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This 21 page Class Notes was uploaded by Abe Jones on Saturday September 12, 2015. The Class Notes belongs to CS 791I at West Virginia University taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/202764/cs-791i-west-virginia-university in ComputerScienence at West Virginia University.
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Date Created: 09/12/15
Medical Image Analysis CS 593791 Computer Science and Electrical Engineering Dept West Virginia University 4th Apiil 2007 39 Outline 0 Vector Field Visualization 39 Outline 0 Vector Field Visualization 0 Color mapping and glyphs 0 Streamlines 0 Line Integral Convolution 39 r kn39iw vwralnmmm Color mapping and glyphs Color Mappmg Color FA Color FA rgb FAVx7 Ivylv lell 4th April 2007 4 20 Vector Field Visualization Color mapping and glyphs Glyphs in 2D and 3D W lllggggfg l Amtm ll CS 593 791 West Virginia University Needle plot and cone plot 4th April 2007 520 l Streamlines Streamlines Problem Find a curve 0s which is tangent to the vector eld V at each point d 67 ms Taylor Series Expansion 0s As 0s TSAS 0As2 We can estimate the next point on the streamline by 0s As 0s VasAs Figure Streamlines iAmJApni 2007 620 l Streamlines A more accurate streamline integrator 3 option for increasing accuracy 0 Decrease the step size As gt Asymptotic error is still 0As 0 Use a higher order Taylor series expansion gt Need to compute higher order derivatives gt Asymptotic error can be 0ASZ or better 0 Evaluate the function at more points gt Asymptotic error can be 0ASZ or better zzmApnr 2007 7 mo l SLreamlmes A more accurate streamline integrator RungeKutta numerical integration highorder error Without directly approximating highorder derivatives Solve the ODE Q fx7y dx First order Taylor series yn yquot 11 Since fxy we have yn1 yn hfxnyn 39 7 a 1 2 Second order Taylor series ynH 7yquot hdx 2 dxz a2 a a 3 Sum 5 imam WC haVeyn1yn hf hzfx fyf I 39MhJApnl 2007 8 20 l Streamlines RungeKutta numerical integration Can we approximate the second order equation using the rst order equation evaluated at different points Ifa E 01thenxn ah E xmxn Ifb E 01thenyn 1711 E ymynirl Can we approximate 1 yn1yn l hf l Ehzm yn1 yn l hafbcn l ahvyn l bk for some values of a a b zzmApni 2007 9 mo I Streamhnes Runge Kutta Yes we need to set parameters a a b so that atom am bhfy f 1102 fyf Setting Oz 17 a b we get the evolution equation h h yn1 yn hfxn 57yquot fxmyn The secondorder RungeKutta scheme is also called the midpoint scheme htthpiil 2007 10 I29 RungeKutta x1 x 1 3 x Midpoint rule Medical Image A l Streamlines RungeKutta 4th order We can achieve 4th order accuracy by evaluating fXy at 4 points each iteration k1 hfxmyn h k k2 hfxnivynl h k k3 hfxnivynz k4 hfxnh7ynk3 1 yn1 yn EUVI Zkz 2163 k4 I AthlApril 2007 11129 l Streamlines Streamtubes Generated by sweeping a circle along the streamline Figure Fiber tracts in white matter Tube radius is proportional to FA ES 537 34714 mg mummies 4th April 2007 13 20 I smamlhes Streamtubes Figure Fiber tracts in White matter 39 Line Integral Convolution 9 Texture based scheme orientation is parallel to vector eld 0 Works by blurring a noise texture along streamlines o Allows for dense visualization of large 2D vector elds 0 Can be easily extended to 3D vector elds I 4LhAp ril 2007 15729 Vector Field Visualization Line Integral Convolution Figure LIC with kernel Widths L 5 L 15 andL 55 CS 593 791 West Virginia Universitv Medical Image Analysis Line Integral Convolution Figure LIC with kernel Widths L 5 L 15 andL 55 S0I L 1x0 T0skso sds S0 L 0 1x0 intensity of the LIC texture at pixel x0 0 T input noise texture o k lter kernel of Width 2L 0 a streamline through point x0 CS 593 791 West Virginia University Medical Image Analysis Line Integral Convolution i n 1 Figure LIC Computation soL xo Tasks0 7 sds 507L 0 I x0 intensity of the LIC texture at pixel x0 0 T input noise texture o k lter kernel of Width 2L 0 a stream line through point x0 I immmil ijwam Line Integral Convolution Color MIC m the human brain 4th April 2007 18 20 Surface Visualization Texture map segmented geometry with LIC Figure Segmented Corpus Callosum Surface Visualization Texture map segmented geometry with LIC Figure Segmented Corpus Callosum