EVERYTHING YOU ALWAYS WANTED TO KNOW ABOUT SNAKES (BUT WERE AFRAID TO ASK) Jim Ivins & John Porrill

Size: px
Start display at page:

Download "EVERYTHING YOU ALWAYS WANTED TO KNOW ABOUT SNAKES (BUT WERE AFRAID TO ASK) Jim Ivins & John Porrill"

Transcription

1 EVERYTHING YOU ALWAYS WANTED TO KNOW ABOUT SNAKES (BUT WERE AFRAID TO ASK) Jim Ivins & John Porri AIVRU Technica Memo #86, Juy 993 (Revised June 995; March 2000) Artificia Inteigence Vision Research Unit University Of Sheffied Engand S0 2TP Reeing and Writhing, of course, to egin with, the Mock Turte repied; and then the different ranches of Arithmetic Amition, Distraction, Ugification, and Derision. Aice s Adventures In Wonderand, y Lewis Carro Pease send feedack and corrections regarding this document to Jim Ivins. E-mai:

2 ABSTRACT Active contour modes known cooquiay as snakes are energy-minimising curves that deform to fit image features. Snakes ock on to neary minima in the potentia energy generated y processing an image. (This energy is minimised y iterative gradient descent according to forces derived using variationa cacuus and Euer-Lagrange Theory.) In addition, interna (smoothing) forces produce tension and stiffness that constrain the ehaviour of the modes; externa forces may e specified y a supervising process or a human user. As is characteristic of gradient descent, the energy minimisation process is unfortunatey prone to osciation uness precautions typicay the use of sma time steps are taken. Active contour modes provide a unified soution to severa image processing proems such as the detection of ight and dark ines, edges, and terminations; they can aso e used in stereo matching, and for segmenting spatia and tempora image sequences. Snakes have often een used in medica research appications; for exampe, in reconstructing threedimensiona features from panar sices of voume data such as NMR or CT images. In addition, many motion tracking systems use snakes to mode moving ojects. The main imitations of the modes are (i) that they usuay ony incorporate edge information (ignoring other image characteristics) possiy comined with some prior expectation of shape; and (ii) that they must e initiaised cose to the feature of interest if they are to avoid eing trapped y other oca minima. KEY WORDS k Cacuus Of Variations k Euer-Lagrange Theory k Gradient Descent k Snakes: Active Contour Modes LICENSE Copyright (C) Jim Ivins & John Porri AIVRU, University of Sheffied, Engand S0 2TP. Veratim copying and distriution of this entire document is permitted in any medium, provided this notice is preserved, ut changing it is not aowed. Page 2

3 INTRODUCTION Low-eve visua tasks such as edge detection and stereo matching are often treated as autonomous ottom-up processes. However, this sequentia approach propagates mistakes to higher processes without providing opportunities for correction. A more attainae goa for ow-eve processing is to provide severa interpretations of the image data, from which higher processes or a human user may choose. Active contour modes first descried y Kass et a (987; 988) provide one possie method for generating these aternative interpretations. Active contour modes are often caed snakes ecause they appear to sither across images (a phenomenon known as hysteresis); they are one exampe of the genera technique of matching a deformae mode to an image using energy minimisation. From any starting point, suject to certain constraints, a snake wi deform into aignment with the nearest saient feature in a suitay processed image; such features correspond to oca minima in the energy generated y processing the image. Snakes thus provide a ow-eve mechanism that seeks appropriate oca minima rather than searching for a goa soution. In addition, high-eve mechanisms can interact with snakes for exampe, to guide them towards features of interest. Unike most other techniques for finding image features, snakes are aways minimising their energy. Changes in high-eve interpretation can therefore affect a snake during the minimisation process, and even in the asence of such changes the mode wi sti exhiit hysteresis in response to a moving stimuus. Snakes do not try to sove the entire proem of finding saient image features; they rey on other mechanisms to pace them somewhere near a desired soution. For exampe, automatic initiaisation procedures can use standard image processing techniques to ocate features of interest that are then refined using snakes. Even in cases where automatic initiaisation is not possie, however, active contour modes can sti e used for image interpretation. An expert user need ony push a snake towards an image feature, and the energy minimisation process wi fit the mode to the data. This ehaviour has een expoited in numerous interactive image processing systems for exampe, see Kass et a (987; 988); Hi et a (992); Porri and Ivins (994). A snake is typicay driven y a potentia energy generated y processing the underying image data. For exampe, Gaussian smoothing foowed y convoution with a Scott (987) is critica of this phiosophy ecause the unspecified high-eve process rarey materiaise except in human form! Page 3

4 gradient-squared operator generates a potentia in which extrema correspond to edges in the origina image. Over a series of iterations the force generated y this energy drives the snake into aignment with the nearest saient edge. However, the snake must aso satisfy some interna constraints for exampe, it must e smooth and continuous in outine. Sometimes the user imposes additiona externa constraints such as attraction or repusion. Current Snake (Time t): New Snake (Time t+): Movement: Edge: Figure : A Cosed Active Contour Mode. This diagram shows a snake with its ends joined so that it forms a cosed oop. Over a series of time steps the snake moves into aignment with the nearest saient feature (in this case an edge). Both interna and externa energy constraints are discussed in Section 2; potentia energy is discussed in Section 3. Section 4 uses these energy terms to derive expicit forces that can e used to drive active contour modes to minimise their energy y iterative gradient descent. The origina (semi-impicit) method proposed y Kass et a (987), which is reated to the expicit use of forces, is then descried in the next two sections. First, the cacuus of variations is used to derive the Euer-Lagrange equation in Section 5; this equation is then used to find the minimum energy condition for an active contour mode. Section 6 expains how to sove the minimum energy equation using a semi-impicit reaxation method ased on a fast matrix inversion agorithm. (Both the expicit and semi-impicit methods use finite differences to compute derivatives as descried in Appendix A; Appendix B contains six mathematica notes that provide simpe ackground information.) Osciation, the main drawack of reaxation methods, is discussed in Section 7. Finay, Section 8 considers the use of inter-snake energy terms in three of the most common appications stereo matching, and segmentation of spatia and tempora image sequences. Page 4

5 2 SNAKE ENERGY FUNCTIONALS A snake is a parametric contour that deforms over a series of iterations (time steps). Each eement x aong the contour therefore depends on two parameters: s = space (curve) parameter x(s, t) t = time (iteration) parameter The contour is infuenced y interna and externa constraints, and y image forces, as outined eow. k Interna forces. Interna constraints give the mode tension and stiffness. k Externa forces. Externa constraints come from high-eve sources such as human operators or automatic initiaisation procedures. k Image forces. Image energy is used to drive the mode towards saient features such as ight and dark regions, edges, and terminations. Representing a snake parametricay as expained in Mathematica Note, x(s) = ( x(s), y(s) ) where s is usuay taken to vary etween 0 and. The tota energy of the mode E snake is given y the sum of the energy for the individua snake eements: E snake = 0 Eeement ( x(s) ) ds (2.) The integra notation used in this section impies an open-ended snake; however, joining the first and ast eements makes the snake into a cosed oop as shown in Figure. Equation 2. can e rewritten in terms of three asic energy functionas: E snake = 0 Eintern (x) ds + 0 Eextern (x) ds + 0 Eimage (x) ds (2.2) The curve parameter s is omitted where no amiguity arises. The gradients of the three energy functionas in Equation 2.2 correspond to the three forces isted aove. The interna and externa energy functionas are considered in more detai eow; image (potentia) energy is deat with in the next section. To impement an active contour mode in computer software the continuous representation is approximated discretey y N snake eements; however, continuous notation is used wherever possie ecause of its greater mathematica eegance. A functiona is a function of one or more functions, giving a scaar resut. Page 5

6 2. INTERNAL (INTRA-SNAKE) ENERGY Using suscripts to indicate derivatives, the interna energy of a snake eement is defined as: E intern (x) = a(s) x s (s) 2 + (s) x ss (s) 2 (2.3) Tension Stiffness This energy contains a first-order term controed y α(s), and a second-order term controed y β(s). The first-order term makes the snake contract ike an eastic and y introducing tension; the second-order term makes it resist ending y producing stiffness. In other words, the parametric curve is predisposed to have constant (preferay zero) veocity and acceeration with respect to its parameter. In the asence of other constraints, an active contour mode simpy coapses to a point ike a strip of infinitey-eastic materia; however, if the ends of the mode are anchored then it forms a straight ine aong which the eements are eveny spaced. Adjusting the weights α(s) and β(s) contros the reative importance of the tension and stiffness terms. For exampe, setting β(s) = 0 in one part of the mode aows it to ecome second-order discontinuous and deveop a corner. For simpicity, the tension and stiffness weightings are assumed to e uniform throughout the remainder of this document, so that α(s) = α and β(s) = β. 2.2 EXTERNAL (EXTRA-SNAKE) ENERGY Both automatic and manua supervision can e used to contro attraction and repusion forces that drive active contour modes to or from specified features. For exampe, a spring-ike attractive force can e generated etween a snake eement and a point i in an image using the foowing externa energy term: E extern (x) = k i x 2 (2.4) This energy is minima (zero) when x = i, and it takes the vaue of k when i x = ± as shown in Figure 2. Mathematica Note 2 reviews the properties of extrema in functions. An externa energy term E extern can aso e used to make part of an image repe an active contour mode: E extern (x) = k i x 2 (2.5) This energy is maxima (infinite) when x = i; it is unity when i x = ± k. Because of the singuarity, the repusion term must e cipped as the denominator approaches zero. Page 6

7 Negating the (positive) constant k in these equations converts attraction to pseudorepusion, and repusion to pseudo-attraction; however, these pseudo energy terms are unusae ecause their minima are infinite. (During energy minimisation, the singuarities competey dominate the ehaviour of an active contour mode, at the expense of a other energy terms.) The forces produced y these energy terms are easiy found y differentiation. (a) Attractive Energy () Repusive Energy E E k 0 + i x k 0 +k i x Figure 2: Attraction And Repusion Energy. These graphs show the attractive and repusive energy terms. Both functionas have maxima vaues that are infinite; the minima are zero. Page 7

8 3 IMAGE (POTENTIAL) ENERGY FUNCTIONALS The potentia energy P generated y processing an image I(x, y) produces a force that can e used to drive snakes towards features of interest. Three different potentia (image) energy functionas are descried eow; these attract snakes to ines, edges, and terminations. The tota potentia energy can e expressed as a weighted comination of these functionas: P = E image = w ine E ine + w edge E edge + w term E term (3.) The nearest oca minimum the potentia energy can e found using gradient descent as descried in Section 4: x d x + dx (3.2) The image forces δx produced y each of the terms in Equation 3. are derived eow, in advance of the main discussion of energy minimisation and forces in Sections 4 6. If just a sma portion of an active contour mode finds a ow-energy image feature then the interna constraints wi pu neighouring eements towards that feature. This effect can e enhanced y spatiay smoothing the potentia energy fied. Typicay, a snake is first aowed to reach equiirium on a very smooth potentia; the urring is then graduay reduced see Witkin et a (986). At very coarse scaes the snake does a poor jo of ocaising features, and fine detai is ost; however, it is attracted to oca minima from far away. Reducing the amount of urring aows the snake to form a more accurate mode of the underying image. 3. REGION FUNCTIONAL The simpest potentia energy is the unprocessed image intensity so that P(x) = I(x): E ine = 0 I ( x(s) ) ds (3.3) According to the sign of w ine in Equation 3., the snake wi e attracted either to ight or dark regions of the image. Using to indicate image gradient, the corresponding image force δx is given y: dx œ žp = ži = I(x) Loca minima in the image intensity can therefore e found y taking sma steps in x: x d x t I(x) (3.4) The positive time step τ is chosen to suit the proem domain; however, it is amost invariay one or two orders of magnitude ess than unity to prevent osciation (see Section 7). For an Page 8

9 extension of this idea for segmenting textures and coours see the work on active region modes y Ivins and Porri (995). 3.2 EDGE FUNCTIONAL By far the most common use for active contour modes is as semi-goa edge-detectors that minimise a potentia energy in which minima correspond to strong edges see Figure 3. (a) Unprocessed Image () Potentia (Edge) Energy Figure 3: Potentia (Edge) Energy. (a) An unprocessed 256-y-256 pixe NMR image. () The potentia energy generated y smoothing the image, convoving it with a simpe gradient operator, and negating the resut (the image has een re-scaed for dispay). Strong edges produce correspondingy ow (dark) oca minima; however, fine detai is ost during the smoothing process, which is necessary to eiminate noise and spread out egitimate edges. Edges can e found with a gradient-ased potentia energy functiona such as: 2 E edge = ži 0 ds (3.5) For exampe, consider a snake eement x = (x, y) with potentia energy P(x) = I(x) 2 ; the image force acting on this eement is given y: dx œ žp = ž ( I 2 ) = 2 I(x) I(x) The term I(x) is the Hessian matrix of second-order image derivatives. Strong edges can therefore e found using: x d x + t I(x) I(x) (3.6) Page 9

10 3.3 TERMINATION FUNCTIONAL The ends of ine segments, and therefore corners, can e found using an energy term ased on the curvature of ines in a sighty smoothed image C(x, y) = G σ (x, y) * I(x, y). If the gradient direction is given y θ = tan (C y / C x ) then the unit vectors aong, and perpendicuar to, the image gradient are given y: Tangent: n = cos h sin h Norma: n z = sin h cos h The curvature of a contour in C(x, y) can e written: E term = žh ž 0 ds = žn z 2 2 C/žn z 0 žc/žn Expanding the derivatives: ds (3.7) E term = 0 C yy C x 2 + C xx C y 2 2C xy C x C y C x 2 + C y 2 3/2 ds (3.8) This energy formua provides a simpe means for attracting snakes towards corners and terminations. Page 0

11 4 GRADIENT DESCENT USING FORCES The previous two sections can e summarised y stating that, at its simpest, the energy E of an active contour mode x(s) is defined as: E( x(s) ) = P(x(s)) ds + 0 Potentia a (s) 2 žs 0 Tension 2 ds + ž 2 x(s) 2 žs 0 2 Stiffness 2 ds (4.) This section considers the task of minimising these energy functionas. First, the genera technique of minimisation y iterative gradient descent is introduced; an equation is then derived to descrie the energy changes that occur when an active contour mode is moved, and this equation is used to cacuate forces for energy minimisation y gradient descent. 4. CONJUGATE GRADIENT DESCENT In genera, an energy function E(x) can e minimised y atering each variae according some sma quantity δx that is guaranteed to reduce the vaue of the function: x x + dx Loca inear approximation gives an expression for the new energy: (4.2) E(x + dx) E(x) + že (4.3) $ dx Ceary, δx must e chosen so that the energy decreases at each iteration. The gradient descent rue is ased on the fact that steps down an energy hypersurface (see Figure 4) can e guaranteed y making sma changes in the direction of the negated gradient: dx œ že The new vaue of the energy function is given y: E(x + dx) E(x) t že 2 (4.4) (4.5) The negative sign and square power (dot product) in this equation guarantee that E wi decrease at each iteration unti the minimum is reached; however, the (sma) time step τ must e chosen carefuy to avoid osciation (see Section 7) and is amost invariae ess than unity. Conjugate gradient descent, as iustrated in Figure 4, finds the nearest oca minimum in an energy hypersuface, with no consideration of goa properties. Unfortunatey, this For simpicity, externa constraints are omitted from the remainder of this document. Page

12 simpicity can ead to proems when there are severa minima cose together ecause a snake can e attracted to a feature (energy minimum) other than that intended y the user. E (a) Energy Hypersurface x2 () Energy Contours x2 High Energy Low Energy x x Figure 4: Conjugate Gradient Descent. This figure shows four aternative paths down a three-dimensiona energy surface. At each iteration the gradient descent agorithm moves the energy vaue towards the nearest oca minimum y making a sma change in the direction given y the negated energy gradient (orthogona to the oca energy contours). The process is repeated unti this gradient (force) is zero, at which point none of the variaes can e atered without increasing the energy. 4.2 ENERGY GRADIENT FOR AN ACTIVE CONTOUR MODEL Before gradient descent can e used to minimise the energy of an active contour mode it is necessary to otain an expression for the corresponding energy gradient which determines the changes that are made to the mode (forces) at each iteration. From Equation 4. the asic energy of a cosed active contour mode is given y: E(x) = Á P(x) ds + a 2 Á x 2 ds + 2 Á x 2 ds x h x(s) x h /žs x h ž 2 x/žs 2 (4.6) Note the use of dashes to indicate derivatives. The ends of this mode are joined so that it forms a cosed oop; in the discrete approximation to this equation, the first and ast of the N snake eements are consecutive so that x(0) x(n). The energy functionas in this version of the equation are integrated around a cosed snake as shown y the integra signs; this removes the need to specify imits. Page 2

13 If the snake changes sighty then the tota energy of the new configuration is: E(x + dx) = Á P(x + dx) ds + a Á 2 x + dx 2 ds + This equation can e simpified using the foowing approximations: P(x + dx) = P(x) + dp(x) P(x) + Á 2 x žp $ dx + dx 2 ds (4.7) x + dx 2 = x$x + 2x$dx + Equation 4.7 therefore simpifies to: dx $ dx x 2 + 2x$dx Negigie E + de Á P(x) + žp $ dx ds (4.8) + a 2 Áx 2 + 2x $ dx ds + 2 Áx 2 + 2x Sutracting 4.6 from 4.8 gives an approximation for the energy change that arises from a sma adjustment to the configuration of the snake: de = Á žp $ dx ds + a Áx $ dx ds + Áx $ dx ds (4.9) This approximation is simpified using integration y parts (see Mathematica Note 5) to eiminate δx and δx : de = Á žp (4.0) $ dx ds a Áx $ dx ds + Áx $ dx ds Equation 4.0 can e factorised to give a simpe expression that incudes the energy gradient: de = Á žp a x + x $ dx ds (4.) Negating this expression gives the oca direction of steepest descent down the energy hypersurface; however, it does not indicate how far to move and must e treated with caution since it is ony a oca description of the surface. $ dx ds 4.3 FORCES Assuming it is not aready at a minimum, the energy of a snake wi decreases at each iteration if δx is a negated fraction (the time step δt, which must e positive) of the energy gradient given y Equation 4.: dx = dt žp a x + x (4.2) Page 3

14 Sustituting this expression ack into Equation 4. gives: de = dt Á žp 2 a x + x The iterative rue for conjugate gradient descent is therefore: x x + dx At the imit of infinitesima steps: žt = a ž2 x žs 2 Tension Force dx dt = že ž4 x žs 4 Stiffness Force ds = a x x dp dx žp Image Force (4.3) (4.4) (4.5) The energy of an active contour mode can therefore e minimised y cacuating this resutant force for, and appying it to, each snake eement in turn. Note that in mechanica systems, force is the product of mass and acceeration: f = m ž2 x žt 2 However, in Equation 4.5 force and veocity are equivaent: f = An active contour mode driven using this equation therefore ehaves as though traveing in a viscous medium such that inertia is negigie and movement with constant veocity requires a constant force to e appied. žt (a) Initia Snake () Fina Snake Figure 5: An Active Contour Mode. This figure shows two views of an MR image (the potentia energy generated y smoothing the image and convoving it with a simpe gradient Page 4

15 operator is shown in Figure 3). (a) An initia snake configuration marked y the user. () The fina snake configuration after energy minimisation y gradient descent; the snake is modeing the skin over the sku. (Note that the snake has een re-parameterised during energy minimisation; this process is not discussed further in this document.) Gradient descent using expicit forces is not the ony way to minimise the energy of an active contour mode. For exampe, dynamic programming was proposed y Amini et a (988) as a method for finding minima that are guaranteed to e goa within some predetermined search range; however, this method suffers from increased computationa compexity and wi not e discussed further in this document. The semi-impicit method originay used y Kass et a (987) is a faster aternative that reies on an efficient matrix inversion agorithm to sove a set of simutaneous equations y reaxation; the soutions to these equations descrie the minimum energy state of an active contour mode. The semi-impicit method is descried in the next two sections. Page 5

16 5 CALCULUS OF VARIATIONS This section uses variationa cacuus to derive the Euer-Lagrange equation, which descries extrema in functionas; this equation is then used to otain an equation that descries the minimum energy condition of an active contour mode. 5. THE EULER-LAGRANGE EQUATION Consider the proem of minimising (or maximising) a functiona E such as: E(y) = a F ( x, y(x), y (x) ) dx (5.) (The independent variae x can e omitted where there is no amiguity). Making a sma change δy to the vaue of the function y generates a corresponding change in E: E(y + dy) = a F ( x, y + dy, y + dy ) dx (5.2) Using the Tayor expansion (see Mathematica Notes 3 and 4) and ignoring terms aove first order: E + de a F + žy dy + dy dx žy Sutracting 5. from 5.3 gives: de a žy dy + dy dx žy Eiminating δy using integration y parts as descried in Mathematica Note 5: de a žy dy d dx žy dy dx At extrema in E a sma change in y produces amost no change in the vaue of the functiona: a žy d dx žy dy dx 0 (5.3) (5.4) (5.5) (5.6) As δy is known to e non-zero, Equation 5.6 gives rise to the Euer-Lagrange Equation which is satisfied at extrema in F: žy d dx žy = 0 (5.7) Of course, the extremum descried y the Euer-Lagrange equation coud e a maximum or a point of infection rather than a minimum. If necessary, the second-order partia derivative (which wi e positive at minima, negative at maxima, and zero at points of infection) can sometimes e cacuated to resove the amiguity. Page 6

17 To summarise, for a sma change δy to a functiona F(x, y, y ): de = a de dy(x) dy dx de dy(x) The first variation δ / δy(x) pays the equivaent roe for functionas that the first derivative d / dx pays for functions, so that δe / δy(x) = 0 at extrema; it can therefore e used to find the minimum energy condition for a snake. = žy d dx žy 5.2 MINIMA IN SNAKE ENERGY FUNCTIONALS From Equation 4. the asic energy of an active contour mode is given y: E( x(s) ) = 0 F ( s, x(s), x (s), x (s) ) ds Consider the effect of a sma change in the vector x: x h x(s) x h /žs x h ž 2 x/žs 2 E(x + dx) = 0 F (x + dx, x + dx, x + dx ) ds (5.8) (5.9) Using the Tayor expansion: E + de 0 F + $ dx + $ dx + Sutracting 5.8 from 5.0: $ dx ds de 0 $ dx + $ dx + $ dx ds Terms in δx and δx are eiminated using integration y parts: de 0 $ dx d ds Factorising: $ dx + de 0 d + d 2 ds ds 2 This yieds the Euer-Lagrange equation for extrema in E: d ds + d 2 ds 2 d 2 ds 2 $ dx ds $ dx ds = 0 (5.0) (5.) (5.2) (5.3) (5.4) Again, this equation descries a types of extrema, not just minima. Fortunatey, when minimising the energy of a snake the amiguity is easiy resoved y changing the sign of each term in the equations of motion. then: If the functiona F is to represent the potentia energy, tension and stiffness of a snake F = P(x) + a 2 x x 2 (5.5) Page 7

18 Assuming α and β are constants, the partia derivatives for Equation 5.4 are as foows: = žp = a x = x Comining 5.4 and 5.5 gives the minima energy condition for a snake: žp a x + x = 0 Energy Gradient (5.6) Unfortunatey, these equations are difficut to sove anayticay ecause x must e known efore P/ x can e found. However, the equations can e soved using semi-impicit reaxation methods as descried in Section 6. Page 8

19 6 SEMI-IMPLICIT MINIMISATION Section 5 showed that, at equiirium, each eement in a snake satisfies a vector equation which states that it does not move during time steps: žt = a ž2 x žs 2 ž4 x žs 4 žp = 0 (6.) The energy of the mode can therefore e minimised y soving a of these equations simutaneousy using semi-impicit reaxation methods. The vector terms in Equation 6. are separae into x and y components. Writing u j where j = 0,, N as a discrete approximation for x(s) or y(s), and using superscript t to denote iteration, Equation 6. ecomes: žu j t žt = a ž2 t u j žs 2 ž4 t u j žs 4 žp žu j t (6.2) t+ 4 4 Fourth 6 Order t t+ t 2 j 2 j j j+ j+2 Second Order Figure 6: Approximating Derivatives With Finite Differences. The second-order derivative (tension force) is approximated over three eements; the fourth-order derivative (stiffness force) is approximated over five eements. In the semi-impicit method these derivatives are regarded as estimates for the next time step. The derivatives in Equation 6.2 are estimated using finite differences as shown in Figure 6 and Appendix A: žu žt d u j t+ u j t dt ž 2 u žs 2 d u j+ t+ + u t+ t+ j 2u j ds 2 ž 4 u žs 4 d u j+2 t+ 4u t+ j+ + 6u t+ j 4u t+ t+ j + u j 2 ds 4 Page 9

20 Note that the second and fourth derivatives are estimated as though at the next time step (t+). Comining these approximations gives the finite difference equation: u j t+ t u j dt = a ds 2 u t+ j+ + u t+ t+ j 2u j ds 4 u j+2 t+ 4u t+ j+ + 6u t+ j 4u t+ t+ j + u j 2 Moving terms that cannot e estimated at time t over to the LHS gives: žp žu j t t+ t+ t+ t+ t+ u j+2 (a + 4)u j+ + ( + 2a + 6)u j (a + 4)u j + u j 2 (6.3) = u j t + dt žp žu j t Note: a h adt/ds2 h dt/ds 4 The RHS of Equation 6.4 can e evauated using the potentia energy at time t: pu t+ j+2 + qu t+ j+ + ru t+ j + qu t+ t+ t+ j + pu j 2 = ũ j ũ j t+ = u j t + dt žp žu j t p h q h a 4 r h + 2a + 6 (6.4) (6.5) This equation eads to a set of 2N simutaneous inear equations (for the x and y co-ordinates of each eement in the snake) that can e written in standard matrix form. r q p p q q r q p p p q r q p p q r q p p p q r q q p p q r t+ u 0 t+ u t+ u 2 t+ u N 3 t+ u N 2 t+ u N = t+ ũ 0 t+ ũ t+ ũ 2 t+ ũ N 3 t+ ũ N 2 t+ ũ N (6.6) M u t+ = ũ t+ The constant vaues making up the matrix M are as foows: p h dt ds 4 q h a dt ds 2 4 dt ds 4 r h + 2a dt ds dt ds 4 This mathematica trick produces a set of equations that descrie the ehaviour of the mode over time. The idea is to move the snake according to the image forces at the current time step, and then to smooth the resuting mode immediatey at the start of the next iteration. At equiirium these processes cance out. Page 20

21 Each row of the matrix can e thought of as a convoution mask for evauating the derivatives; the vectors represent the positions of the snake eements, oth efore and after adjustment to conform with the interna forces. Mutipying oth sides of Equation 6.6 y the inverse of M gives the fina soution (see Mathematica Note 6): u t+ = M u t + dt žp žu t (6.7) Note that M is a cycic symmetric pentadiagona anded matrix which can e inverted using the agorithm descried y Benson and Evans (973; 977) making the soution of Equation 6.7 an O(N) process rather than O(N 3 ). If the tension and stiffness parameters and the numer of eements are constant then the inverse matrix need ony e cacuated once. #define N const apha=.0, eta=0.5; // tension, stiffness const ds=.0, ds2=ds*ds, dt=0.05; // space, time doue x[n], y[n]; // snake // code to create snake here do { // externa step for(int j=0; j<n; j++) { x[j] += dt * fx(x[j], y[j]); // image force y[j] += dt * fy(x[j], y[j]); } // interna step a=apha*dt/ds2; =eta*dt/ds2 // NB: constants? p=; q=-a-4; r=+2a+6; pentadiagona_sove(p, q, r, x, n); pentadiagona_sove(p, q, r, y, n); } whie(!equiirium); Agorithm : Semi-Impicit Energy Minimisation. This C-stye pseudo-code outines the semi-impicit agorithm for minimising the energy of an active contour mode. The co-ordinates of the N snake eements are specified y the arrays x[s] and y[s]. The externa step moves each eement according to the image forces computed using the (undefined) functions fx() and fy(). The interna step then smoothes the mode y soving a set of equations in the form of a pentadiagona anded matrix (see Equation 6.6). The process is repeated unti equiirium is detected in some way. (The effects of the interna and externa steps cance out at equiirium.) Page 2

22 The semi-impicit energy minimisation process is summarised in Agorithm. Each iteration takes impicit Euer steps with respect to the interna energy, and expicit Euer steps with respect to the externa and image energy. The minimisation process is therefore stae in the presence of very high tension and stiffness. Furthermore, with ordinary reaxation methods the propagation of forces aong a snake is sow; however, the semi-impicit procedure aows forces to trave aritrary distances in a singe O(N) iteration. Page 22

Chapter 1 Structural Mechanics

Chapter 1 Structural Mechanics Chapter Structura echanics Introduction There are many different types of structures a around us. Each structure has a specific purpose or function. Some structures are simpe, whie others are compex; however

More information

Face Hallucination and Recognition

Face Hallucination and Recognition Face Haucination and Recognition Xiaogang Wang and Xiaoou Tang Department of Information Engineering, The Chinese University of Hong Kong {xgwang1, xtang}@ie.cuhk.edu.hk http://mmab.ie.cuhk.edu.hk Abstract.

More information

An Idiot s guide to Support vector machines (SVMs)

An Idiot s guide to Support vector machines (SVMs) An Idiot s guide to Support vector machines (SVMs) R. Berwick, Viage Idiot SVMs: A New Generation of Learning Agorithms Pre 1980: Amost a earning methods earned inear decision surfaces. Linear earning

More information

3.5 Pendulum period. 2009-02-10 19:40:05 UTC / rev 4d4a39156f1e. g = 4π2 l T 2. g = 4π2 x1 m 4 s 2 = π 2 m s 2. 3.5 Pendulum period 68

3.5 Pendulum period. 2009-02-10 19:40:05 UTC / rev 4d4a39156f1e. g = 4π2 l T 2. g = 4π2 x1 m 4 s 2 = π 2 m s 2. 3.5 Pendulum period 68 68 68 3.5 Penduum period 68 3.5 Penduum period Is it coincidence that g, in units of meters per second squared, is 9.8, very cose to 2 9.87? Their proximity suggests a connection. Indeed, they are connected

More information

Chapter 3: e-business Integration Patterns

Chapter 3: e-business Integration Patterns Chapter 3: e-business Integration Patterns Page 1 of 9 Chapter 3: e-business Integration Patterns "Consistency is the ast refuge of the unimaginative." Oscar Wide In This Chapter What Are Integration Patterns?

More information

Secure Network Coding with a Cost Criterion

Secure Network Coding with a Cost Criterion Secure Network Coding with a Cost Criterion Jianong Tan, Murie Médard Laboratory for Information and Decision Systems Massachusetts Institute of Technoogy Cambridge, MA 0239, USA E-mai: {jianong, medard}@mit.edu

More information

Normalization of Database Tables. Functional Dependency. Examples of Functional Dependencies: So Now what is Normalization? Transitive Dependencies

Normalization of Database Tables. Functional Dependency. Examples of Functional Dependencies: So Now what is Normalization? Transitive Dependencies ISM 602 Dr. Hamid Nemati Objectives The idea Dependencies Attributes and Design Understand concepts normaization (Higher-Leve Norma Forms) Learn how to normaize tabes Understand normaization and database

More information

The Radix-4 and the Class of Radix-2 s FFTs

The Radix-4 and the Class of Radix-2 s FFTs Chapter 11 The Radix- and the Cass of Radix- s FFTs The divide-and-conuer paradigm introduced in Chapter 3 is not restricted to dividing a probem into two subprobems. In fact, as expained in Section. and

More information

Multi-Robot Task Scheduling

Multi-Robot Task Scheduling Proc of IEEE Internationa Conference on Robotics and Automation, Karsruhe, Germany, 013 Muti-Robot Tas Scheduing Yu Zhang and Lynne E Parer Abstract The scheduing probem has been studied extensivey in

More information

SAT Math Must-Know Facts & Formulas

SAT Math Must-Know Facts & Formulas SAT Mat Must-Know Facts & Formuas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas

More information

Artificial neural networks and deep learning

Artificial neural networks and deep learning February 20, 2015 1 Introduction Artificia Neura Networks (ANNs) are a set of statistica modeing toos originay inspired by studies of bioogica neura networks in animas, for exampe the brain and the centra

More information

A quantum model for the stock market

A quantum model for the stock market A quantum mode for the stock market Authors: Chao Zhang a,, Lu Huang b Affiiations: a Schoo of Physics and Engineering, Sun Yat-sen University, Guangzhou 5175, China b Schoo of Economics and Business Administration,

More information

Finance 360 Problem Set #6 Solutions

Finance 360 Problem Set #6 Solutions Finance 360 Probem Set #6 Soutions 1) Suppose that you are the manager of an opera house. You have a constant margina cost of production equa to $50 (i.e. each additiona person in the theatre raises your

More information

Learning framework for NNs. Introduction to Neural Networks. Learning goal: Inputs/outputs. x 1 x 2. y 1 y 2

Learning framework for NNs. Introduction to Neural Networks. Learning goal: Inputs/outputs. x 1 x 2. y 1 y 2 Introduction to Neura Networks Learning framework for NNs What are neura networks? Noninear function approimators How do they reate to pattern recognition/cassification? Noninear discriminant functions

More information

Inductance. Bởi: OpenStaxCollege

Inductance. Bởi: OpenStaxCollege Inductance Bởi: OpenStaxCoege Inductors Induction is the process in which an emf is induced by changing magnetic fux. Many exampes have been discussed so far, some more effective than others. Transformers,

More information

Maintenance activities planning and grouping for complex structure systems

Maintenance activities planning and grouping for complex structure systems Maintenance activities panning and grouping for compex structure systems Hai Canh u, Phuc Do an, Anne Barros, Christophe Berenguer To cite this version: Hai Canh u, Phuc Do an, Anne Barros, Christophe

More information

Advanced ColdFusion 4.0 Application Development - 3 - Server Clustering Using Bright Tiger

Advanced ColdFusion 4.0 Application Development - 3 - Server Clustering Using Bright Tiger Advanced CodFusion 4.0 Appication Deveopment - CH 3 - Server Custering Using Bri.. Page 1 of 7 [Figures are not incuded in this sampe chapter] Advanced CodFusion 4.0 Appication Deveopment - 3 - Server

More information

WHITE PAPER BEsT PRAcTIcEs: PusHIng ExcEl BEyond ITs limits WITH InfoRmATIon optimization

WHITE PAPER BEsT PRAcTIcEs: PusHIng ExcEl BEyond ITs limits WITH InfoRmATIon optimization Best Practices: Pushing Exce Beyond Its Limits with Information Optimization WHITE Best Practices: Pushing Exce Beyond Its Limits with Information Optimization Executive Overview Microsoft Exce is the

More information

The guaranteed selection. For certainty in uncertain times

The guaranteed selection. For certainty in uncertain times The guaranteed seection For certainty in uncertain times Making the right investment choice If you can t afford to take a ot of risk with your money it can be hard to find the right investment, especiay

More information

CONTRIBUTION OF INTERNAL AUDITING IN THE VALUE OF A NURSING UNIT WITHIN THREE YEARS

CONTRIBUTION OF INTERNAL AUDITING IN THE VALUE OF A NURSING UNIT WITHIN THREE YEARS Dehi Business Review X Vo. 4, No. 2, Juy - December 2003 CONTRIBUTION OF INTERNAL AUDITING IN THE VALUE OF A NURSING UNIT WITHIN THREE YEARS John N.. Var arvatsouakis atsouakis DURING the present time,

More information

Introduction to XSL. Max Froumentin - W3C

Introduction to XSL. Max Froumentin - W3C Introduction to XSL Max Froumentin - W3C Introduction to XSL XML Documents Stying XML Documents XSL Exampe I: Hamet Exampe II: Mixed Writing Modes Exampe III: database Other Exampes How do they do that?

More information

Teamwork. Abstract. 2.1 Overview

Teamwork. Abstract. 2.1 Overview 2 Teamwork Abstract This chapter presents one of the basic eements of software projects teamwork. It addresses how to buid teams in a way that promotes team members accountabiity and responsibiity, and

More information

5. Introduction to Robot Geometry and Kinematics

5. Introduction to Robot Geometry and Kinematics V. Kumar 5. Introduction to Robot Geometry and Kinematics The goa of this chapter is to introduce the basic terminoogy and notation used in robot geometry and kinematics, and to discuss the methods used

More information

Art of Java Web Development By Neal Ford 624 pages US$44.95 Manning Publications, 2004 ISBN: 1-932394-06-0

Art of Java Web Development By Neal Ford 624 pages US$44.95 Manning Publications, 2004 ISBN: 1-932394-06-0 IEEE DISTRIBUTED SYSTEMS ONLINE 1541-4922 2005 Pubished by the IEEE Computer Society Vo. 6, No. 5; May 2005 Editor: Marcin Paprzycki, http://www.cs.okstate.edu/%7emarcin/ Book Reviews: Java Toos and Frameworks

More information

Physics 100A Homework 11- Chapter 11 (part 1) The force passes through the point A, so there is no arm and the torque is zero.

Physics 100A Homework 11- Chapter 11 (part 1) The force passes through the point A, so there is no arm and the torque is zero. Physics A Homework - Chapter (part ) Finding Torque A orce F o magnitude F making an ange with the x axis is appied to a partice ocated aong axis o rotation A, at Cartesian coordinates (,) in the igure.

More information

3.3 SOFTWARE RISK MANAGEMENT (SRM)

3.3 SOFTWARE RISK MANAGEMENT (SRM) 93 3.3 SOFTWARE RISK MANAGEMENT (SRM) Fig. 3.2 SRM is a process buit in five steps. The steps are: Identify Anayse Pan Track Resove The process is continuous in nature and handed dynamicay throughout ifecyce

More information

ELEVATING YOUR GAME FROM TRADE SPEND TO TRADE INVESTMENT

ELEVATING YOUR GAME FROM TRADE SPEND TO TRADE INVESTMENT Initiatives Strategic Mapping Success in The Food System: Discover. Anayze. Strategize. Impement. Measure. ELEVATING YOUR GAME FROM TRADE SPEND TO TRADE INVESTMENT Foodservice manufacturers aocate, in

More information

Order-to-Cash Processes

Order-to-Cash Processes TMI170 ING info pat 2:Info pat.qxt 01/12/2008 09:25 Page 1 Section Two: Order-to-Cash Processes Gregory Cronie, Head Saes, Payments and Cash Management, ING O rder-to-cash and purchase-topay processes

More information

Betting on the Real Line

Betting on the Real Line Betting on the Rea Line Xi Gao 1, Yiing Chen 1,, and David M. Pennock 2 1 Harvard University, {xagao,yiing}@eecs.harvard.edu 2 Yahoo! Research, pennockd@yahoo-inc.com Abstract. We study the probem of designing

More information

CERTIFICATE COURSE ON CLIMATE CHANGE AND SUSTAINABILITY. Course Offered By: Indian Environmental Society

CERTIFICATE COURSE ON CLIMATE CHANGE AND SUSTAINABILITY. Course Offered By: Indian Environmental Society CERTIFICATE COURSE ON CLIMATE CHANGE AND SUSTAINABILITY Course Offered By: Indian Environmenta Society INTRODUCTION The Indian Environmenta Society (IES) a dynamic and fexibe organization with a goba vision

More information

Vibration Reduction of Audio Visual Device Mounted on Automobile due to Gap Vibration

Vibration Reduction of Audio Visual Device Mounted on Automobile due to Gap Vibration Vibration Reduction of Audio Visua Device Mounted on Automobie due to Gap Vibration Nobuyuki OKUBO, Shinji KANADA, Takeshi TOI CAMAL, Department of Precision Mechanics, Chuo University 1-13-27 Kasuga,

More information

A Supplier Evaluation System for Automotive Industry According To Iso/Ts 16949 Requirements

A Supplier Evaluation System for Automotive Industry According To Iso/Ts 16949 Requirements A Suppier Evauation System for Automotive Industry According To Iso/Ts 16949 Requirements DILEK PINAR ÖZTOP 1, ASLI AKSOY 2,*, NURSEL ÖZTÜRK 2 1 HONDA TR Purchasing Department, 41480, Çayırova - Gebze,

More information

SELECTING THE SUITABLE ERP SYSTEM: A FUZZY AHP APPROACH. Ufuk Cebeci

SELECTING THE SUITABLE ERP SYSTEM: A FUZZY AHP APPROACH. Ufuk Cebeci SELECTING THE SUITABLE ERP SYSTEM: A FUZZY AHP APPROACH Ufuk Cebeci Department of Industria Engineering, Istanbu Technica University, Macka, Istanbu, Turkey - ufuk_cebeci@yahoo.com Abstract An Enterprise

More information

CONDENSATION. Prabal Talukdar. Associate Professor Department of Mechanical Engineering IIT Delhi E-mail: prabal@mech.iitd.ac.in

CONDENSATION. Prabal Talukdar. Associate Professor Department of Mechanical Engineering IIT Delhi E-mail: prabal@mech.iitd.ac.in CONDENSATION Praba Taukdar Associate Professor Department of Mechanica Engineering IIT Dehi E-mai: praba@mech.iitd.ac.in Condensation When a vapor is exposed to a surface at a temperature beow T sat, condensation

More information

Simultaneous Routing and Power Allocation in CDMA Wireless Data Networks

Simultaneous Routing and Power Allocation in CDMA Wireless Data Networks Simutaneous Routing and Power Aocation in CDMA Wireess Data Networks Mikae Johansson *,LinXiao and Stephen Boyd * Department of Signas, Sensors and Systems Roya Institute of Technoogy, SE 00 Stockhom,

More information

Pricing Internet Services With Multiple Providers

Pricing Internet Services With Multiple Providers Pricing Internet Services With Mutipe Providers Linhai He and Jean Warand Dept. of Eectrica Engineering and Computer Science University of Caifornia at Berkeey Berkeey, CA 94709 inhai, wr@eecs.berkeey.edu

More information

A Similarity Search Scheme over Encrypted Cloud Images based on Secure Transformation

A Similarity Search Scheme over Encrypted Cloud Images based on Secure Transformation A Simiarity Search Scheme over Encrypted Coud Images based on Secure Transormation Zhihua Xia, Yi Zhu, Xingming Sun, and Jin Wang Jiangsu Engineering Center o Network Monitoring, Nanjing University o Inormation

More information

Figure 1. A Simple Centrifugal Speed Governor.

Figure 1. A Simple Centrifugal Speed Governor. ENGINE SPEED CONTROL Peter Westead and Mark Readman, contro systems principes.co.uk ABSTRACT: This is one of a series of white papers on systems modeing, anaysis and contro, prepared by Contro Systems

More information

Business schools are the academic setting where. The current crisis has highlighted the need to redefine the role of senior managers in organizations.

Business schools are the academic setting where. The current crisis has highlighted the need to redefine the role of senior managers in organizations. c r o s os r oi a d s REDISCOVERING THE ROLE OF BUSINESS SCHOOLS The current crisis has highighted the need to redefine the roe of senior managers in organizations. JORDI CANALS Professor and Dean, IESE

More information

1B11 Operating Systems. Input/Output and Devices

1B11 Operating Systems. Input/Output and Devices University Coege London 1B11 Operating Systems Input/Output and s Prof. Steve R Wibur s.wibur@cs.uc.ac.uk Lecture Objectives How do the bits of the I/O story fit together? What is a device driver? 1B11-5

More information

Early access to FAS payments for members in poor health

Early access to FAS payments for members in poor health Financia Assistance Scheme Eary access to FAS payments for members in poor heath Pension Protection Fund Protecting Peope s Futures The Financia Assistance Scheme is administered by the Pension Protection

More information

Fast Robust Hashing. ) [7] will be re-mapped (and therefore discarded), due to the load-balancing property of hashing.

Fast Robust Hashing. ) [7] will be re-mapped (and therefore discarded), due to the load-balancing property of hashing. Fast Robust Hashing Manue Urueña, David Larrabeiti and Pabo Serrano Universidad Caros III de Madrid E-89 Leganés (Madrid), Spain Emai: {muruenya,darra,pabo}@it.uc3m.es Abstract As statefu fow-aware services

More information

Lecture 7 Datalink Ethernet, Home. Datalink Layer Architectures

Lecture 7 Datalink Ethernet, Home. Datalink Layer Architectures Lecture 7 Dataink Ethernet, Home Peter Steenkiste Schoo of Computer Science Department of Eectrica and Computer Engineering Carnegie Meon University 15-441 Networking, Spring 2004 http://www.cs.cmu.edu/~prs/15-441

More information

Australian Bureau of Statistics Management of Business Providers

Australian Bureau of Statistics Management of Business Providers Purpose Austraian Bureau of Statistics Management of Business Providers 1 The principa objective of the Austraian Bureau of Statistics (ABS) in respect of business providers is to impose the owest oad

More information

FRAME BASED TEXTURE CLASSIFICATION BY CONSIDERING VARIOUS SPATIAL NEIGHBORHOODS. Karl Skretting and John Håkon Husøy

FRAME BASED TEXTURE CLASSIFICATION BY CONSIDERING VARIOUS SPATIAL NEIGHBORHOODS. Karl Skretting and John Håkon Husøy FRAME BASED TEXTURE CLASSIFICATION BY CONSIDERING VARIOUS SPATIAL NEIGHBORHOODS Kar Skretting and John Håkon Husøy University of Stavanger, Department of Eectrica and Computer Engineering N-4036 Stavanger,

More information

SAT Math Facts & Formulas

SAT Math Facts & Formulas Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences:

More information

Betting Strategies, Market Selection, and the Wisdom of Crowds

Betting Strategies, Market Selection, and the Wisdom of Crowds Betting Strategies, Market Seection, and the Wisdom of Crowds Wiemien Kets Northwestern University w-kets@keogg.northwestern.edu David M. Pennock Microsoft Research New York City dpennock@microsoft.com

More information

GWPD 4 Measuring water levels by use of an electric tape

GWPD 4 Measuring water levels by use of an electric tape GWPD 4 Measuring water eves by use of an eectric tape VERSION: 2010.1 PURPOSE: To measure the depth to the water surface beow and-surface datum using the eectric tape method. Materias and Instruments 1.

More information

Take me to your leader! Online Optimization of Distributed Storage Configurations

Take me to your leader! Online Optimization of Distributed Storage Configurations Take me to your eader! Onine Optimization of Distributed Storage Configurations Artyom Sharov Aexander Shraer Arif Merchant Murray Stokey sharov@cs.technion.ac.i, {shraex, aamerchant, mstokey}@googe.com

More information

Market Design & Analysis for a P2P Backup System

Market Design & Analysis for a P2P Backup System Market Design & Anaysis for a P2P Backup System Sven Seuken Schoo of Engineering & Appied Sciences Harvard University, Cambridge, MA seuken@eecs.harvard.edu Denis Chares, Max Chickering, Sidd Puri Microsoft

More information

READING A CREDIT REPORT

READING A CREDIT REPORT Name Date CHAPTER 6 STUDENT ACTIVITY SHEET READING A CREDIT REPORT Review the sampe credit report. Then search for a sampe credit report onine, print it off, and answer the questions beow. This activity

More information

A Description of the California Partnership for Long-Term Care Prepared by the California Department of Health Care Services

A Description of the California Partnership for Long-Term Care Prepared by the California Department of Health Care Services 2012 Before You Buy A Description of the Caifornia Partnership for Long-Term Care Prepared by the Caifornia Department of Heath Care Services Page 1 of 13 Ony ong-term care insurance poicies bearing any

More information

Virtual trunk simulation

Virtual trunk simulation Virtua trunk simuation Samui Aato * Laboratory of Teecommunications Technoogy Hesinki University of Technoogy Sivia Giordano Laboratoire de Reseaux de Communication Ecoe Poytechnique Federae de Lausanne

More information

Automatic Projector Display Surface Estimation. Using Every-Day Imagery

Automatic Projector Display Surface Estimation. Using Every-Day Imagery Automatic Projector Dispay Surface Estimation Using Every-Day Imagery Ruigang Yang and Greg Wech Department of Computer Science Univeristy of North Caroina at Chape Hi Abstract Projector-based dispay systems

More information

the points are called control points approximating curve

the points are called control points approximating curve Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.

More information

Hybrid Process Algebra

Hybrid Process Algebra Hybrid Process Agebra P.J.L. Cuijpers M.A. Reniers Eindhoven University of Technoogy (TU/e) Den Doech 2 5600 MB Eindhoven, The Netherands Abstract We deveop an agebraic theory, caed hybrid process agebra

More information

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:

More information

With the arrival of Java 2 Micro Edition (J2ME) and its industry

With the arrival of Java 2 Micro Edition (J2ME) and its industry Knowedge-based Autonomous Agents for Pervasive Computing Using AgentLight Fernando L. Koch and John-Jues C. Meyer Utrecht University Project AgentLight is a mutiagent system-buiding framework targeting

More information

Pricing and Revenue Sharing Strategies for Internet Service Providers

Pricing and Revenue Sharing Strategies for Internet Service Providers Pricing and Revenue Sharing Strategies for Internet Service Providers Linhai He and Jean Warand Department of Eectrica Engineering and Computer Sciences University of Caifornia at Berkeey {inhai,wr}@eecs.berkeey.edu

More information

NCH Software FlexiServer

NCH Software FlexiServer NCH Software FexiServer This user guide has been created for use with FexiServer Version 1.xx NCH Software Technica Support If you have difficuties using FexiServer pease read the appicabe topic before

More information

Key Features of Life Insurance

Key Features of Life Insurance Key Features of Life Insurance Life Insurance Key Features The Financia Conduct Authority is a financia services reguator. It requires us, Aviva, to give you this important information to hep you to decide

More information

Chapter 2 Traditional Software Development

Chapter 2 Traditional Software Development Chapter 2 Traditiona Software Deveopment 2.1 History of Project Management Large projects from the past must aready have had some sort of project management, such the Pyramid of Giza or Pyramid of Cheops,

More information

SABRe B2.1: Design & Development. Supplier Briefing Pack.

SABRe B2.1: Design & Development. Supplier Briefing Pack. SABRe B2.1: Design & Deveopment. Suppier Briefing Pack. 2013 Ros-Royce pc The information in this document is the property of Ros-Royce pc and may not be copied or communicated to a third party, or used

More information

Life Contingencies Study Note for CAS Exam S. Tom Struppeck

Life Contingencies Study Note for CAS Exam S. Tom Struppeck Life Contingencies Study Note for CAS Eam S Tom Struppeck (Revised 9/19/2015) Introduction Life contingencies is a term used to describe surviva modes for human ives and resuting cash fows that start or

More information

Dynamic Pricing Trade Market for Shared Resources in IIU Federated Cloud

Dynamic Pricing Trade Market for Shared Resources in IIU Federated Cloud Dynamic Pricing Trade Market or Shared Resources in IIU Federated Coud Tongrang Fan 1, Jian Liu 1, Feng Gao 1 1Schoo o Inormation Science and Technoogy, Shiiazhuang Tiedao University, Shiiazhuang, 543,

More information

Income Protection Options

Income Protection Options Income Protection Options Poicy Conditions Introduction These poicy conditions are written confirmation of your contract with Aviva Life & Pensions UK Limited. It is important that you read them carefuy

More information

Precise assessment of partial discharge in underground MV/HV power cables and terminations

Precise assessment of partial discharge in underground MV/HV power cables and terminations QCM-C-PD-Survey Service Partia discharge monitoring for underground power cabes Precise assessment of partia discharge in underground MV/HV power cabes and terminations Highy accurate periodic PD survey

More information

CLOUD service providers manage an enterprise-class

CLOUD service providers manage an enterprise-class IEEE TRANSACTIONS ON XXXXXX, VOL X, NO X, XXXX 201X 1 Oruta: Privacy-Preserving Pubic Auditing for Shared Data in the Coud Boyang Wang, Baochun Li, Member, IEEE, and Hui Li, Member, IEEE Abstract With

More information

eg Enterprise vs. a Big 4 Monitoring Soution: Comparing Tota Cost of Ownership Restricted Rights Legend The information contained in this document is confidentia and subject to change without notice. No

More information

Network/Communicational Vulnerability

Network/Communicational Vulnerability Automated teer machines (ATMs) are a part of most of our ives. The major appea of these machines is convenience The ATM environment is changing and that change has serious ramifications for the security

More information

Applying graph theory to automatic vehicle tracking by remote sensing

Applying graph theory to automatic vehicle tracking by remote sensing 0 0 Appying graph theory to automatic vehice tracking by remote sensing *Caros Lima Azevedo Nationa Laboratory for Civi Engineering Department of Transportation Av. Do Brasi, Lisbon, 00-0 Portuga Phone:

More information

The Simple Pendulum. by Dr. James E. Parks

The Simple Pendulum. by Dr. James E. Parks by Dr. James E. Parks Department of Physics and Astronomy 401 Niesen Physics Buidin The University of Tennessee Knoxvie, Tennessee 37996-100 Copyriht June, 000 by James Edar Parks* *A rihts are reserved.

More information

GREEN: An Active Queue Management Algorithm for a Self Managed Internet

GREEN: An Active Queue Management Algorithm for a Self Managed Internet : An Active Queue Management Agorithm for a Sef Managed Internet Bartek Wydrowski and Moshe Zukerman ARC Specia Research Centre for Utra-Broadband Information Networks, EEE Department, The University of

More information

Best Practices for Push & Pull Using Oracle Inventory Stock Locators. Introduction to Master Data and Master Data Management (MDM): Part 1

Best Practices for Push & Pull Using Oracle Inventory Stock Locators. Introduction to Master Data and Master Data Management (MDM): Part 1 SPECIAL CONFERENCE ISSUE THE OFFICIAL PUBLICATION OF THE Orace Appications USERS GROUP spring 2012 Introduction to Master Data and Master Data Management (MDM): Part 1 Utiizing Orace Upgrade Advisor for

More information

THE CAUSES OF IBC (INTERMEDIATE BULK CONTAINER) LEAKS AT CHEMICAL PLANTS AN ANALYSIS OF OPERATING EXPERIENCE

THE CAUSES OF IBC (INTERMEDIATE BULK CONTAINER) LEAKS AT CHEMICAL PLANTS AN ANALYSIS OF OPERATING EXPERIENCE THE CAUSES OF IBC (INTERMEDIATE BULK CONTAINER) LEAKS AT CHEMICAL PLANTS AN ANALYSIS OF OPERATING EXPERIENCE Christopher J. Beae (FIChemE) Ciba Expert Services, Charter Way, Maccesfied, Cheshire, SK10

More information

WHITE PAPER UndERsTAndIng THE VAlUE of VIsUAl data discovery A guide To VIsUAlIzATIons

WHITE PAPER UndERsTAndIng THE VAlUE of VIsUAl data discovery A guide To VIsUAlIzATIons Understanding the Vaue of Visua Data Discovery A Guide to Visuaizations WHITE Tabe of Contents Executive Summary... 3 Chapter 1 - Datawatch Visuaizations... 4 Chapter 2 - Snapshot Visuaizations... 5 Bar

More information

ASYMPTOTIC DIRECTION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS arxiv:math/0512388v2 [math.pr] 11 Dec 2007

ASYMPTOTIC DIRECTION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS arxiv:math/0512388v2 [math.pr] 11 Dec 2007 ASYMPTOTIC DIRECTION FOR RANDOM WALKS IN RANDOM ENVIRONMENTS arxiv:math/0512388v2 [math.pr] 11 Dec 2007 FRANÇOIS SIMENHAUS Université Paris 7, Mathématiques, case 7012, 2, pace Jussieu, 75251 Paris, France

More information

Week 3: Consumer and Firm Behaviour: The Work-Leisure Decision and Profit Maximization

Week 3: Consumer and Firm Behaviour: The Work-Leisure Decision and Profit Maximization AROEOOIS 2006 Week 3: onsumer and Firm Behaviour: The Work-Leisure Decision and Profit aximization Questions for Review 1. How are a consumer s preferences over goods represented? By utiity functions:

More information

SNMP Reference Guide for Avaya Communication Manager

SNMP Reference Guide for Avaya Communication Manager SNMP Reference Guide for Avaya Communication Manager 03-602013 Issue 1.0 Feburary 2007 2006 Avaya Inc. A Rights Reserved. Notice Whie reasonabe efforts were made to ensure that the information in this

More information

1 Basic concepts in geometry

1 Basic concepts in geometry 1 asic concepts in geometry 1.1 Introduction We start geometry with the simpest idea a point. It is shown using a dot, which is abeed with a capita etter. The exampe above is the point. straight ine is

More information

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 12, DECEMBER 2013 1

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 12, DECEMBER 2013 1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 12, DECEMBER 2013 1 Scaabe Muti-Cass Traffic Management in Data Center Backbone Networks Amitabha Ghosh, Sangtae Ha, Edward Crabbe, and Jennifer

More information

Thermal properties. Heat capacity atomic vibrations, phonons temperature dependence contribution of electrons

Thermal properties. Heat capacity atomic vibrations, phonons temperature dependence contribution of electrons Therma properties Heat capacity atomic vibrations, phonons temperature dependence contribution of eectrons Therma expansion connection to anharmonicity of interatomic potentia inear and voume coefficients

More information

PREFACE. Comptroller General of the United States. Page i

PREFACE. Comptroller General of the United States. Page i - I PREFACE T he (+nera Accounting Office (GAO) has ong beieved that the federa government urgenty needs to improve the financia information on which it bases many important decisions. To run our compex

More information

The Use of Cooling-Factor Curves for Coordinating Fuses and Reclosers

The Use of Cooling-Factor Curves for Coordinating Fuses and Reclosers he Use of ooing-factor urves for oordinating Fuses and Recosers arey J. ook Senior Member, IEEE S& Eectric ompany hicago, Iinois bstract his paper describes how to precisey coordinate distribution feeder

More information

Pay-on-delivery investing

Pay-on-delivery investing Pay-on-deivery investing EVOLVE INVESTment range 1 EVOLVE INVESTMENT RANGE EVOLVE INVESTMENT RANGE 2 Picture a word where you ony pay a company once they have deivered Imagine striking oi first, before

More information

Solutions for Review Problems

Solutions for Review Problems olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

More information

COMPARISON OF DIFFUSION MODELS IN ASTRONOMICAL OBJECT LOCALIZATION

COMPARISON OF DIFFUSION MODELS IN ASTRONOMICAL OBJECT LOCALIZATION COMPARISON OF DIFFUSION MODELS IN ASTRONOMICAL OBJECT LOCALIZATION Františe Mojžíš Department of Computing and Contro Engineering, ICT Prague, Technicá, 8 Prague frantise.mojzis@vscht.cz Abstract This

More information

LADDER SAFETY Table of Contents

LADDER SAFETY Table of Contents Tabe of Contents SECTION 1. TRAINING PROGRAM INTRODUCTION..................3 Training Objectives...........................................3 Rationae for Training.........................................3

More information

Measuring operational risk in financial institutions

Measuring operational risk in financial institutions Measuring operationa risk in financia institutions Operationa risk is now seen as a major risk for financia institutions. This paper considers the various methods avaiabe to measure operationa risk, and

More information

Older people s assets: using housing equity to pay for health and aged care

Older people s assets: using housing equity to pay for health and aged care Key words: aged care; retirement savings; reverse mortgage; financia innovation; financia panning Oder peope s assets: using housing equity to pay for heath and aged care The research agenda on the ageing

More information

Arbitrary High Order Finite Volume Schemes for Seismic Wave Propagation on Unstructured Meshes in 2D and 3D

Arbitrary High Order Finite Volume Schemes for Seismic Wave Propagation on Unstructured Meshes in 2D and 3D Geophys. J. Int. ( 4, Arbitrary High Order Finite Voume Schemes for Seismic Wave Propagation on Unstructured Meshes in D and 3D Michae Dumbser,, Martin Käser, Josep de a Puente 3 Department of Civi and

More information

Federal Financial Management Certificate Program

Federal Financial Management Certificate Program MANAGEMENT CONCEPTS Federa Financia Management Certificate Program Training to hep you achieve the highest eve performance in: Accounting // Auditing // Budgeting // Financia Management ENROLL TODAY! Contract

More information

Licensed to: CengageBrain User

Licensed to: CengageBrain User Licensed to: Licensed to: This is an eectronic version of the print textbook. Due to eectronic rights restrictions, some third party content may be suppressed. Editoria review has deemed that any suppressed

More information

Lexmark ESF Applications Guide

Lexmark ESF Applications Guide Lexmark ESF Appications Guide Hep your customers bring out the fu potentia of their Lexmark soutions-enabed singe-function and mutifunction printers Lexmark Appications have been designed to hep businesses

More information

ABSTRACT. Categories and Subject Descriptors. General Terms. Keywords 1. INTRODUCTION. Jun Yin, Ye Wang and David Hsu

ABSTRACT. Categories and Subject Descriptors. General Terms. Keywords 1. INTRODUCTION. Jun Yin, Ye Wang and David Hsu Jun Yin, Ye Wang and David Hsu ABSTRACT Prompt feedback is essentia for beginning vioin earners; however, most amateur earners can ony meet with teachers and receive feedback once or twice a week. To hep

More information

Enhanced continuous, real-time detection, alarming and analysis of partial discharge events

Enhanced continuous, real-time detection, alarming and analysis of partial discharge events DMS PDMG-RH DMS PDMG-RH Partia discharge monitor for GIS Partia discharge monitor for GIS Enhanced continuous, rea-time detection, aarming and anaysis of partia discharge events Unrivaed PDM feature set

More information

Advertising opportunities with the Irish National Teachers Organisation (INTO)

Advertising opportunities with the Irish National Teachers Organisation (INTO) Advertising opportunities with the Irish Nationa Teachers Organisation (INTO) 1. InTouch Magazine 2. Website 3. E Newsetter 4. INTO Members Diary: Advertising and sponsorship opportunities 5. Tips for

More information

A New Statistical Approach to Network Anomaly Detection

A New Statistical Approach to Network Anomaly Detection A New Statistica Approach to Network Anomay Detection Christian Caegari, Sandrine Vaton 2, and Michee Pagano Dept of Information Engineering, University of Pisa, ITALY E-mai: {christiancaegari,mpagano}@ietunipiit

More information

Budgeting Loans from the Social Fund

Budgeting Loans from the Social Fund Budgeting Loans from the Socia Fund tes sheet Pease read these notes carefuy. They expain the circumstances when a budgeting oan can be paid. Budgeting Loans You may be abe to get a Budgeting Loan if:

More information