Conjugate-Gradient Preconditioning Methods for Shift-Variant PET Image Reconstruction Abstract-Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e. for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method. Tomographic image reconstruction using statistical methods can provide more accurate system models, statistical models, and physical constraints than the conventional filtered backprojection (FBP) method. Most statistical methods for image reconstruction require minimizing an objective function related to the measurement statistics. For realistic image sizes, direct minimization methods are computationally intractable, so iterative methods are required. For objective functions that are quadratic, or at least convex and locally quadratic, conjugate-gradient (CG) algorithms are appealing for reasons of convergence rate, simplicity, and potential for parallelization [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] . This paper describes new preconditioning methods that accelerate the convergence of gradient-based iterative methods for penalized weighted least-squares tomographic image reconstruction. The proposed methods generalize those described in [8, 11] . The methods also apply to other inverse problems such as image restoration.. Gradient-based iterations often converge slowly for poorly-conditioned problems and for large-scale problems where the Newton-Raphson method is impractical due to Scott D. Booth was with the Dept. of Bioengineering, The University of Michigan, Ann Arbor, MI. He is now with the University of Virginia Medical Center, Charlottesville, VA 22906. This work was supported in part by NIH grants CA-60711 and CA-54362 and by the Whitaker Foundation.. the size of the Hessian matrix. Broadly speaking, the goal when preconditioning is to induce a coordinate transformation that improves the condition number of a problem, which generally leads to faster convergence. Several "generic" preconditioners for CG methods are described in textbooks, e.g. [12] . The simplest and perhaps most prevalent preconditioners are diagonal scaling matrices [12] , versions of which have been applied to image reconstruction 1 [4, [6] [7] [8] 14] . Although diagonal preconditioners can accelerate convergence in many optimization problems, these preconditioners cannot provide the fastest convergence rate for imaging problems since they ignore the off-diagonal structure of the Hessian of the objective function (the correlation between pixels and the 1=r response of tomographic systems).. Some simple (or perhaps over-simplified) imaging problems are nearly shift-invariant, i.e., the matrix 2 G 0 G, which is part of the Hessian of the objective function, is approximately block-Toeplitz, where G is a system matrix described in (1) below. For these problems the diagonal preconditioner is ineffective, but appropriate circulant preconditioners can provide very remarkable improvements in convergence rate. (See [15] for a recent thorough review of this subject.) Circulant preconditioners are particularly appealing since one can use the fast-Fourier transform (FFT) for efficient implementation. Several optimal circulant preconditioners are available for Toeplitz problems [15] [16] [17] [18] [19] . Such circulant preconditioners, also called "Fourier" preconditioners, have been applied to both tomographic image reconstruction [20] and image restoration problems [21, 22] . Figure 3 , described in Section V below, illustrates the well-known effectiveness of circulant preconditioners for shift-invariant problems. Circu-1 Curiously, the most prevalent preconditioner in the emission tomography literature is not the usual diagonal matrix (8) formed from the Hessian, but is rather the particular estimate-dependent diagonal matrix that is implicit in the popular expectation-maximization (EM) algorithm. (For emission tomography the EM algorithm is equivalent to a gradient-ascent iteration with a certain estimate-dependent diagonal preconditioner [13] .) Kaufman [4] and Mumcuoglu et al. [6] incorporated this diagonal preconditioner into conjugategradient algorithms. The estimate-dependence of this preconditioner affects the conjugacy of the direction vectors, leading Lalush et al. to advocate an iterationindependent diagonal preconditioner [3, 14] . lant preconditioners have also been applied to total variation methods for nonlinear image restoration [23, 24] .. Unfortunately, many imaging problems are shift variant, for the following reasons. Firstly, many imaging systems produce heteroscedastic measurements, particularly in quantum-limited applications such as emission tomography and photon-limited optical imaging [25, 26] . In these applications, the measurement noise covariance is a diagonal matrix with very nonuniform entries, due to both nonuniform Poisson variance and to physical effects such as detector efficiency and attenuation [27] . Therefore the Fisher information term G 0 WG within the Hessian (see (5) and Fig. 1 below) is shift variant and thus poorly approximated by any circulant preconditioner. Secondly, incorporating a nonquadratic edge-preserving penalty function into the objective function to perform regularization introduces an additional form of shift variance into the Hessian (5). Finally, for some imaging systems (e.g. SPECT, 3D PET, and helical CT), even if we were to disregard the nonuniform noise variance, the matrix G 0 G is still inherently shift-variant due to the system geometry and/or spatial variations in detector response. (See [28] for an image restoration method for shift-variant imaging systems.) Thus, neither diagonal nor circulant preconditioning is well-suited to shift-variant imaging problems. Since statistical methods for image reconstruction yield higherquality images than FBP reconstruction but at a price of increased computation, it is important to develop methods for accelerating convergence of the iterative algorithms. This paper generalizes the quadratic method described in [8] by developing improved preconditioners that accommodate the shift-variance caused by nonuniform noise and nonquadratic penalties.. Section II reviews the image reconstruction problem and the preconditioned conjugate gradient iteration. Section III summarizes the preconditioners. Section IV describes a new method for the CG line-search step. Section V reports empirical results that demonstrate significantly improved convergence rates in PET reconstruction with real data. We have described new conjugate-gradient preconditioning methods for shift-variant imaging problems, and have presented representative results that demonstrate that the proposed preconditioners lead to significantly improved convergence rates over previous diagonal and circulant preconditioners for PET image reconstruction.. The development of the preconditioners used several heuristics about the properties of the Hessian for image reconstruction problems. Although the results show that the methods work, the derivation is somewhat less theoretically satisfying than the optimality results known for circulant preconditioners for Toeplitz problems [15] [16] [17] [18] [19] . Whether such results can be extended to the more complicated structure of shift-variant problems remains an inviting challenge.. Circulant-based methods are perhaps inherently poorly suited to shift-variant problems. Our preconditioner partially circumvents this drawback by essentially using separate Fourier transforms for each pixel (i.e. for each column of the Hessian), which is then simplified through the interpolation approximation (29) . A more direct approach might begin with bases that better accommodate shift-variance, such as wavelets.. Although this paper focuses on image reconstruction, the preconditioning methods are also useful for calculating analytical metrics related to imaging system performance, such as bias, variance, and spatial resolution [30, 52, 55] . Many of these calculations require iterative methods for solving large-scale linear systems of equations involving the Hessian of the objective function.. Although the preconditioner development was presented for 2D imaging, the entire argument also applies to higher dimensions, such as 3D PET imaging, or "2.5D" imaging, in which each image in a stack is imaged separately but a 3D penalty is used during reconstruction to provide regularization both within and between planes. We expect these preconditioners will be particularly useful for fully 3D PET [10] . The inherent shift-variance of G 0 G for such systems will add to the challenge. For 3D SPECT reconstruction, one may be able to form suitable preconditioners by adapting linear methods such as in [56] . We plan to investigate such methods in the near future. Normalized l 2 distance to solution x 1 versus iteration n for the quadratically-penalized unweighted least squares (QPULS) objective function. Shown is CG with no preconditioning, with the diagonal preconditioner (8) , and with the circulant preconditioner (12) . In the QPULS case, the Hessian is nearly block-circulant, so the circulant preconditioner provides remarkable acceleration. Normalized l 2 distance to solution x 1 versus iteration n for the nonquadratically-penalized weighted least squares (NPWLS) objective function. Shown is CG with no preconditioning, with the diagonal preconditioner (8) , with the circulant preconditioner (12) , and with the proposed M 9 preconditioner (30). The proposed preconditioner provides significant acceleration in convergence rate over conventional preconditioners.

O O O O O B O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


STEERABILITY OF HERMITE KERNEL Steerability is a useful and important property of "kernel" functions. It enables certain complicated operations involving orientation manipulation on images to be executed with high e±-ciency. Thus, we focus our attention on the steerability of Hermite polynomials and their versions modulated by the Gaussian function with di®erent powers, de¯ned as the Hermite kernel. Certain special cases of such kernel, Hermite polynomials, Hermite functions and Gaussian derivatives are discussed in detail. Correspondingly, these cases demonstrate that the Hermite kernel is a powerful and e®ective tool for image processing. Furthermore, the steerability of the Hermite kernel is proved with the help of a property of Hermite polynomials revealing the rule concerning the product of two Hermite polynomials after coordination rotation. Consequently, any order of the Hermite kernel inherits steerability. Moreover, a couple sets of an explicit interpolation function and basis function can be directly obtained. We provide some examples to verify steerability of the Hermite kernel. Experimental results show the e®ectiveness of steerability and its potential applications in the¯elds of image processing and computer vision. Since the steerable¯lters were¯rst proposed by Freeman and Adelson in 1991, they became widely utilized for image processing. 9, 14, 22, 23, 28, 30 Steerable¯lters are characterized as a class of¯lters in which a¯lter of arbitrary orientation is synthesized as a linear combination of a set of \basis¯lters". A mathematical explanation of steerable¯lter is presented in Eq. (1). where k j ðÞ and g j ðr; Þ serve as interpolation functions and basis functions, respectively. Steerable¯lters allow e®ective and simple practical implementations of deformable¯lters.. Historically, the idea of steerable¯lters can be found in the work of Danielsson and Knutsson without explicitly mentioning its conception. 4, 10 It was Freeman and Adelson who coined the concept of \steerability". They introduced steerable¯lters based on Gaussian derivatives and concluded that a product of a polynomial and a rotationally symmetric window function results in a steerable function. 9 Together with steerability in orientation, Simoncelli et al. also investigated steerability in translation and scale. The terminology \shiftability" is proposed for steerability in di®erent parameters resulting in various deformations. Simoncelli and his colleagues also addressed joint shiftability, covering steerability in both orientation and position, as well as in both scale and position simultaneously. 22 A class of steerable wedgē lters were designed by Simoncelli and Farid. Ambiguity on the orientation map was e±ciently eliminated by using steerable wedge¯lters. 24 The Lie-group theory is used to develop steerable¯lters as well. Typically, steerable¯lters were designed by Fourier decompositions, e®ectively generalized by Michaelis and Sommer with the Lie-group theory. 13 Teo and Hel-Or based handling of problems related to computational aspects of steerable functions on the Lie-group theory. 29 Comparatively, Beil proposed a method to derive a minimum number of basis functions based on Cartesian tensor calculus. Rotation invariants were expressed by tensor calculus. These invariant components thus enable construction of a steerable function. 3 More recently, the design of multi-steerable¯lters was formulated by Muhlich et al. They demonstrated that the multi-steerable¯lters can be constructed by a correct combination of single steerable¯lters. The polynomial techniques for generating multisteerable¯lters and some applications such as junction analysis were also given. 14 Approximation of a given deformed function by basis functions is investigated in relationship with steerable¯lters. Perona de¯ned the quality of approximation and proposed a technique useful for¯nding the best basis functions to approximate the given function within a prede¯ned error or to exactly represent the function. 16 , 17 Yu et al. proposed approximate steerability which decomposes the signal locally in the spatial domain with Gaussian masks as basis¯lters. Their method yields both higher orientation resolution and lower complexity. 33 On the other hand, Hermite polynomials and their weighted versions are powerful tools and thus are widely used in image processing and computer vision. Here, we call Hermite polynomials and their weighted versions the \Hermite kernel" for convenience. Theoretically, the kernel function is rigorously de¯ned in relation to the inner product and Hilbert space. Some kernel functions such as the linear kernel, polynomial kernel and Gaussian kernel are frequently used in practice. For an instance, support vector machines are one of the most well-known algorithms based on the kernel functions. 27 The Hermite kernel is a kind of \pseudo" kernel function according to the de¯nition. However, we can still¯nd wide applications of the Hermite kernel.. Certain terminologies such as \Hermite transform", 12 \Shapelets transform", 20, 11 \Hermite moments", 26 and \GaussianÀHermite moments" 25, 31, 32 are all based on the Hermite kernel. In this paper we o®er a solution of the steerability of the Hermite kernel. We are capable of decomposing a rotated kernel function of any order by the speci¯c interpolation functions and basis functions, no matter what the kernel represents: a Hermite polynomial, a Hermite function, a Gaussian derivative or any other Hermite polynomial based function. Freeman and Adelson formulated a method to compute interpolation functions of the pth partial derivative of the Gaussian function G p;0 . Comparatively, our method o®ers computation of interpolation functions for any Hermite kernel. It is not necessary for one index (either p or q) to equal 0, nor does the kernel have to be con¯ned to Gaussian derivatives. Actually, the kernel can be either Hermite polynomials, Hermite functions, or even the Hermite polynomial-related versions in the 2D space.. The paper is organized as follows: we present the construction of a steerable function and the de¯nition of the Hermite kernel in Sec. 2. Further, we discuss special cases of the Hermite kernel, which correspond to Hermite polynomials, Hermite functions and Gaussian derivatives, respectively. The derivations of the interpolation and basis functions are formulated in Sec. 3. Subsequently in Sec. 4, we provide applications of such steerability in image processing. Section 5 concludes the paper. In this paper, we de¯ne the Hermite kernel as Hermite polynomials modulated by the Gaussian envelope of di®erent powers. Both Hermite functions and Gaussian

B O B I B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B I O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O B O O B I O O O O O O O O O B I O O O O O O O O B I O O O O O O O O B I O B O O O O O O O O O O O O O O O O O O O O O O B O O B I O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O


A Low-complexity Region-based Video Coder Using Backward Morphological Motion Field Segmentation We introduce a novel region-based video compression framework based on morphology to e ciently capture motion correspondences between consecutive frames in an image sequence. Our coder is built on the observation that the motion eld associated with typical image sequences can be segmented into component motion sub eld \clusters" associated with distinct objects or regions in the scene, and further that these clusters can be e ciently captured using morphological operators in a \backward" framework that avoids the need to send region shape information. Region segmentation is performed directly on the motion eld by introducing a small \core" for each cluster that captures the essential features of the cluster and reliably represents its motion behavior. Cluster matching is used in lieu of the conventional block matching methods of standard video coders to de ne a cluster motion representation paradigm. Furthermore, a region-based pel-recursive approach is applied to nd the re nement motion eld for each cluster and the cluster motion prediction error image is coded by a novel adaptive scalar quantization method. Experimental results reveal a 10-20% reduction in prediction error energy and 1-3 dB gain in nal reconstructed PSNR over the standard MPEG-1 coder at typical bitrates of 500 kbits/sec to 1 Mb/sec on standard test sequences, while also requiring lower computational complexity. We have introduced a region-based video compression framework that uses morphological operation to e ciently capture motion correspondences between consecutive frames in an image sequence. Our coder is built on the observation that the motion eld can be segmented into component motion sub eld \clusters" associated with distinct regions. Further, these clusters can be e ciently captured using morphological operations in a \backward" framework that avoids the need to explicitly specify object boundaries. Region segmentation is performed directly on the motion eld by introducing a small \core" for each cluster that characterizes the essential features of the cluster and reliably represents its motion behavior. A motion vector estimated from the cluster \core" is then used to guide the cluster growing. Cluster matching is used in place of block matching to de ne the cluster motion representation paradigm. The clusters thus obtained are then merged and pruned by a R-D criterion, and are propagated and modi ed across frames to take advantage of the temporal consistency of object shapes. Furthermore, a region-based pel-recursive approach is presented to nd a re nement motion vector for each cluster and the cluster motion prediction error signal is coded by a novel adaptive scalar quantization method. Experimental coding results reveal performance gains in both motion compensated prediction error energy and nal residue-coded PSNR using our proposed algorithm on standard video test sequences over the MPEG-1 standard, while also requiring less computational complexity. Susceptibility to quantization noise as a result of the backward nature of our motion compensation framework can possibly cause loss of performance. We propose to attack this problem as our future research, by considering a combination of forward motion compensation and morphological motion eld segmentation, and addresssing the issues related to occlusion, uncovering and global motion. respectively. The associated mean squared variance were calculated to be 0.531, 0.132, and 0.0545 respectively. Note that a half-pel search was performed and the statistics was generated for both x and y components. The set of pixels used for estimating a motion re nement of pixel a 2 C when a 7x7 window is used. Here both the black and gray dots stand for pixels belonging to C, while black dots represent the pixels that have been identi ed to be included in C earlier than a in the cluster growing process de ned in Section 3. Figure 12 : A comparison between the performance of our region-based transformless residue coding algorithm and that of standard DCT residue coding. Here the nal coded PSNR values for Football sequence at 1Mb/s are plotted against frame number. The solid line stands for the performance of Algorithm 2 with a GOP size of 10, while the dotted line represents that of an algorithm that combines our region-based motion compensation and standard DCT residue coding. In other words, the algorithms that generate the two curves are identical in the region-based motion estimation & compensation part, but di er in the residue coding part.

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Parametrizing Del Pezzo surfaces of degree 8 using Lie algebras For a Del Pezzo surface of degree 8 given over the rationals we decide whether there is a rational parametrization of the surface and construct one in the affirmative case. We define and use the Lie algebra of the surface to reach the aim. The algorithm has been implemented in Magma. In this paper we decide whether a given Del Pezzo surface of degree 8 has a rational parametrization over Q, and find one in the affirmative case. There are two kinds of Del Pezzo surfaces of degree 8: blowups of P 2 in one point and twists of P 1 × P 1 . Here we deal with both of them. The problem is a particular case of a more general one, namely parametrization of surfaces over the rational numbers. There one reduces to several base cases. Except some trivial cases (e.g. P 2 ), there are Del Pezzo surfaces of degrees 5 till 9, and conic fibrations. The latter are solved in [10] . Rational parametrization of Del Pezzo surfaces of degree 5 is discussed in [11] , and of degree 7 can be found in e.g. [7] . Parametrization of degree 9 is solved in [3] . Hence the last unsolved class are Del Pezzo surfaces of degree 6.. For Del Pezzo surfaces of degree 6 the Hasse principle holds (cf [7] ). For a given prime p we can decide (e.g. using Magma [1]) whether there is a point on the surface over the local field Q p . But there still remains the problem of finding a finite set of "bad primes" and finding a rational point provided that we have points over local fields.. Here we use another approach. Similarly as in [3] , the parametrization problem is reduced to a problem concerning Lie algebras and their representations. The Lie algebra approach appears to be preferable even in situations where also other methods are known, such as the parametrization problem of blowups of the plane in one point, see [7] .. The paper is structured as follows. In section 2, we introduce the Lie algebra of a variety and give an algorithm for computing it. Section 3 is dealing with "identification problems"; the main focus will be to reduce the problem of identifying a variety (i.e. constructing an isomorphism if exists) to the problem of identifying a Lie algebra. This will solve some instances of our parametrization problem of Del Pezzo surfaces of degree 8. The remaining instances are dealt with in section 4.. The second and third author were supported by SFB Grant F1303 of the Austrian Science Fund (FWF).

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A comparison of two entropy stable discontinuous Galerkin spectral element approximations for the shallow water equations with non-constant topography In this work, we compare and contrast two provably entropy stable and high-order accurate nodal discontinuous Galerkin spectral element methods applied to the one dimensional shallow water equations for problems with non-constant bottom topography. Of particular importance for numerical approximations of the shallow water equations is the well-balanced property. The well-balanced property is an attribute that a numerical approximation can preserve a steady-state solution of constant water height in the presence of a bottom topography. Numerical tests are performed to explore similarities and differences in the two high-order schemes. Fluid flows in lakes, rivers, and near oceanic shores are of great interest in hydrology, oceanography and climate modeling. Common to all of these flows is the fact that vertical scales of motion are much smaller than the horizontal scales. By this and the assumption of hydrostatic balance [30] , the incompressible Navier-Stokes equations can be simplified to the so-called shallow water equations, which in one dimension are (1.1). Here, h is the height of the water and v is the velocity. The constant g is the acceleration due to gravity and the function b = b(x) represents the bottom topography of the surface over which the fluid flows. The shallow water system with topography (1.1) amounts to a system of balance laws. If the bottom topography is flat, i.e. b ≡ constant, then Eq. (1.1) reduces to the standard shallow water equations without topography, which is a strictly hyperbolic system of conservation laws,. It is well-known that solutions of the conservation law (1.2) as well as solutions of the balance law (1.1), may develop shock discontinuities in finite time, independent of the smoothness of the Remark 1. (Nonlinear Correction Terms) For numerical schemes constructed from a skew-symmetric form it is not obvious that the approximation conserves momentum discretely. However, from the Lax-Wendroff theorem, conservation is a fundamental requirement to correctly predict shock speeds. Fisher et al. [10] proved important properties for diagonal norm summation-by-parts (SBP) operators, such as the discontinuous Galerkin spectral element method with Gauss-Lobatto quadrature [13, 14] . If we subtract the discrete flux (2.2) from (2.1), we obtain an error in the associated product rule (2.3). In [14] , it was shown that the discrete mean value of terms with the structure of (2.3) is exactly zero. In this work we compare the provable stability and well-balancedness of two entropy conserving, high-order DG approximations for the shallow water equations. We found that the that the base entropy conservative algorithm of Carpenter et al. applied to the shallow water equations with entropy conserving/stable numerical fluxes is remarkably robust despite large oscillations at flow discontinuities. Also, when entropy stabilization was included at interior nodes the amount of dissipation added appeared optimal, in an entropy preserving sense. As we found in numerical experiments the entropy stabilized skew-symmetric scheme was slightly more dissipative and widened shocks more so than the algorithm of Carpenter et al.. Unfortunately, the procedure of Carpenter et al. is not well-balanced. It was found that for a certain moving water steady state equilibrium the lack of well-balancedness was insignificant. The ESDGSEM and SSSEC-DG schemes could recover the moving water steady state to machine precision. However, we found in numerical examples that the lack of well-balancedness of the SSSEC-DG scheme leads to significant errors in quasi-steady state computations. Also, for demanding test cases with low water height we found that the lack of well-balancedness lead to unstable behavior for the SSSEC-DG scheme. We note that the method was not designed for the solution of the shallow water equations, so a lack of well-balancedness is not surprising. Through numerical experimentation we found that a scheme that fails to maintain the "lake at rest" condition generates spurious waves near the bottom topography which propagate through the domain. The correction term found by the authors to maintain well-balancedness causes the loss of provable stability for the scheme. It is an open question of can a simple correction term be found which also maintains the provable entropy stability.. Inspired by the work of Carpenter et al., we explored stability for a skew-symmetric, entropy conserving scheme designed for the shallow water equations. We showed provable entropy stability, but with the additional feature of well-balancedness for an arbitrary, continuous bottom topography. We applied this scheme to several known, challenging quasi-steady state and shock problems to demonstrate the scheme's robustness and ability to model complex phenomena in the shallow water equations.

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Toward a testbed for evaluating computational trust models: experiments and analysis We propose a generic testbed for evaluating social trust models and we show how existing models can fit our tesbed. To showcase the flexibility of our design, we implemented a prototype and evaluated three trust algorithms, namely EigenTrust, PeerTrust and Appleseed, for their vulnerabilites to attacks and compliance to various trust properties. For example, we were able to exhibit discrepancies between EigenTrust and PeerTrust, as well as trade-offs between resistance to slandering attacks versus self-promotion. Trust is a tool used in the decision-making process and it can be computed. There are many models based on social trust that attempt to aid agents in making rational decisions. However, these models vary in terms of their input and output requirements. This makes evaluations against a common set of attacks difficult. We provided details of our prototype and evaluated trust algorithms for vulnerabilities to attacks and adherence to trust properties. We were able to show that our model could indeed accommodate various algorithms, and that the specification of experiments in terms of the model was straightforward and easy to express programmatically. The experiments that we ran allowed us to make the following observations:. • We were able to exhibit discrepancies between the outputs of EigenTrust and PeerTrust due to the way the initial normalized local trust values are calculated in EigenTrust; EigenTrust ignores the excess positive feedbacks once they outnumber negative feedbacks; • There are tradeoffs between sensitivity to self-promotion and sensitivity to slandering. For example, an algorithm that is overly sensitive to slandering might take too long to incorporate bad feedback, and this allows bad behavior to go unnoticed for a while. In general, the difficulty is that one cannot always detect an attack just by looking at feedback history, since multiple interpretations are always possible. Clarification may come with the accumulation of feedback data, but multi-agent systems might not be able to survive to see that accumulation, unless a certain number of pre-trusted nodes can be provided to bootstrap the system. • We were able to express and verify basic trust properties, such as weak transitivity and the fact that trust is more easily lost than gained. These should be building blocks upon which more complex properties could be expressed.. We have identified the following limitations and future work:. • Since the feedback history graph limits itself to agent-to-agent transaction ratings, recommender systems such as Credence [10] that use agent-to-object ratings cannot be included in the testbed.. • Trust systems that rely on different agent types (advisors, trusters, trustees, etc.) such as in [17] cannot be accommodated in our model.. • Our framework cannot accommodate trust algorithms, such as REGRET [27] or FIRE [28] , that use inputs other than feedback history.. • Different agents might have different reputations in different contexts. Our testbed does not explicitly represent this context. One would have to create separate graphs for each context (thus making the assumption that the contexts are independent from one another), and this might be an oversimplification in cases where trust in one context partially influences trust in another context. We note however that in most of the trust algorithms we have classified the notion of context is not explicitly taken into account either.. • Agent behavior simulation: it would be desirable to use agent simulations to automatically generate feedback history graphs (stage 1 of the workflow). Such a feature would be useful for designing large-scale experiments with each agent acting in a different manner.. • Support for distrust: distrust indicates how much an agent is not trusted (opposite of trustworthiness) and algorithms such as the distrust-aware version of Appleseed compute both trust and distrust propagation. Because our reputation graphs do not include distrust information, such algorithms cannot be evaluated using our testbed.. In addition to addressing the above limitations in our model, future work involves implementing more trust algorithms, building a user interface for our testbed and performing large-scale experiments. In particular, experiments using large datasets from websites such as Epinions, Advogato, and Facebook can yield interesting results. 1 In the remainder of the paper we will simply refer to the trust systems or models that have no specific name by the name of the first author of the paper we are citing, so [5] will be referred to as "Aberer". With the growth of online community-based systems such as peer-to-peer file-sharing applications, e-commerce and social networking websites, there is an increasing need to provide computational trust mechanisms to determine which users or agents are honest and which ones are malicious. Many models calculate trust by relying on analyzing a history of interactions. The calculations can range from the simple averaging of ratings on eBay to flow-based scores in the Advogato website. Thus for a researcher to evaluate and compare his or her latest model against existing ones, a comprehensive test tool is needed. However, our research shows that the tools that exist to assist researchers are not flexible enough to include different trust models and their evaluations. Moreover, these tools use their own set of application-dependent metrics to evaluate a reputation system. This means that a number of trust models cannot be evaluated for vulnerabilities against certain types of attacks. Thus, there is still a need for a generic testbed to evaluate and compare computational trust models. A prototype was designed and built in Java to test the model described in Section 'Problem description and model'. The two main components of the testbed are graphs and algorithms. A Graph can be a feedback history graph, a reputation graph or a trust graph. These graphs follow our model as described in the previous section.. The graphs can be populated either programatically or by providing a file following the Attribute-Relation File Format (ARFF) used in the Weka machine learning toolkit (ARFF was chosen for its simplicity, but other formats could obviously be accommodated as well in the future if needed). For instance, a feedback history graph is populated with a file containing a list of relations, each containing 3 attributes: source agent identifier, sink agent identifier and the feedback value. An example of a ARFF file for populating a feedback history graph is given in Listing 1:. Listing 1 An example ARFF file for populating a feedback history graph @relation feedback @attribute assessorID string @attribute assesseeID string @attribute feedbackValue numeric @data 0,1,0.6 0,1,0.7 0,1,0.2 In our prototype, even though the algorithms are implemented using object-orientation, they are ultimately presented to the experimenter as functions that take a graph as input and return a graph as the output, to fully conform to the model we have presented, and to make the experiments simple and intuitive to express. Other utility functions, such as evaluation, discretization and comparison, are also provided. See Listing 2 for an example.. Listing 2 How to create a simple experiment in our testbed import static trust.Algorithms. * ; FHG fhg = new FHG("ex.arff"); eigentrust(discretize(fhg)); //the experiment! The testbed's source code is open and is available on Google Code 4 . To ensure that our algorithm implementations match the authors' intentions and our own understanding, we also provide a set of unit tests taken from examples found in the literature as well as our own additional scenarios.

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Constructing graphs with no immersion of large complete graphs In 1989, Lescure and Meyniel proved, for $d=5, 6$, that every $d$-chromatic graph contains an immersion of $K_d$, and in 2003 Abu-Khzam and Langston conjectured that this holds for all $d$. In 2010, DeVos, Kawarabayashi, Mohar, and Okamura proved this conjecture for $d = 7$. In each proof, the $d$-chromatic assumption was not fully utilized, as the proofs only use the fact that a $d$-critical graph has minimum degree at least $d - 1$. DeVos, Dvo\v{r}\'ak, Fox, McDonald, Mohar, and Scheide show the stronger conjecture that a graph with minimum degree $d-1$ has an immersion of $K_d$ fails for $d=10$ and $d\geq 12$ with a finite number of examples for each value of $d$, and small chromatic number relative to $d$, but it is shown that a minimum degree of $200d$ does guarantee an immersion of $K_d$. ::: In this paper we show that the stronger conjecture is false for $d=8,9,11$ and give infinite families of examples with minimum degree $d-1$ and chromatic number $d-3$ or $d-2$ that do not contain an immersion of $K_d$. Our examples can be up to $(d-2)$-edge-connected. We show, using Haj\'os' Construction, that there is an infinite class of non-$(d-1)$-colorable graphs that contain an immersion of $K_d$. We conclude with some open questions, and the conjecture that a graph $G$ with minimum degree $d - 1$ and more than $\frac{|V(G)|}{1+m(d+1)}$ vertices of degree at least $md$ has an immersion of $K_d$. A classic question in graph theory is: if a graph has chromatic number d does the graph "contain" a complete graph on d vertices in some way? We know that the containment is not as a subgraph, just consider an odd cycle. In the 1940s Hajós [13] conjectured that the necessary containment was subdivision. A graph H is a subdivision of a graph G, if G contains a subgraph which is isomorphic to a graph that can be obtained from H by subdividing some of the edges. Hajós' Conjecture is true for d ≤ 4 [8] , however, Catlin [3] showed it is not true for d ≥ 7 by giving a family of counterexamples. Hajós' Conjecture remains open for d ∈ {5, 6}.. Another property to consider in trying to answer this question is graph minor. A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from a subgraph of G by contracting edges. We will say G has a minor of H. In 1943, Hadwiger [12] conjectured that every loopless d-chromatic graph has a minor of K d . Hajós' Conjecture is true for d ≤ 4, so Hadwiger's Conjecture is true for these values. Wagner [20] showed that the d = 5 case of Hadwiger's Conjecture is equivalent to the Four Color Theorem, and Robertson, Seymour, and Thomas [19] proved the case when d = 6; the cases d ≥ 7 are still open.. Given the difficulties in proving (or disproving) Hadwiger's Conjecture, we choose to explore a different type of containment, namely, graph immersion. The concept of immersion was introduced by Nash-Williams [17] , as a weakening of graph subdivision, when he conjectured that for every countable sequence G i (i = 1, 2, . . .) of graphs, there exist j > i ≥ 1 such that there is an immersion of G i in G j . This conjecture was proved by Robertson and Seymour in [18] . The well-quasi-ordering of graphs by immersion has also been discussed in [11, 14, 16] . Immersions in digraphs have been studied in [4] and [7] , and immersion in other contexts can be found in [1, 9] and [10] . In this paper we will consider immersions of complete graphs.. All graphs in this paper are simple unless otherwise stated. We will follow the notation of West in [21] .

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Classification of surgical processes using dynamic time warping In the creation of new computer-assisted intervention systems, Surgical Process Models (SPMs) are an emerging concept used for analyzing and assessing surgical interventions. SPMs represent Surgical Processes (SPs) which are formalized as symbolic structured descriptions of surgical interventions using a pre-defined level of granularity and a dedicated terminology. In this context, one major challenge is the creation of new metrics for the comparison and the evaluation of SPs. Thus, correlations between these metrics and pre-operative data are used to classify surgeries and highlight specific information on the surgery itself and on the surgeon, such as his/her level of expertise. In this paper, we explore the automatic classification of a set of SPs based on the Dynamic Time Warping (DTW) algorithm. DTW is used to compute a similarity measure between two SPs that focuses on the different types of activities performed during surgery and their sequencing, by minimizing time differences. Indeed, it turns out to be a complementary approach to the classical methods that only focus on differences in the time and the number of activities. Experiments were carried out on 24 lumbar disc herniation surgeries to discriminate the surgeons level of expertise according to a prior classification of SPs. Supervised and unsupervised classification experiments have shown that this approach was able to automatically identify groups of surgeons according to their level of expertise (senior and junior), and opens many perspectives for the creation of new metrics for comparing and evaluating surgeries.Keywords: surgical process models, classification, clustering, dynamic time warping, surgery evaluation This is the author's version of an article published in the Journal of Biomedical Informatics. The final authenticated version is available online at: http://dx The analysis and modeling of surgical procedures has recently emerged in the medical engineering field. Surgical procedures can be broken down into four main levels of granularity, from high to low [1] : phases, steps, tasks and motions. A surgical intervention can be described using a formal and structured language to create a Surgical Process (SP) at a fixed granularity level. Thus, Surgical Process Models (SPMs) are used to study, evaluate and analyze surgical activities in the Operating Room (OR). In this field, a recent and important challenge has been the design of new methods to compare and group similar SPs in order to identify relevant patterns that can be correlated with other pre-operative data in order to highlight specific information on the surgery. The main issue in such analysis is the definition of similarity metrics between SPs that reveal objective and quantitative differences at every granularity level of the surgical procedure. Indeed, SPs from the same intervention type can have high variability, which can be caused by many parameters such as the different operating techniques, the intrinsic difficulty of the surgical procedure or the surgeons expertise. Consequently, similarity measures have to be designed to accurately assess the similarity between SPs according to their content (i.e. the different activities performed by the surgeon) and their sequencing (i.e. the order in which the activities are performed).. In a recent work, Riffaud et al. [2] computed similarity metrics and performed statistical analysis for comparing groups of senior and junior surgeons (i.e. experienced and inexperienced surgeons). The metrics used were (i) General parameters of the procedure: the operating time for the whole procedure and for each step, (ii) General parameters of the surgeons activity: the number of activities performed with either the right or the left hand and the number of changes in microscope position, (iii) Specific parameters of the surgeons activity: all the gestures performed by the surgeon, the instruments used and the anatomical structure treated. Some of these metrics were found to be statistically different in a significant way when comparing the junior and senior groups. These metrics are of interest but provide no clues as to differences in terms of sequentiality. If the exact same activities were performed in a random order, the evaluation would have been the same. Consequently, in this paper we have introduced a new approach by exploring the use of the Dynamic Time Warping (DTW) [3] algorithm to evaluate similarities between SPs. DTW is used to measure the similarity between two sequences which may vary in time or speed. As SPs have been acquired in different environments, they can easily vary in time, which makes DTW particularly suitable for comparing SPs. Using DTW to compare SPs allows us to focus on the sequencing of the activities comprising the SPs. Indeed, DTW makes it possible to reduce the importance of time variations in the comparison and to focus on the number of activities and their organization in the timeline of the surgery.. Using this similarity metric, we addressed the problem of the automatic classification of SPs in supervised and unsupervised ways. We focused our evaluation on the correlation between automatic classification and the surgeons level of expertise. We present experiments using 24 SPs in lumbar disc herniation surgery, half of which were performed by senior surgeons and half by junior surgeons. Evaluation studies shown that our approach was able to automatically identify these two clusters of surgeons based on the comparison of the SPs using DTW. Furthermore, our approach was also able to go further by identifying sub-clusters of surgeons.. This kind of application is of great importance to identify and understand surgical behaviors. Indeed, one of the important challenges is to understand the parameters that influence the way surgery is performed. The goal is to better understand the practice of surgery and to provide significant feedbacks to surgeons before, during and after the intervention. The work presented in this paper highlights a significant application in surgical behaviors identification, and open many perspectives in this field.. The paper is organized as follows: in Section 2, we present work related to SP processing and surgery similarity metrics. In Section 3, we present the proposed approach and formalize the classification of SPs. In Section 4, we present several experiments which highlight the relevance of the proposed method. Lastly, in Section 5 we present a discussion and Section 6 concludes the paper. The creation of new metrics for the comparison and the evaluation of SPs is a major challenge. In this paper, we have proposed a new surgery metric based on the DTW algorithm that enables us to focus the analysis on the different types of activity performed during the surgery and their sequencing, rather than on the time differences. Supervised and unsupervised classification experiments have allowed us to establish that DTW similarity metrics were capable of discriminating groups of SPs, and that correlations between these groups and pre-operative data then enable us to highlight specific information. Results on the classification of surgeons level of expertise were shown. One possibility for improving the analysis would be to introduce semantics into the surgery similarity metrics. In this research, at each time step, a binary comparison of two surgical activities is performed. The idea would be to introduce a semantic matrix in order to link each activity using different distance values in a predefined similarity scale for a more complex analysis of SPs. The results presented below are composed of two steps. In the first step (section 4.1) we carried out experiments on the data presented in section 3.6 to highlight the ability of our method to identify relevant patterns. In a second step (section 4.2), we present results on the noise added to the data to study the robustness of the metric to noise data. Figure 4 presents the dendrogram of the AHC for the closure step, which is the only step performed by junior and senior surgeons alone. Along with the clustering, index plots of the right-hand activities of each surgery are shown below. Three clusters visually emerged from the analysis. When cutting the dendrogram to create 2 clusters, 12 surgeries can be extracted from each cluster. The first cluster (left part, in green and blue) contains 10 surgeries performed by seniors and 2 by juniors. For the second cluster (right part, in red), there are 10 surgeries performed by juniors and 2 by seniors. Additionally, within the first cluster, a sub-classification can be found (between red and blue parts), where each sub-cluster contains five surgeries performed by seniors and 1 by a junior. By keeping the two main clusters, an accuracy of 83.33% is found, considering that 20 surgeries out of 24 are classified in the right cluster. Figure 5 illustrates the evolution of the three criteria (i.e. 3NN accuracy, clustering accuracy and sum of the distance matrix) on the noisy datasets (section 3.7) according to the noise level. Figures 5 (a) and (b) respectively show the evolution of the accuracy of the 3NN classifier and the clustering accuracy. The accuracies decrease with the increase of noise in the dataset. This trend highlights the fact that the similarity measure using DTW accurately evaluates similarity between SPs. Indeed, the noise increase disrupts the performance of SP similarity. Figure 5 (c) shows the evolution of the sum of the distance matrix (Eq. 13). This sum increases with the level of noise, which means that adding noise tends to decrease the similarity between the SPs. This result also shows that this similarity measure is relevant as it reveals that if the content of two SPs is different, they will be dissimilar according to our measure. However, one can also note that these results highlight that the method is sensitive to noise. Indeed, with more than 10% of noise the accuracy begins to be low. The strong influence of the noise is mostly due to the use of a binary distance between the activity (see Eq. 3). Indeed, if only one component of an activity is different, the two activities will be evaluated as different. To reduce the influence of noise, other distances between activities could be used, as for example a fuzzy distance weighting each component of the activities. Figure 7 presents the similarity matrices between the 24 SPMs without noise (a), with 10% of noise (b) and with 60% noise (c). The first 12 rows/columns of the matrices correspond to juniors and the remaining 12 rows/columns correspond to the seniors. The histogram of the distance was stretched and normalized in order to compute gray levels. In these matrices, the darker the cell, the more dissimilar the SPs. Table 2 presents the evolution of the average distance in percentage between the junior and the senior groups according to three levels of noise. Figure 6 shows the evolution of the percentages for all the levels of noise. Given two levels of noise, the percentage is computed by studying the difference between the distance matrices of the two levels and by computing the geometric mean of the percentage of increase of the distance. These values show that the distances within each group (junior and senior) tend to increase faster than the distance between the junior and the senior groups when the noise increase. This can be explained by the fact that the SPs belonging to the same group are more similar, and a change in an activity resulting from the introduction of noise is likely to affect the alignment provided by DTW. On the contrary, when measuring the distance between SPs from two different groups, the introduction of noise is less likely to affect the similarity as a change in one component of two activities already containing one difference will have no effect. Given the advent of the latest technologies in the Operating Room (OR), an important need has emerged for tools to assess and evaluate the impact of these new technologies. Within this field, the development of new methods for objective surgical skill evaluation is an important issue [4, 5] . Surgical skills can be assessed based on five factors: knowledge, decision making, technical skills, communication skills and leadership skills. From these five factors, many studies have been conducted for developing objective methods of technical skill evaluation. A comprehensive review can be found in [6] .. One approach to surgical skills evaluation [7] is to consider the patient's outcome in assessing surgeons. Unfortunately, this metric is highly variable and dependent upon the patients specific characteristics. Additionally, patient outcome is usually a multi-factor criterion requiring long term follow-up. Even if outcome-based metrics are straightforward to use, they are not objective enough and they do not study the differences in the surgical procedure in detail. Another approach uses human grading techniques. The underlying idea is to ask to a senior surgeon to provide an evaluation rating scale using dedicated check-lists during the observation of an intervention. Several scores have been proposed: Objective Structured Assessment of Technical Skills (OSATS) [8] , Objective Structured Clinical Examinations (OSCE) [9] , and the Global Rating Scale (GRS) [10] have shown good results. However, this method has proven to be very time-consuming and also very observerdependant. Motion has also been investigated to analyze dexterity by tracking the surgeon's hand [11] , arm [12] , and instruments [13] , using various and complementary tracking systems [14] or in the context of robotic assistance [15] . This work has focused on motion pattern analysis, for instance using time series analysis of the different motions. The main drawback of such approaches is their low level of granularity, which does not give insight into the surgical scenario followed.. Lastly, on-line and off-line recordings of surgeries have been of growing interest for analyzing procedures and assessing surgeons. Recordings can be performed using sensor devices or directly by an observer. This data extraction process can be supported by two level of knowledge: activity recording can be performed either according to common standards of surgical procedures, including standard surgical terms [16, 17] , or according to fixed protocol created by local experts. In this second case, the first step consists in building up one's own vocabulary. New terminology is employed and provides knowledge representation that is proper to the surgeons own experience and to the specific surgical environment. In this context, SP recordings can be driven by specific Surgical Process Models (SPMs), including complex dedicated ontologies. The need for modelbased systems to assist and monitor Computer-Assisted Surgery (CAS) has been discussed by Jannin et al. [18] , who also mentioned the importance of foresight within a surgical procedure by modeling the surgery into a sequence of major steps. A model was proposed in the context of neurosurgical interventions, based on a UML class diagram and a textual description for breaking down the procedure. SPMs have also been introduced as an added-value for the description of surgeries during an SP recording by the group in Leipzig. Burgert et al. [19] proposed an explicit and formal description in an ontology based on General Ontological Language (GOL) for representing surgical interventions. Another study [20] focused on the description of concepts and technologies for the acquisition of surgical workflows by monitoring surgical interventions. They introduced a universal adaptable recording scheme describing the subdivision of the surgical interventions into detailed records of manual work steps. New software was implemented (i.e. a surgical workflow editor: the ICCAS editor system) to record processes during the intervention. They also introduced methods for computing generic SPMs that could serve to generate and compare surgical procedures [21, 22] . Recently, Bouarfa et al. [23] presented a mechanism for dataset pre-processing before HMM training. Their objective was to infer high-level tasks (i.e. defined as surgical steps) from a set of observable low-level tasks (e.g. picking up instruments or putting down instruments). They stressed that the information extracted from the OR must be discriminant, invariant to task distortion, compact in size and easy to monitor.. Lastly, Combi et al. [24] studied clinical activities in terms of clinical workflows. They studied clinical processes which are comprised of clinical activities to be done by given actors in a given order satisfying given temporal constraints. They modeled workflows as a set of activities and proposed a similarity function taking into account the order and duration of the activities. On a more general level, Vankipuram et al. [25] studied workflows composed of recordings of the motion and location of clinical teams. These data are then used to model activities in critical care environments. These recent studies highlight the serious need and the emerging trend of tools able to take into account the sequentiality of activities.. The study of processes is also related to the field of process mining [26] , which traditionally has been used to extract models from event logs and to check or extend existing models. One of the goals of process mining is to create a model which ideally represents a set of processes. Then, a single process can be checked against the model to identify problems. The field of process mining [27] is composed of different tasks, as the discovery of the models, the check of the conformance of one process against a model or the extension of a model. Even though the process mining approach introduces interesting and important concepts, it does not generally tackle the definition of metrics to compare processes. Furthermore, the concepts introduced are generic and an adaptation to a specific domain is always necessary. The work presented in this paper is related to this field in the way that the general concepts are similar. However, we introduce in this paper a methodology which is specific to surgical processes.

B O O O O B I I O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I B B B I I B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A Basic Study on Stimulation Pulse Width Determination for Dynamic Presentation of Electrocutaneous Sensation Patterns In this study, we aimed to realize a method of presenting dynamic electrocutaneous sensation patterns as a user interface for persons with disabilities. In this study, the previously used pulse width (200 µs) was used in pattern recognition testing on the forearm and was compared with the short pulse width (80 µs) used for stimulation. A cathodic pulse with a frequency of 100 pps, which was suggested to be suitable in our previous study, was used. Experimental results on neurologically intact subjects suggested that the previously used pulse width was suitable and that a short pulse width would not be effective for stabilizing the stimulation amplitude or for improving the pattern recognition ability of our method. In this study, we focused on a method of graphical tactile display using electrocutaneous sensation patterns for persons with visual or auditory impairments or as a new method of information presentation to assist rehabilitation. In our previous study [1] , a method of dynamic pattern presentation was proposed, in which patterns were presented by moving the position of electrical stimulation on the forearm. The proposed method was found to enable the transmission of more information than the previous static pattern presentation method [2] .. However, the parameters of stimulation pulse trains used for the dynamic pattern presentation were not examined sufficiently. The polarity of the stimulation pulse and the pulse frequency were tested in our previous experiments, in which a cathodic pulse with a frequency of 50 or 100 pps was suggested to be useful for presenting dynamic patterns [3] . On the other hand, only a pulse width of 200 µs was used in our previous studies.. In this paper, electrical stimulation using a short pulse width (80 µs) was examined to clarify whether or not the previously used pulse width (200 µs) was suitable, because a short pulse width was considered to be effective for inducing an electrocutaneous sensation. That is, a shorter pulse width allows more spatially selective stimulation of nerve fibers [4] . It also reduces variations in the current threshold [5] . In this paper, pattern recognition experiments using different pulse widths were performed to determine whether or not the previously used pulse width (200 µs) was suitable. For our dynamic pattern presentation method, it was suggested that the previously used pulse width was satisfactory and that a short pulse width would have no effect on reducing the variation of stimulation voltage and on improving pattern recognition ability. An effective training method is necessary to improve the pattern recognition ratio for practical applications of dynamic pattern presentation using electrocutaneous sensation. There is no difference in the average value of CV of the stimulation amplitude between different pulse widths and no clear decrease in CV upon using a short pulse width. In addition, no clear improvement of CRR by using a short pulse width was observed. From these results, it is considered that there is no advantage in using a short pulse width in our method. A pulse width longer than 200 µs was not tested in this study, because it might decrease the pain threshold and the dynamic range, defined as the set of values between the threshold amplitude and maximum amplitude [6] . Possible reasons why the shorter pulse width did not reduce the variation of stimulation amplitude are as follows. First, the stimulation amplitude was determined by each subject as the value at which he/she could comfortably detect the sensation. Second, the voltage stimulation amplitude is discussed in this study but the variation of the threshold for current stimulation was investigated in the previous study [5] . Third, a shorter pulse width has been suggested to increase the dynamic range [4] .. The reasons why there was no difference in the value of CRR between different pulse widths are not clear. However, an increased stimulation amplitude may cause a decrease in CRR because it activates nerve fibers over a large area. In all the subjects, ratios of the average stimulation amplitude of the nine electrodes with a pulse width of 80 µs to that with a width of 200 µs were almost 1.0 or higher. This was because the subjects determined the amplitude of which the sensation was detected comfortably. Therefore, it is possible that the increased stimulation pulse amplitude reduced the effect of the short pulse width on improving the spatial selectivity of activating nerve fibers.. As seen in our previous experiment [1] , it was possible to improve CRR by repeating the recognition experiment. In this study, the subjects were trained before each recognition test. Although training might improve CRR, half the subjects did not have a value of CRR larger than 50% in most sessions. Subjects A and B, who had higher values of CRR, participated in our previous study [3] . Although subject C participated in our previous study [1] , the value of CRR was not high. Subjects D, E and F participated in our experiments for the first time. By considering the results for various subjects, it is suggested that a difference in pulse width does not affect the pattern recognition ability because none of the subjects showed a clear improvement in the value of CRR. First, the coefficient of variation (CV) of the stimulation amplitudes was calculated for each session. In the calculation of CV, the stimulation voltage amplitude for one electrode was removed in one session (subj.A, 80 µs (2nd)) because it was extremely large compared with those of the other eight electrodes. There was no major difference in the value of CV between the two pulse widths and between experimental sessions (0.21±0.05 for the short pulse width and 0.20±0.04 for the long pulse width). Figure 3 . where CV(80 µs) and CV(200 µs) denote CV for pulse width of 80 µs and 200 µs, respectively. The value of CV decreased in 5 out of 12 experiment days. However, no clear decrease in the value of CV for a short pulse width was observed in this result. Figure 4 shows the correct recognition ratio (CRR) in each session in order of experiments. Figure 5 shows the rate of change of CRR, calculated by. where CRR(80 µs) and CRR(200 µs) denote CRR for pulse widths of 80 µs and 200 µs, respectively. There was no correlation between CRR and the appearance ratio for all the subjects (r = 0.125 ± 0.08). As shown in Fig. 5 , CRR was improved by using the short pulse width on 3 out of 12 experiment days. However, by comparing the results of the two sessions in each day, Fig. 4 suggests that subjects A, C and D had a larger CRR for a 200 µs pulse width, subjects B and E had a larger CRR in the second session in each day, and subject F had a larger CRR in the first session in each day. That is, it is not clear that the value of CRR improved using the short pulse width.

O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O B I I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A case for the coexistence of heterogeneous wireless networks As the ISM spectrum becomes crowded by various network devices, such as WiFi and ZigBee, the coexistence between them poses a critical challenge, due to their heterogeneous MAC/PHY layers. Recent measurement studies have shown moderate to high WiFi traffic to severely impair coexisting ZigBee's performance. We propose a new mechanism, called cooperative busy tone (CBT), that exploits the inherent cooperation among ZigBee nodes to harmonize their coexistence with WiFi devices. CBT employs a separate ZigBee node to emit a busy-tone concurrently with the desired data transmission, thereby improving the visibility of ZigBee devices to WiFi. It adopts an innovative way to concurrently schedule the busy-tone and a data packet without causing interference between them. To optimize CBT, we establish a theoretical framework that models the performance of coexisting ZigBee and WiFi networks, and analyzes the impact of key design parameters in CBT. Our analytical results and testbed implementation demonstrate CBT's significant throughput improvement over the legacy ZigBee protocol, with negligible performance loss to WiFi. The past decade has witnessed the proliferation of wireless devices (e.g., ZigBee and WiFi) running on the ISM band, and the coexistence between them becomes increasingly critical. As an interference avoidance mechanism, CSMA becomes ineffective for coordinating heterogeneous devices, which may adopt asynchronous time slots, different scheduling modes (e.g., TDMA vs. CSMA), disparate transmission/interference Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. In an enterprise WiFi WLAN and a co-located 90-node ZigBee WPAN for building energy management, more than a half of the ZigBee links were observed to suffer connection loss during peak hours due to WiFi interference [7] . The harmful coexistence has also been observed in previous small-to medium-scale experimental deployments [4, 5, 8] .. A further microscopic study [9] has revealed the root cause of such harmful coexistence. First, ZigBee's transmit power is 20dB lower than WiFi's, yielding a smaller spatial footprint and hence, its poor visibility to WiFi. The ZigBee's MAC-layer time slot is also 16 times longer, and it can easily be preempted by WiFi in the middle of a rx/tx transition (e.g., sensing-to-transmission or data-to-ACK transition), thus causing collision, even if they can sense each other. In addition, besides CSMA, ZigBee allows for TDMA mode, which operates without carrier sensing, and may arbitrarily collide with an ongoing WiFi transmission.. Based on the above observations, we propose a new mechanism, called Cooperative Busy Tone (CBT), to facilitate ZigBee's coexistence with WiFi. Unlike the traditional CSMA that relies on a data packet as an implicit carrier signal (busy-tone), CBT assigns a separate ZigBee node called signaler, as a proxy to perform the carrier signaling, i.e., sending a busy-tone packet concurrently with the data packet. The signaler may have a higher power than the ZigBee transmitter, thus allowing the WiFi nodes to sense the ZigBee's presence indirectly by detecting the busy-tone ( Fig. 1(a) ). The busy-tone continues throughout the data and ACK roundtrip, thus preventing the WiFi's preemption in the rx/tx switching gap ( Fig. 1(b) ). The core component in CBT is an innovative way to concurrently schedule the busy-tone and a data packet without causing interference between them.. To understand the performance of CBT, we have developed an analytical framework that models a heterogeneous network where ZigBee and WiFi coexist. Our analysis quan- tifies the network performance, with and without running CBT. We have also prototyped CBT based on TinyOS and the GNURadio library [1] . Our experiments on the MicaZ motes and USRP2 [3] software radio platform further corroborate the feasibility and effectiveness of CBT.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O