algorithm provides good moving object tracking capabilities. Such capabilities are used for segmenting and estimating the moving object velocity and segmentation in a future frame. The proposed algorithm is employed for frame prediction. REFERENCES [1] Y. Altunbasak and A. M. Tekalp, “Occlusion-adaptive, content-based mesh design and forward tracking,” IEEE Trans. Image Processing, vol. 6, pp. 1270–1280, Sept. 1997. [2] K. J. Bradshaw, I. D. Reid, and D. W. Murray, “The active recovery of 3-D motion trajectories and their use in prediction,” IEEE Trans. Pattern Anal. Machine Intell., vol. 19, pp. 219–224, Mar. 1997. [3] S. M. Smith and J. M. Brady, “ASSET-2: Real-time motion segmentation and shape tracking,” IEEE Trans. Pattern Anal. Machine Intell., vol. 17, pp. 814–820, Aug. 1995. [4] J. Weber and J. Malik, “Rigid body segmentation and shape description from dense optical flow under weak perspective,” IEEE Trans. Pattern Anal. Machine Intell., vol. 19, pp. 139–143, Feb. 1997.
Stéphane Bonnet, Françoise Peyrin, Francis Turjman, and Rémy Prost Abstract—In this paper, the use of nonseparable wavelets for tomographic reconstruction is investigated. Local tomography is also presented. The algorithm computes both the quincunx approximation and detail coefficients of a function from its projections. Simulation results showed that nonseparable wavelets provide a reconstruction improvement versus separable wavelets. Index Terms—Local tomography, McClellan transformation, nonseparable wavelets.
I. INTRODUCTION Computerized tomography (CT) consists of recovering a function from a set of its projections and relies on the inversion of the Radon transform. According to the nature of the data set, this problem may be ill-posed. The use of wavelets for inverse problems in general, and CT in particular, presents several interesting features to stabilize the inversion process [1]. As a matter of fact, wavelets may bring valuable solutions to the problem of local tomography [2]–[4]. The relationships between the continuous wavelet transform and the Radon transform have first been established in several independent works [5], [6]. Olson was the first to devise a reconstruction scheme from a customized sampling of the Radon transform [2]. Delaney [3] and Rashid-Farrokhi [4] proposed a multiresolution tomographic reconstruction algorithm to recover the two-dimensional (2-D) separable discrete wavelet transform (2-D DWT) of the image from its projections, and applied it to local tomography. Both algorithms are based on 2-D wavelets, constructed by tensor products of one-dimensional (1-D) wavelets. The 2-D separable wavelets impose a rectangular tiling of the Manuscript received December 28, 1998; revised February 27, 2000. S. Bonnet was supported by a grant from Siemens, France. This work is in the scope of the scientific topics of the PRC-GDR ISIS research group of the French National Center for Scientific Research (CNRS). The associate editor coordinating the review of this manuscript and approving it for publication was Prof. William Clem Karl. S. Bonnet and R. Prost are with CREATIS, CNRS Research Unit, 69621 Villeurbanne Cedex, France (e-mail: [email protected]). F. Peyrin is with CREATIS, CNRS Research Unit, 69621 Villeurbanne Cedex, France. She is also with ESRF, 38043 Grenoble Cedex, France. F. Turjman is with CREATIS, CNRS Research Unit, 69621 Villeurbanne Cedex, France. He is also with Hôpital Neurologique, 69500 Bron, France. Publisher Item Identifier S 1057-7149(00)06139-X.
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Let R the Radon operator of an image f
2 L2(IR2 ) be defined by
Rf (t; ) = R f (t) = p f (t) =
(x)(t 0 hx; 2i) dx:
f
IR
(1)
The filtered backprojection (FBP) algorithm relies on the inversion formula [8] f
= R# 3R f:
(2)
3 is the Lambda-operator used in the context of 3-tomography [9] and is defined by (3p)(! ) = j! j2^ p(! ), R# is the backprojection operator # given by R q (x) = 0 q (hx; 2i) d . II. NONSEPARABLE MULTIRESOLUTION ANALYSIS Multiresolution analysis (MRA) using separable or nonseparable wavelet basis may be expressed by a general formalism based on a 2 222 . This matrix imposes conditions on the dilation matrix sampling and scaling schemes. The most common dilation matrix is the quincunx matrix defined as: Q = (1 1; 1 0 1). A separable sublattice is given by a diagonal matrix such as: S = 2 . A 2-D MRA with dilation matrix is a generalization to L2 (IR2 ) of the 1-D MRA proposed by Mallat (see [10], [11]). In 2-D, a critically sampled filter bank that would achieve perfect reconstruction requires a sampling rate of N = jdet j. Consequently, a nonseparable biorthogonal MRA of 2-D signals requires a pair of dual scaling func~ g and (N 0 1) pairs of dual mother wavelets f9; 9~ g. The tions f8; 8 dilated and translated versions of 8 are defined according to by
D
Fig. 1. Diamond-shaped McClellan transformation applied on the “near-coiflet” wavelet: (a) scaling filter, (b) wavelet filter, (c) 2-D scaling filter, and (d) 2-D wavelet filter.
D D
I
D
D
8j;k (x) = jdet Dj0(j=2) 8(D0j x 0 k)
j
2
;
D k2
2:
(3) For a fixed scale j , the set f8j;k ; 2 2 g forms a non-orthogonal basis of the approximation space Vj of L2 (IR2 ) . Moreover, in a 2-D MRA, the scaling function 8 satisfies the two-scale equation [12]
k
8(x) =
k
Fig. 2. Nonseparable wavelet tomographic reconstruction from local projection data.
frequency plane, which is not well suited to the radial band-limited assumption of the image. In a preliminary work, we first proposed the idea of using nonseparable wavelets in tomography [7]. In this paper, we develop a framework for multiresolution tomography, which both includes the separable and nonseparable cases. This new proposal generalizes previous algorithms and provides the material for nonseparable multiresolution tomography. The application of this reconstruction scheme to 2-D wavelets generated via McClellan transformation is considered. This new approach allows to respect the geometry of the system by tiling the frequency plane in a diamond-shaped fashion that is more respectful to the radial band-limited assumptions. Let L2 (IRn ) denote the vector spaces of measurable, square-integrable n-D functions. The Fourier transform of a function f 2 L2 (IRn ) is defined by: f^(! ) = IR f ( ) e02ihx;!i d where boldfaced letters denote n-D vectors (i.e., in IR2 , = (x; y )t ) and h ; i is the usual inner product in IRn . Continuous and discrete convolution operators are both denoted by 3. The vector 2 = (cos ; sin )t is used to characterize the unit circle in IR2 .
x
x
x
k2
h
[k]8(Dx 0 k)
(4)
where h[ ] are the coefficients of a 2-D low-pass filter. i The set of wavelets f9j; k ; i = 1 1 1 1 (N 0 1)g forms a non orthogonal basis of Wj , the non-orthogonal complement of Vj in Vj 01 . Thus any function f 2 L2 (IR2 ) can be expanded in a nonseparable biorthonormal wavelet series as f
(x) =
k2
+
hf; 8J;ki 8~ J;k(x) N
01
=1 j J k2
i
hf; 9j;i k i9~ j;i k (x)
(5)
where J is the depth of the decomposition. The inner products hf; 8j;k i correspond to projections of f onto subspace Vj and represent the discrete approximation coefficients of f at resolution j . III. NONSEPARABLE MULTIRESOLUTION TOMOGRAPHIC RECONSRUCTION A. Nonseparable Wavelet-Based Inversion Formula In this section, we propose a general multiresolution inversion formula for tomography. The framework developed here will include the previous algorithms ([3] and [4]) since it will also be applied using a separable wavelet basis. The discrete approximation of f at resolution j can also be interpreted as a continuous convolution evaluated at point ( j ): j;0 )( j ) where 8 ( ) = 8(0 ). Aj f [ ] = hf; 8j;n i = (f 3 8
n
Dn
x
Dn x
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Fig. 3. (a) Nonseparable approximation coefficient, (b)–(c) nonseparable detail coefficients, (d) reconstructed image from the three subimages, (e) separable approximation coefficients, (f)–(h) separable detail coefficients, and (i) reconstructed image from the four subimages.
Applying inversion formula (2), the coefficients may be rewritten as
n] = (f 3 R#3R 8 j;0 )(Dj n):
(6)
n] = R#(R f 3 3R 8 j;0)(Dj n):
(7)
Aj f [
Due to the property [5]: f 3R# g as
= R# (R f 3g ), (6) can be restated
Aj f [
The approximation coefficients can then be evaluated by backprojecting on a given sublattice (defined by ) the convolution of the Radon transform of f with the ramp-filtered Radon transform of the
D
dilated reversed scaling function 8j;0 . Since the same result holds for the wavelet functions, this method allows the computation of the detail coefficients. This algorithm generalizes a wavelet-analogue of the FBP algorithm using nonseparable wavelet bases. A synthesis step is then necessary to recover the original image at a given resolution from these coefficients (usually the finest approximation is obtained for j = 0). Note that the separable case is obtained by considering = S and then 8j;0 ( ) = 2 (x) 2 (y ) with 2 (x) = 20j=2 (20j x). In Fourier space, the modified ramp-filter is derived from (7) as
D D
x
j j 2 2j (2j ! cos ) 2 (2j ! sin ):
h (! ) = ! k;j
(8)
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Fig. 4. (a) Nonseparable local approximation coefficients, (b) nonseparable local detail coefficients, (c) reconstructed region-of-interest from the two subimages, (d) zoom of the reconstructed region-of-interest, and (e) zoom of the original ROI.
B. Quincunx Multiresolution Tomographic Algorithm From here, we restrict ourselves to the choice of a quincunx dila~ ; 9; 9~ g be MRA biorthogonal wavelets with the tion matrix. Let f8; 8 ~ ; g; g~g defined as in (4). The discrete approximaassociated filters fh; h tion at resolution j can also be computed via the 2-D discrete filtering by Aj f
[n] = (A f 3 h j )[Dj n] ( )
0
n
n
(j )
[n] =
k2
[k] h j 0 [n 0 Dj0 k]:
h
(
1)
1
(10)
As in [3], it is straightforward to derive an inversion formula for discretely sampled data p [n] = R f (nTp ) to recover the discrete
[n] = p~h p~h ; [m] = (p 3 kh Aj f
(9)
where A0 f is the full-resolution discrete image. Considering h0 [ ] = [ ], the equivalent low-pass filter h(j ) after j iterations is given recursively by h
image A0 f [n1 ; n2 ] = f (n1 To ; n2 To ). Here, Tp denotes the radial spacing of the projection and To the sampling rate of the image. Using discrete time Fourier transform (DTFT) and following the same scheme as (6) and (7), we obtain
0
;
To Tp
)[ ]
hDj n; 2i
d
; m :
(11) (12)
The discrete filtering (12) is defined in Fourier space by Kh
( ) = TTpo j!j 2 H j j!j 12
; !
( )
To ! Tp
cos ; TTpo ! sin ; (13)
where H (j ) (resp. Kh ; ) denotes the discrete 2-D Fourier transform of the low-pass filter h(j ) (resp. kh ; ). This filtering step is similar
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to the standard FBP’s one, but the discrete filters kh ; include the wavelet and scaling filters. In addition, if radial filters are used, the angular dependency, which is compulsory with separable bases, is eliminated. One possible way in that direction is to generate nearly radial symmetric 2-D filters using McClellan transformation [13]. In the case of quincunx decimation, the transformation has to preserve the perfect reconstruction properties from a pair of 1-D biorthogonal filters [11]. This transformation results in 2-D nonseparable diamond-shaped filters. Fig. 1 illustrates this transformation applied to the “near-coiflet” scaling and wavelet filter. Moreover as we are using 2-D zero-phase filters, DTFT is purely real and from symmetry considerations, the filtering operation cost can be further reduced. C. Local Tomography The utility of wavelets in local tomography is motivated by the observation that 1-D wavelet functions with sufficient vanishing moments remain essentially local after ramp-filtering [2]. Fortunately, the chosen diamond McClellan transformation also preserves zeros at aliasing frequencies [12]. Thus, quincunx wavelets keep identically their good properties of space-frequency localization after ramp-filtering and so they are able to localize the Radon transform as in the separable case [4]. Nevertheless, increasing the number of vanishing moments enlarges the support of the filter and reduces the efficiency of the localization. Under those assumptions, the support of the modified scaling ramp filter kh ; will essentially have the same radius of support as the 2-D scaling filter h(j ) . A schematic diagram for local tomography is presented in Fig. 2 with a decomposition over one level.
Fig. 5.
Energy error in the ROI (16 pixels) versus nonlocal data. TABLE I PPSNR
OF THE RECONSTRUCTED DIFFERENT WAVELETS
ROI
FOR
IV. RESULTS Simulations were carried out with the Shepp-Logan brain phantom. The reconstructions are calculated on a 256 2 256 pixel grid using a set of 256 projections uniformly spaced over [0; [ with 256 samples per projections. We have used the “near-coiflet” wavelet with five taps ([14, Table III]) because of its good property of localization of the Radon transform, demonstrated in [4]. We applied the algorithm presented in Section III-B using two levels of decomposition. Fig. 3(a)–(c) shows the quincunx approximation A2 f and detail coefficients D2 f; D1 f reconstructed from global Radon transform data using (12). The quality of the reconstructed image, shown in Fig. 3(d) is evaluated by measuring the mean square error (MSE) with the image obtained with FBP, considered as the reference. The MSE is equal to 3:12 2 1003 . To compare our approach with the separable case, we applied the separable algorithm for one level of decomposition with the same choice of 1-D filter. The approximation coefficients are shown in Fig. 3(e) and the three detail coefficients in Fig. 3(f)–(h). The quality of the reconstructed image in Fig. 3(i) is visually the same (both images Fig. 3(d)–(i) are displayed in the range [00:2 1:2]) but the MSE in the separable case has slightly increased to 4:34 2 1003 . An example of local tomographic reconstruction is presented on Fig. 4. The region-of-interest is a centered disk of radius 32 pixels. We used the Radon transform data passing through this ROI in addition with an extra margin of 16 pixels to recover the high-frequency component [Fig. 4(b)]. We used a constant extrapolation of this region-of-exposure (ROE) to obtain the low-frequency component shown in Fig. 4(a). As a matter of fact, this extrapolation allows to reduce oscillations at the edge of the ROI [8]. Fig. 4(c) shows the ROI reconstructed from the approximation and detail images using our nonseparable algorithm. A zoom of the ROI is presented in Fig. 4(d) in comparison to the original ROI in Fig. 4(e). The quality of reconstruction is again very good and the radiation exposure is around 37% of the standard FBP. The reconstructed image corresponds
actually to the addition of the original phantom and a constant bias. This phenomenon arises naturally in local tomography and has already been well-studied in [8] and [4]. In local reconstruction, the MSE between the ROI reconstructed by our algorithm using only local projections and the one obtained using the standard FBP with global projections is calculated after diminution of the constant bias as MSE =
1
N n2ROI
c (fFBP (n)
0 fc
n))2
NSM (
(14)
where N is the number of pixels in the ROI, f c denotes the zero-mean function f and FBP or nonseparable multiresolution (NSM) refers to the reconstruction method. We have also measured the peak-to-peak signal-to-noise ratio (PPSNR) defined as: 20 log10 dyn= (MSE) where dyn is the dynamic range in the ROI reconstructed by FBP. Fig. 5 shows the plot of the PPSNR versus the amount of non local data, collected to recover the centered ROI. This amount is equal to the difference between the ROE and the ROI in terms of pixels. As expected, the energy error in the ROI is reduced when we increase the region-of-interest and consequently the amount of nonlocal data. Finally, we compared the quality of local reconstructions according to the PPSNR obtained using the nonseparable algorithm with the one achieved by the separable algorithm of [4]. Simulations were performed over the same ROI (radius of 16 pixels) and the same ROE (margin of 12 pixels) in both cases with different choice of wavelets. Moreover, to demonstrate the importance of the number of vanishing moments of , we exchanged for each given wavelet, the role of the ~ g. The filter pairs that have been studied are the spline filter pairs fh; h filters “bior2.4” ([14, Table I]), the spline variant with less dissimilar lengths “bior4.4” ([14, Table II]) and the “near-coiflet.” The examination of Table I first shows that the quality of local reconstruction
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is always better when the nonseparable algorithm is used (around 1 dB) compared with the separable approach. Nevertheless, the quality is high in both cases for any kinds of wavelets. Secondly, we notice that the case where the analyzing wavelet has more vanishing moments performs best. Indeed, if the pair of spline filters “bior2.4” is chosen (i.e and ~ have respectively 2 and 4 vanishing moments), our reconstruction algorithm leads to a PPSNR of 38.4 dB while exchanging the filters order leads to a better PPSNR of 40.12 dB. Finally, a similar study with the filter “bior4.4” ( and ~ have both four vanishing moments), shows that the case where is more regular performs best. These results are in agreement with the requirement imposed on wavelets in Section III-C.
Capacity of Full Frame DCT Image Watermarks Mauro Barni, Franco Bartolini, Alessia De Rosa, and Alessandro Piva
Abstract—The evaluation of the number of bits that can be hidden within an image through digital watermarking is a crucial topic, which has been addressed only for additive watermarks. The evaluation of watermark capacity is very important because it allows to put a theoretical upper bound on the amount of information that can be hidden into an image by a given watermarking procedure, regardless of the watermark extraction technique. It is the purpose of this work to suggest a methodology for the evaluation of the watermark capacity in a nonadditive, non-Gaussian framework, and to discuss the results we obtained by applying it to a set of standard images. Index Terms—Nonadditive DCT watermarks, watermark channel capacity, .
V. CONCLUSION In this paper, we have presented a general framework for multiresolution tomographic reconstruction. This approach allows to recover the nonseparable wavelet and scaling coefficients of a function from its projections. The separable case can be obtained for a particular choice of the dilation matrix . McClellan transformation has been used to construct a 2-D multiresolution analysis and its application in the field of tomography has been described. Furthermore, local tomography using these nonseparable bases is possible and comparisons with the separable approach shows a slight improvement in terms of PPSNR.
D
REFERENCES [1] A. Aldroubi and M. Unser, Wavelets in Medicine and Biology. Boca Raton: CRC, 1996. [2] T. Olson and J. DeStefano, “Wavelet localization of the radon transform,” IEEE Trans. Signal Processing, vol. 42, pp. 2055–2067, Aug. 1994. [3] A. H. Delaney and Y. Bresler, “Multiresolution tomographic reconstruction using wavelets,” IEEE Trans. Image Processing, vol. 4, pp. 799–813, June 1995. [4] F. Rashid-Farrokhi, K. J. R. Liu, C. A. Berenstein, and D. Walnut, “Wavelet-based multiresolution local tomography,” IEEE Trans. Image Processing, vol. 6, pp. 1412–1430, Oct. 1997. [5] C. A. Berenstein and D. Walnut, “Local inversion of the radon transform in even dimensions using waveletes,” in 75 Years of the Radon Transform. Cambridge, MA: International, 1994, pp. 38–58. [6] F. Peyrin, M. Zaim, and R. Goutte, “Construction of wavelet decompositions for tomographic images,” J. Math. Imag. Vis., vol. 3, pp. 105–121, 1993. [7] F. Peyrin and M. Zaim, “Wavelet transform and tomography: Continuous and discrete approaches,” Wavelets Med. Biol., vol. 1, pp. 209–230. [8] F. Natterer, The Mathematics of Computerized Tomography. New York: Wiley, 1986. [9] A. Faridani, E. L. Ritman, and K. T. Smith, “Local tomography,” SIAM J. Appl. Math., vol. 52, pp. 459–484, Apr. 1992. [10] S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell., vol. 11, pp. 674–693, July 1989. [11] A. Karoui and R. Vaillancourt, “McClellan transformation and the construction of biorthogonal wavelete bases of L (R ),” CRM, Montréal, P.Q., Canada, Tech. Rep. 2222, Nov. 1994. [12] J. Kovaˇcevic´ and M. Vetterli, “Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R ,” IEEE Trans. Inform. Theory, vol. 38, pp. 533–555, Mar. 1992. [13] J. H. McClellan, “The design of two-dimensional digital filters by transformations,” in Proc. 7th Annu. Princeton Conf. Information Science Systems, 1973, pp. 247–251. [14] M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Processing, vol. 1, pp. 205–220, Apr. 1992.
I. INTRODUCTION Digital watermarking has recently received great attention as a tool for copyright protection. A great deal of research has been carried out mainly addressing the development of robust, yet unperceivable, watermarking strategies [1]. Other important issues, though, have to be investigated to make watermark-based copyright protection feasible. Among them, the evaluation of the maximum number of information bits that can be hidden within a piece of data of given size plays a mayor role. This problem is usually addressed by looking at the watermarking process as a communication task consisting of two main steps: watermark embedding, in which the signal, i.e., the watermark, is transmitted over the channel (hereafter the watermark-channel) the host data acts the part of, and watermark recovery, in which the signal is received and extracted from data. Thus the maximum number of information bits that can be hidden is the capacity of the watermark-channel. The importance of capacity evaluation stands in the fact that it represents an upper bound on the amount of information that can be transmitted through the channel (i.e., that can be hidden into the image) regardless of the particular decoder structure. In communication systems, capacity depends on the information signal power and on channel characteristics. Similarly, as we will see, for image watermarking capacity depends on the watermark (i.e., the information signal) strength, on image statistics, and on the way the watermark is embedded (these last two features representing the channel characteristics). A straightforward approach to watermark capacity evaluation consists in modeling the watermark-channel as an additive white Gaussian noise (AWGN) channel [2], [3], so that Shannon’s theorem on channel coding can be used. According to the AWGN model, the watermark is added to a given set of features extracted from the host data. In the case of image watermarking, such features can be pixel intensities, transformed-domain coefficients, block-DCT coefficients etc. If the AWGN model is adopted, the watermark capacity for each use of the channel,
Manuscript received May 7, 1999; revised February 15, 2000. This work was supported in part by the Italian Ministry of the University and the Scientific and Technological Research. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Naohisa Ohta. M. Barni is with the Department of Information Engineering, University of Siena, 53100-Siena, Italy (e-mail: [email protected]). F. Bartolini, A. De Rosa, and A. Piva are with the Department of Electronics and Telecommunications, University of Florence, 50139-Firenze, Italy (e-mail: [email protected]; [email protected]; [email protected]). Publisher Item Identifier S 1057-7149(00)06132-7.
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[5] A. Kumar, Y. B.-Shalom, and E. Oron, “Precision tracking based on segmentation with optimal layering for imaging sensors,” IEEE Trans. Pattern Anal. Machine Intell., vol. 17, pp. 182–188, Feb. 1995. [6] A. G. Bors¸ and I. Pitas, “Optical flow estimation and moving object segmentation based on median radial basis function network,” IEEE Trans. Image Processing, vol. 7, pp. 693–702, May 1998. [7] J. Konrad and E. Dubois, “Bayesian estimation of motion vector fields,” IEEE Trans. Pattern Anal. Machine Intell., vol. 14, pp. 910–927, Sept. 1992. [8] A. G. Bors¸ and I. Pitas, “Median radial basis function neural network,” IEEE Trans. Neural Networks, vol. 7, pp. 1351–1364, Nov. 1996. [9] V. Vapnik, Estimation of Dependences Based on Empirical Data. New York: Springer-Verlag, 1982. [10] B. Widrow and S. D. Stearns, Adaptive Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1985.
Tomographic Reconstruction Using Nonseparable Wavelets Fig. 11. PSNR of the predicted frame in the “Hamburg taxi” image sequence. “–” denotes the PSNR of the proposed tracking algorithm. “- -” represents the PSNR prediction considering the initial MRBF model on 4 4 pixel blocks. “-.” denotes the PSNR between the actual frame and that used for prediction.
2
Indy Workstation. The trained network, can be used for those successive frames which match the model according to a criterion [6]. In this case, 95 s are required for segmenting the moving objects and the optical flow for 20 frames when using 424 pixel blocks. When employing tracking as described in this study, only 68 s are necessary for the same frames using pixel resolution segmentation. In the first case only 3040 vectors had been processed while in the second case their number was 48 640. The segmentation provided by the tracking algorithm is quite good as it can be observed from the experimental results and provides a good basis for prediction-based frame reconstruction. The prediction PSNR of the tracking algorithm is better than when considering the initial MRBF model for segmenting all the frames and assuming just the previous moving object features for reconstruction, as it can be observed from Fig. 11. VI. CONCLUSION We propose a moving object tracking algorithm derived from the Bayesian theory. The optical flow and the segmentation features are jointly modeled by the MRBF network in the initial stage. The occluding and unlabeled regions are detected and classified appropriately. The proposed algorithm provides good moving object tracking capabilities. Such capabilities are used for segmenting and estimating the moving object velocity and segmentation in a future frame. The proposed algorithm is employed for frame prediction. REFERENCES [1] Y. Altunbasak and A. M. Tekalp, “Occlusion-adaptive, content-based mesh design and forward tracking,” IEEE Trans. Image Processing, vol. 6, pp. 1270–1280, Sept. 1997. [2] K. J. Bradshaw, I. D. Reid, and D. W. Murray, “The active recovery of 3-D motion trajectories and their use in prediction,” IEEE Trans. Pattern Anal. Machine Intell., vol. 19, pp. 219–224, Mar. 1997. [3] S. M. Smith and J. M. Brady, “ASSET-2: Real-time motion segmentation and shape tracking,” IEEE Trans. Pattern Anal. Machine Intell., vol. 17, pp. 814–820, Aug. 1995. [4] J. Weber and J. Malik, “Rigid body segmentation and shape description from dense optical flow under weak perspective,” IEEE Trans. Pattern Anal. Machine Intell., vol. 19, pp. 139–143, Feb. 1997.
Stéphane Bonnet, Françoise Peyrin, Francis Turjman, and Rémy Prost Abstract—In this paper, the use of nonseparable wavelets for tomographic reconstruction is investigated. Local tomography is also presented. The algorithm computes both the quincunx approximation and detail coefficients of a function from its projections. Simulation results showed that nonseparable wavelets provide a reconstruction improvement versus separable wavelets. Index Terms—Local tomography, McClellan transformation, nonseparable wavelets.
I. INTRODUCTION Computerized tomography (CT) consists of recovering a function from a set of its projections and relies on the inversion of the Radon transform. According to the nature of the data set, this problem may be ill-posed. The use of wavelets for inverse problems in general, and CT in particular, presents several interesting features to stabilize the inversion process [1]. As a matter of fact, wavelets may bring valuable solutions to the problem of local tomography [2]–[4]. The relationships between the continuous wavelet transform and the Radon transform have first been established in several independent works [5], [6]. Olson was the first to devise a reconstruction scheme from a customized sampling of the Radon transform [2]. Delaney [3] and Rashid-Farrokhi [4] proposed a multiresolution tomographic reconstruction algorithm to recover the two-dimensional (2-D) separable discrete wavelet transform (2-D DWT) of the image from its projections, and applied it to local tomography. Both algorithms are based on 2-D wavelets, constructed by tensor products of one-dimensional (1-D) wavelets. The 2-D separable wavelets impose a rectangular tiling of the Manuscript received December 28, 1998; revised February 27, 2000. S. Bonnet was supported by a grant from Siemens, France. This work is in the scope of the scientific topics of the PRC-GDR ISIS research group of the French National Center for Scientific Research (CNRS). The associate editor coordinating the review of this manuscript and approving it for publication was Prof. William Clem Karl. S. Bonnet and R. Prost are with CREATIS, CNRS Research Unit, 69621 Villeurbanne Cedex, France (e-mail: [email protected]). F. Peyrin is with CREATIS, CNRS Research Unit, 69621 Villeurbanne Cedex, France. She is also with ESRF, 38043 Grenoble Cedex, France. F. Turjman is with CREATIS, CNRS Research Unit, 69621 Villeurbanne Cedex, France. He is also with Hôpital Neurologique, 69500 Bron, France. Publisher Item Identifier S 1057-7149(00)06139-X.
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Let R the Radon operator of an image f
2 L2(IR2 ) be defined by
Rf (t; ) = R f (t) = p f (t) =
(x)(t 0 hx; 2i) dx:
f
IR
(1)
The filtered backprojection (FBP) algorithm relies on the inversion formula [8] f
= R# 3R f:
(2)
3 is the Lambda-operator used in the context of 3-tomography [9] and is defined by (3p)(! ) = j! j2^ p(! ), R# is the backprojection operator # given by R q (x) = 0 q (hx; 2i) d . II. NONSEPARABLE MULTIRESOLUTION ANALYSIS Multiresolution analysis (MRA) using separable or nonseparable wavelet basis may be expressed by a general formalism based on a 2 222 . This matrix imposes conditions on the dilation matrix sampling and scaling schemes. The most common dilation matrix is the quincunx matrix defined as: Q = (1 1; 1 0 1). A separable sublattice is given by a diagonal matrix such as: S = 2 . A 2-D MRA with dilation matrix is a generalization to L2 (IR2 ) of the 1-D MRA proposed by Mallat (see [10], [11]). In 2-D, a critically sampled filter bank that would achieve perfect reconstruction requires a sampling rate of N = jdet j. Consequently, a nonseparable biorthogonal MRA of 2-D signals requires a pair of dual scaling func~ g and (N 0 1) pairs of dual mother wavelets f9; 9~ g. The tions f8; 8 dilated and translated versions of 8 are defined according to by
D
Fig. 1. Diamond-shaped McClellan transformation applied on the “near-coiflet” wavelet: (a) scaling filter, (b) wavelet filter, (c) 2-D scaling filter, and (d) 2-D wavelet filter.
D D
I
D
D
8j;k (x) = jdet Dj0(j=2) 8(D0j x 0 k)
j
2
;
D k2
2:
(3) For a fixed scale j , the set f8j;k ; 2 2 g forms a non-orthogonal basis of the approximation space Vj of L2 (IR2 ) . Moreover, in a 2-D MRA, the scaling function 8 satisfies the two-scale equation [12]
k
8(x) =
k
Fig. 2. Nonseparable wavelet tomographic reconstruction from local projection data.
frequency plane, which is not well suited to the radial band-limited assumption of the image. In a preliminary work, we first proposed the idea of using nonseparable wavelets in tomography [7]. In this paper, we develop a framework for multiresolution tomography, which both includes the separable and nonseparable cases. This new proposal generalizes previous algorithms and provides the material for nonseparable multiresolution tomography. The application of this reconstruction scheme to 2-D wavelets generated via McClellan transformation is considered. This new approach allows to respect the geometry of the system by tiling the frequency plane in a diamond-shaped fashion that is more respectful to the radial band-limited assumptions. Let L2 (IRn ) denote the vector spaces of measurable, square-integrable n-D functions. The Fourier transform of a function f 2 L2 (IRn ) is defined by: f^(! ) = IR f ( ) e02ihx;!i d where boldfaced letters denote n-D vectors (i.e., in IR2 , = (x; y )t ) and h ; i is the usual inner product in IRn . Continuous and discrete convolution operators are both denoted by 3. The vector 2 = (cos ; sin )t is used to characterize the unit circle in IR2 .
x
x
x
k2
h
[k]8(Dx 0 k)
(4)
where h[ ] are the coefficients of a 2-D low-pass filter. i The set of wavelets f9j; k ; i = 1 1 1 1 (N 0 1)g forms a non orthogonal basis of Wj , the non-orthogonal complement of Vj in Vj 01 . Thus any function f 2 L2 (IR2 ) can be expanded in a nonseparable biorthonormal wavelet series as f
(x) =
k2
+
hf; 8J;ki 8~ J;k(x) N
01
=1 j J k2
i
hf; 9j;i k i9~ j;i k (x)
(5)
where J is the depth of the decomposition. The inner products hf; 8j;k i correspond to projections of f onto subspace Vj and represent the discrete approximation coefficients of f at resolution j . III. NONSEPARABLE MULTIRESOLUTION TOMOGRAPHIC RECONSRUCTION A. Nonseparable Wavelet-Based Inversion Formula In this section, we propose a general multiresolution inversion formula for tomography. The framework developed here will include the previous algorithms ([3] and [4]) since it will also be applied using a separable wavelet basis. The discrete approximation of f at resolution j can also be interpreted as a continuous convolution evaluated at point ( j ): j;0 )( j ) where 8 ( ) = 8(0 ). Aj f [ ] = hf; 8j;n i = (f 3 8
n
Dn
x
Dn x
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Fig. 3. (a) Nonseparable approximation coefficient, (b)–(c) nonseparable detail coefficients, (d) reconstructed image from the three subimages, (e) separable approximation coefficients, (f)–(h) separable detail coefficients, and (i) reconstructed image from the four subimages.
Applying inversion formula (2), the coefficients may be rewritten as
n] = (f 3 R#3R 8 j;0 )(Dj n):
(6)
n] = R#(R f 3 3R 8 j;0)(Dj n):
(7)
Aj f [
Due to the property [5]: f 3R# g as
= R# (R f 3g ), (6) can be restated
Aj f [
The approximation coefficients can then be evaluated by backprojecting on a given sublattice (defined by ) the convolution of the Radon transform of f with the ramp-filtered Radon transform of the
D
dilated reversed scaling function 8j;0 . Since the same result holds for the wavelet functions, this method allows the computation of the detail coefficients. This algorithm generalizes a wavelet-analogue of the FBP algorithm using nonseparable wavelet bases. A synthesis step is then necessary to recover the original image at a given resolution from these coefficients (usually the finest approximation is obtained for j = 0). Note that the separable case is obtained by considering = S and then 8j;0 ( ) = 2 (x) 2 (y ) with 2 (x) = 20j=2 (20j x). In Fourier space, the modified ramp-filter is derived from (7) as
D D
x
j j 2 2j (2j ! cos ) 2 (2j ! sin ):
h (! ) = ! k;j
(8)
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Fig. 4. (a) Nonseparable local approximation coefficients, (b) nonseparable local detail coefficients, (c) reconstructed region-of-interest from the two subimages, (d) zoom of the reconstructed region-of-interest, and (e) zoom of the original ROI.
B. Quincunx Multiresolution Tomographic Algorithm From here, we restrict ourselves to the choice of a quincunx dila~ ; 9; 9~ g be MRA biorthogonal wavelets with the tion matrix. Let f8; 8 ~ ; g; g~g defined as in (4). The discrete approximaassociated filters fh; h tion at resolution j can also be computed via the 2-D discrete filtering by Aj f
[n] = (A f 3 h j )[Dj n] ( )
0
n
n
(j )
[n] =
k2
[k] h j 0 [n 0 Dj0 k]:
h
(
1)
1
(10)
As in [3], it is straightforward to derive an inversion formula for discretely sampled data p [n] = R f (nTp ) to recover the discrete
[n] = p~h p~h ; [m] = (p 3 kh Aj f
(9)
where A0 f is the full-resolution discrete image. Considering h0 [ ] = [ ], the equivalent low-pass filter h(j ) after j iterations is given recursively by h
image A0 f [n1 ; n2 ] = f (n1 To ; n2 To ). Here, Tp denotes the radial spacing of the projection and To the sampling rate of the image. Using discrete time Fourier transform (DTFT) and following the same scheme as (6) and (7), we obtain
0
;
To Tp
)[ ]
hDj n; 2i
d
; m :
(11) (12)
The discrete filtering (12) is defined in Fourier space by Kh
( ) = TTpo j!j 2 H j j!j 12
; !
( )
To ! Tp
cos ; TTpo ! sin ; (13)
where H (j ) (resp. Kh ; ) denotes the discrete 2-D Fourier transform of the low-pass filter h(j ) (resp. kh ; ). This filtering step is similar
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to the standard FBP’s one, but the discrete filters kh ; include the wavelet and scaling filters. In addition, if radial filters are used, the angular dependency, which is compulsory with separable bases, is eliminated. One possible way in that direction is to generate nearly radial symmetric 2-D filters using McClellan transformation [13]. In the case of quincunx decimation, the transformation has to preserve the perfect reconstruction properties from a pair of 1-D biorthogonal filters [11]. This transformation results in 2-D nonseparable diamond-shaped filters. Fig. 1 illustrates this transformation applied to the “near-coiflet” scaling and wavelet filter. Moreover as we are using 2-D zero-phase filters, DTFT is purely real and from symmetry considerations, the filtering operation cost can be further reduced. C. Local Tomography The utility of wavelets in local tomography is motivated by the observation that 1-D wavelet functions with sufficient vanishing moments remain essentially local after ramp-filtering [2]. Fortunately, the chosen diamond McClellan transformation also preserves zeros at aliasing frequencies [12]. Thus, quincunx wavelets keep identically their good properties of space-frequency localization after ramp-filtering and so they are able to localize the Radon transform as in the separable case [4]. Nevertheless, increasing the number of vanishing moments enlarges the support of the filter and reduces the efficiency of the localization. Under those assumptions, the support of the modified scaling ramp filter kh ; will essentially have the same radius of support as the 2-D scaling filter h(j ) . A schematic diagram for local tomography is presented in Fig. 2 with a decomposition over one level.
Fig. 5.
Energy error in the ROI (16 pixels) versus nonlocal data. TABLE I PPSNR
OF THE RECONSTRUCTED DIFFERENT WAVELETS
ROI
FOR
IV. RESULTS Simulations were carried out with the Shepp-Logan brain phantom. The reconstructions are calculated on a 256 2 256 pixel grid using a set of 256 projections uniformly spaced over [0; [ with 256 samples per projections. We have used the “near-coiflet” wavelet with five taps ([14, Table III]) because of its good property of localization of the Radon transform, demonstrated in [4]. We applied the algorithm presented in Section III-B using two levels of decomposition. Fig. 3(a)–(c) shows the quincunx approximation A2 f and detail coefficients D2 f; D1 f reconstructed from global Radon transform data using (12). The quality of the reconstructed image, shown in Fig. 3(d) is evaluated by measuring the mean square error (MSE) with the image obtained with FBP, considered as the reference. The MSE is equal to 3:12 2 1003 . To compare our approach with the separable case, we applied the separable algorithm for one level of decomposition with the same choice of 1-D filter. The approximation coefficients are shown in Fig. 3(e) and the three detail coefficients in Fig. 3(f)–(h). The quality of the reconstructed image in Fig. 3(i) is visually the same (both images Fig. 3(d)–(i) are displayed in the range [00:2 1:2]) but the MSE in the separable case has slightly increased to 4:34 2 1003 . An example of local tomographic reconstruction is presented on Fig. 4. The region-of-interest is a centered disk of radius 32 pixels. We used the Radon transform data passing through this ROI in addition with an extra margin of 16 pixels to recover the high-frequency component [Fig. 4(b)]. We used a constant extrapolation of this region-of-exposure (ROE) to obtain the low-frequency component shown in Fig. 4(a). As a matter of fact, this extrapolation allows to reduce oscillations at the edge of the ROI [8]. Fig. 4(c) shows the ROI reconstructed from the approximation and detail images using our nonseparable algorithm. A zoom of the ROI is presented in Fig. 4(d) in comparison to the original ROI in Fig. 4(e). The quality of reconstruction is again very good and the radiation exposure is around 37% of the standard FBP. The reconstructed image corresponds
actually to the addition of the original phantom and a constant bias. This phenomenon arises naturally in local tomography and has already been well-studied in [8] and [4]. In local reconstruction, the MSE between the ROI reconstructed by our algorithm using only local projections and the one obtained using the standard FBP with global projections is calculated after diminution of the constant bias as MSE =
1
N n2ROI
c (fFBP (n)
0 fc
n))2
NSM (
(14)
where N is the number of pixels in the ROI, f c denotes the zero-mean function f and FBP or nonseparable multiresolution (NSM) refers to the reconstruction method. We have also measured the peak-to-peak signal-to-noise ratio (PPSNR) defined as: 20 log10 dyn= (MSE) where dyn is the dynamic range in the ROI reconstructed by FBP. Fig. 5 shows the plot of the PPSNR versus the amount of non local data, collected to recover the centered ROI. This amount is equal to the difference between the ROE and the ROI in terms of pixels. As expected, the energy error in the ROI is reduced when we increase the region-of-interest and consequently the amount of nonlocal data. Finally, we compared the quality of local reconstructions according to the PPSNR obtained using the nonseparable algorithm with the one achieved by the separable algorithm of [4]. Simulations were performed over the same ROI (radius of 16 pixels) and the same ROE (margin of 12 pixels) in both cases with different choice of wavelets. Moreover, to demonstrate the importance of the number of vanishing moments of , we exchanged for each given wavelet, the role of the ~ g. The filter pairs that have been studied are the spline filter pairs fh; h filters “bior2.4” ([14, Table I]), the spline variant with less dissimilar lengths “bior4.4” ([14, Table II]) and the “near-coiflet.” The examination of Table I first shows that the quality of local reconstruction
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is always better when the nonseparable algorithm is used (around 1 dB) compared with the separable approach. Nevertheless, the quality is high in both cases for any kinds of wavelets. Secondly, we notice that the case where the analyzing wavelet has more vanishing moments performs best. Indeed, if the pair of spline filters “bior2.4” is chosen (i.e and ~ have respectively 2 and 4 vanishing moments), our reconstruction algorithm leads to a PPSNR of 38.4 dB while exchanging the filters order leads to a better PPSNR of 40.12 dB. Finally, a similar study with the filter “bior4.4” ( and ~ have both four vanishing moments), shows that the case where is more regular performs best. These results are in agreement with the requirement imposed on wavelets in Section III-C.
Capacity of Full Frame DCT Image Watermarks Mauro Barni, Franco Bartolini, Alessia De Rosa, and Alessandro Piva
Abstract—The evaluation of the number of bits that can be hidden within an image through digital watermarking is a crucial topic, which has been addressed only for additive watermarks. The evaluation of watermark capacity is very important because it allows to put a theoretical upper bound on the amount of information that can be hidden into an image by a given watermarking procedure, regardless of the watermark extraction technique. It is the purpose of this work to suggest a methodology for the evaluation of the watermark capacity in a nonadditive, non-Gaussian framework, and to discuss the results we obtained by applying it to a set of standard images. Index Terms—Nonadditive DCT watermarks, watermark channel capacity, .
V. CONCLUSION In this paper, we have presented a general framework for multiresolution tomographic reconstruction. This approach allows to recover the nonseparable wavelet and scaling coefficients of a function from its projections. The separable case can be obtained for a particular choice of the dilation matrix . McClellan transformation has been used to construct a 2-D multiresolution analysis and its application in the field of tomography has been described. Furthermore, local tomography using these nonseparable bases is possible and comparisons with the separable approach shows a slight improvement in terms of PPSNR.
D
REFERENCES [1] A. Aldroubi and M. Unser, Wavelets in Medicine and Biology. Boca Raton: CRC, 1996. [2] T. Olson and J. DeStefano, “Wavelet localization of the radon transform,” IEEE Trans. Signal Processing, vol. 42, pp. 2055–2067, Aug. 1994. [3] A. H. Delaney and Y. Bresler, “Multiresolution tomographic reconstruction using wavelets,” IEEE Trans. Image Processing, vol. 4, pp. 799–813, June 1995. [4] F. Rashid-Farrokhi, K. J. R. Liu, C. A. Berenstein, and D. Walnut, “Wavelet-based multiresolution local tomography,” IEEE Trans. Image Processing, vol. 6, pp. 1412–1430, Oct. 1997. [5] C. A. Berenstein and D. Walnut, “Local inversion of the radon transform in even dimensions using waveletes,” in 75 Years of the Radon Transform. Cambridge, MA: International, 1994, pp. 38–58. [6] F. Peyrin, M. Zaim, and R. Goutte, “Construction of wavelet decompositions for tomographic images,” J. Math. Imag. Vis., vol. 3, pp. 105–121, 1993. [7] F. Peyrin and M. Zaim, “Wavelet transform and tomography: Continuous and discrete approaches,” Wavelets Med. Biol., vol. 1, pp. 209–230. [8] F. Natterer, The Mathematics of Computerized Tomography. New York: Wiley, 1986. [9] A. Faridani, E. L. Ritman, and K. T. Smith, “Local tomography,” SIAM J. Appl. Math., vol. 52, pp. 459–484, Apr. 1992. [10] S. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell., vol. 11, pp. 674–693, July 1989. [11] A. Karoui and R. Vaillancourt, “McClellan transformation and the construction of biorthogonal wavelete bases of L (R ),” CRM, Montréal, P.Q., Canada, Tech. Rep. 2222, Nov. 1994. [12] J. Kovaˇcevic´ and M. Vetterli, “Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R ,” IEEE Trans. Inform. Theory, vol. 38, pp. 533–555, Mar. 1992. [13] J. H. McClellan, “The design of two-dimensional digital filters by transformations,” in Proc. 7th Annu. Princeton Conf. Information Science Systems, 1973, pp. 247–251. [14] M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. Image Processing, vol. 1, pp. 205–220, Apr. 1992.
I. INTRODUCTION Digital watermarking has recently received great attention as a tool for copyright protection. A great deal of research has been carried out mainly addressing the development of robust, yet unperceivable, watermarking strategies [1]. Other important issues, though, have to be investigated to make watermark-based copyright protection feasible. Among them, the evaluation of the maximum number of information bits that can be hidden within a piece of data of given size plays a mayor role. This problem is usually addressed by looking at the watermarking process as a communication task consisting of two main steps: watermark embedding, in which the signal, i.e., the watermark, is transmitted over the channel (hereafter the watermark-channel) the host data acts the part of, and watermark recovery, in which the signal is received and extracted from data. Thus the maximum number of information bits that can be hidden is the capacity of the watermark-channel. The importance of capacity evaluation stands in the fact that it represents an upper bound on the amount of information that can be transmitted through the channel (i.e., that can be hidden into the image) regardless of the particular decoder structure. In communication systems, capacity depends on the information signal power and on channel characteristics. Similarly, as we will see, for image watermarking capacity depends on the watermark (i.e., the information signal) strength, on image statistics, and on the way the watermark is embedded (these last two features representing the channel characteristics). A straightforward approach to watermark capacity evaluation consists in modeling the watermark-channel as an additive white Gaussian noise (AWGN) channel [2], [3], so that Shannon’s theorem on channel coding can be used. According to the AWGN model, the watermark is added to a given set of features extracted from the host data. In the case of image watermarking, such features can be pixel intensities, transformed-domain coefficients, block-DCT coefficients etc. If the AWGN model is adopted, the watermark capacity for each use of the channel,
Manuscript received May 7, 1999; revised February 15, 2000. This work was supported in part by the Italian Ministry of the University and the Scientific and Technological Research. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Naohisa Ohta. M. Barni is with the Department of Information Engineering, University of Siena, 53100-Siena, Italy (e-mail: [email protected]). F. Bartolini, A. De Rosa, and A. Piva are with the Department of Electronics and Telecommunications, University of Florence, 50139-Firenze, Italy (e-mail: [email protected]; [email protected]; [email protected]). Publisher Item Identifier S 1057-7149(00)06132-7.
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