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


 Hope Davis
 2 years ago
 Views:
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. Emai:
2 ABSTRACT Active contour modes known cooquiay as snakes are energyminimising 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 EuerLagrange 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 EuerLagrange 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 Loweve visua tasks such as edge detection and stereo matching are often treated as autonomous ottomup processes. However, this sequentia approach propagates mistakes to higher processes without providing opportunities for correction. A more attainae goa for oweve 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 oweve mechanism that seeks appropriate oca minima rather than searching for a goa soution. In addition, higheve 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 higheve 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 higheve process rarey materiaise except in human form! Page 3
4 gradientsquared 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 (semiimpicit) 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 EuerLagrange 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 semiimpicit reaxation method ased on a fast matrix inversion agorithm. (Both the expicit and semiimpicit 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 intersnake 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 higheve 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 openended 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 (INTRASNAKE) 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 firstorder term controed y α(s), and a secondorder term controed y β(s). The firstorder term makes the snake contract ike an eastic and y introducing tension; the secondorder 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 infiniteyeastic 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 secondorder 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 (EXTRASNAKE) 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 springike 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 pseudoattraction; 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 owenergy 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 semigoa edgedetectors 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 256y256 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 rescaed 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 gradientased 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 secondorder 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 threedimensiona 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 reparameterised 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 semiimpicit 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 semiimpicit method is descried in the next two sections. Page 5
16 5 CALCULUS OF VARIATIONS This section uses variationa cacuus to derive the EuerLagrange 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 EULERLAGRANGE 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 nonzero, Equation 5.6 gives rise to the EuerLagrange Equation which is satisfied at extrema in F: žy d dx žy = 0 (5.7) Of course, the extremum descried y the EuerLagrange equation coud e a maximum or a point of infection rather than a minimum. If necessary, the secondorder 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 EuerLagrange 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 semiimpicit reaxation methods as descried in Section 6. Page 8
19 6 SEMIIMPLICIT 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 semiimpicit 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 secondorder derivative (tension force) is approximated over three eements; the fourthorder derivative (stiffness force) is approximated over five eements. In the semiimpicit 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 coordinates 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=a4; r=+2a+6; pentadiagona_sove(p, q, r, x, n); pentadiagona_sove(p, q, r, y, n); } whie(!equiirium); Agorithm : SemiImpicit Energy Minimisation. This Cstye pseudocode outines the semiimpicit agorithm for minimising the energy of an active contour mode. The coordinates 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 semiimpicit 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 semiimpicit procedure aows forces to trave aritrary distances in a singe O(N) iteration. Page 22
Sorting, Merge Sort and the DivideandConquer Technique
Inf2B gorithms and Data Structures Note 7 Sorting, Merge Sort and the DivideandConquer Technique This and a subsequent next ecture wi mainy be concerned with sorting agorithms. Sorting is an extremey
More informationChapter 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 informationFace 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 informationAn 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 information3.5 Pendulum period. 20090210 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 informationChapter 3: ebusiness Integration Patterns
Chapter 3: ebusiness Integration Patterns Page 1 of 9 Chapter 3: ebusiness Integration Patterns "Consistency is the ast refuge of the unimaginative." Oscar Wide In This Chapter What Are Integration Patterns?
More informationSecure 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 Emai: {jianong, medard}@mit.edu
More informationNormalization 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 (HigherLeve Norma Forms) Learn how to normaize tabes Understand normaization and database
More informationThe Radix4 and the Class of Radix2 s FFTs
Chapter 11 The Radix and the Cass of Radix s FFTs The divideandconuer paradigm introduced in Chapter 3 is not restricted to dividing a probem into two subprobems. In fact, as expained in Section. and
More informationSAT Math MustKnow Facts & Formulas
SAT Mat MustKnow 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 informationA 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 Yatsen University, Guangzhou 5175, China b Schoo of Economics and Business Administration,
More informationMultiRobot Task Scheduling
Proc of IEEE Internationa Conference on Robotics and Automation, Karsruhe, Germany, 013 MutiRobot Tas Scheduing Yu Zhang and Lynne E Parer Abstract The scheduing probem has been studied extensivey in
More informationArtificial 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 informationLearning 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 informationFinance 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 informationInductance. 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 informationMaintenance 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 informationAdvanced 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 informationIntroduction 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 informationWHITE 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 informationCONTRIBUTION 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 informationThe 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 informationArt of Java Web Development By Neal Ford 624 pages US$44.95 Manning Publications, 2004 ISBN: 1932394060
IEEE DISTRIBUTED SYSTEMS ONLINE 15414922 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 information5. 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 informationPhysics 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 informationELEVATING 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 information3.3 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 informationTeamwork. 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 informationOrdertoCash Processes
TMI170 ING info pat 2:Info pat.qxt 01/12/2008 09:25 Page 1 Section Two: OrdertoCash Processes Gregory Cronie, Head Saes, Payments and Cash Management, ING O rdertocash and purchasetopay processes
More informationBetting 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@yahooinc.com Abstract. We study the probem of designing
More informationCERTIFICATE 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 informationSELECTING 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 informationCONDENSATION. Prabal Talukdar. Associate Professor Department of Mechanical Engineering IIT Delhi Email: prabal@mech.iitd.ac.in
CONDENSATION Praba Taukdar Associate Professor Department of Mechanica Engineering IIT Dehi Emai: praba@mech.iitd.ac.in Condensation When a vapor is exposed to a surface at a temperature beow T sat, condensation
More informationA 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 informationSimultaneous 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 informationVibration 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 11327 Kasuga,
More informationA 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 informationFigure 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 information1B11 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? 1B115
More informationFast Robust Hashing. ) [7] will be remapped (and therefore discarded), due to the loadbalancing property of hashing.
Fast Robust Hashing Manue Urueña, David Larrabeiti and Pabo Serrano Universidad Caros III de Madrid E89 Leganés (Madrid), Spain Emai: {muruenya,darra,pabo}@it.uc3m.es Abstract As statefu fowaware services
More informationBusiness 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 informationEarly 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 informationLecture 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 15441 Networking, Spring 2004 http://www.cs.cmu.edu/~prs/15441
More informationAustralian 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 informationFRAME 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 N4036 Stavanger,
More informationPricing 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 informationSAT 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 informationGWPD 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 andsurface datum using the eectric tape method. Materias and Instruments 1.
More informationBetting Strategies, Market Selection, and the Wisdom of Crowds
Betting Strategies, Market Seection, and the Wisdom of Crowds Wiemien Kets Northwestern University wkets@keogg.northwestern.edu David M. Pennock Microsoft Research New York City dpennock@microsoft.com
More informationVirtual 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 informationMarket 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 informationREADING 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 informationCalculation of helicopter maneuverability in forward flight based on energy method
COMPUTER MODELLING & NEW TECHNOLOGIES 214 18(5) 554 Cacuation of heicopter maneuverabiity in forward fight based on energy method Abstract Nanjian Zhuang, Jinwu Xiang, Zhangping Luo *, Yiru Ren Schoo
More informationA Description of the California Partnership for LongTerm Care Prepared by the California Department of Health Care Services
2012 Before You Buy A Description of the Caifornia Partnership for LongTerm Care Prepared by the Caifornia Department of Heath Care Services Page 1 of 13 Ony ongterm care insurance poicies bearing any
More informationPricing 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 informationAutomatic Projector Display Surface Estimation. Using EveryDay Imagery
Automatic Projector Dispay Surface Estimation Using EveryDay Imagery Ruigang Yang and Greg Wech Department of Computer Science Univeristy of North Caroina at Chape Hi Abstract Projectorbased dispay systems
More informationTake 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 informationHybrid 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 informationRAJALAKSHMI 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 informationWith the arrival of Java 2 Micro Edition (J2ME) and its industry
Knowedgebased Autonomous Agents for Pervasive Computing Using AgentLight Fernando L. Koch and JohnJues C. Meyer Utrecht University Project AgentLight is a mutiagent systembuiding framework targeting
More informationDivide and Conquer Approach
Divide and Conquer Approac Deiverabes Divide and Conquer Paradigm nteger Mutipication Strassen Matrix Mutipication Cosest Pair of points nfinite Wa Probem 6/7/01 8:58 PM Copyrigt @ gdeepak.com Divide and
More informationKey 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 informationChapter 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 informationSABRe B2.1: Design & Development. Supplier Briefing Pack.
SABRe B2.1: Design & Deveopment. Suppier Briefing Pack. 2013 RosRoyce pc The information in this document is the property of RosRoyce pc and may not be copied or communicated to a third party, or used
More informationPrecise assessment of partial discharge in underground MV/HV power cables and terminations
QCMCPDSurvey 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 informationDynamic 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 informationCLOUD service providers manage an enterpriseclass
IEEE TRANSACTIONS ON XXXXXX, VOL X, NO X, XXXX 201X 1 Oruta: PrivacyPreserving Pubic Auditing for Shared Data in the Coud Boyang Wang, Baochun Li, Member, IEEE, and Hui Li, Member, IEEE Abstract With
More informationIncome 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 informationNCH 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 informationthe 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 informationThe 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 37996100 Copyriht June, 000 by James Edar Parks* *A rihts are reserved.
More informationeg 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 informationNetwork/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 informationApplying 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, 000 Portuga Phone:
More informationTHE 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 informationWHITE 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 informationASYMPTOTIC 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 informationThermal 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 informationSNMP Reference Guide for Avaya Communication Manager
SNMP Reference Guide for Avaya Communication Manager 03602013 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 informationArbitrary 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 informationComputing the depth of an arrangement of axisaligned rectangles in parallel
Computing the depth of an arrangement of axisaigned rectanges in parae Hemut At Ludmia Scharf Abstract We consider the probem of computing the depth of the arrangement of n axisaigned rectanges in the
More informationThe Use of CoolingFactor Curves for Coordinating Fuses and Reclosers
he Use of ooingfactor 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 informationLicensed 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 informationBest 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 informationPREFACE. 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 informationGREEN: 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 UtraBroadband Information Networks, EEE Department, The University of
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More informationSolutions 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 information1 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 informationLADDER SAFETY Table of Contents
Tabe of Contents SECTION 1. TRAINING PROGRAM INTRODUCTION..................3 Training Objectives...........................................3 Rationae for Training.........................................3
More informationLearning from evaluations Processes and instruments used by GIZ as a learning organisation and their contribution to interorganisational learning
Monitoring and Evauation Unit Learning from evauations Processes and instruments used by GIZ as a earning organisation and their contribution to interorganisationa earning Contents 1.3Learning from evauations
More informationMeasuring 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 informationPayondelivery investing
Payondeivery 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 informationHistory and Definition of CNC 100
History and Definition of CNC 100 Wecome to the Tooing University. This course is designed to be used in conjunction with the onine version of this cass. The onine version can be found at http://www.tooingu.com.
More informationOlder 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 informationWeek 3: Consumer and Firm Behaviour: The WorkLeisure Decision and Profit Maximization
AROEOOIS 2006 Week 3: onsumer and Firm Behaviour: The WorkLeisure Decision and Profit aximization Questions for Review 1. How are a consumer s preferences over goods represented? By utiity functions:
More informationOn target: ensuring geometric accuracy in radiotherapy
On target: ensuring geometric accuracy in radiotherapy The Roya Coege of Radioogists Institute of Physics and Engineering in Medicine Society and Coege of Radiographers Contents Foreword 6 Executive summary
More informationEnhanced continuous, realtime detection, alarming and analysis of partial discharge events
DMS PDMGRH DMS PDMGRH Partia discharge monitor for GIS Partia discharge monitor for GIS Enhanced continuous, reatime detection, aarming and anaysis of partia discharge events Unrivaed PDM feature set
More informationLife 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 informationIEEE 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 MutiCass Traffic Management in Data Center Backbone Networks Amitabha Ghosh, Sangtae Ha, Edward Crabbe, and Jennifer
More information