1. P. Venkataraman Mechanical Engineering Rochester Institute of TechnologyCEIS University Technology Showcase, February 12, 2009 Reduce and Super Smooth your Scattered Surface Data with the Bezier Filter Mechanical Engineering Reduce and Super Smooth your Surface Data with the Bezier Filter

2. P. Venkataraman Mechanical Engineering Rochester Institute of TechnologyCEIS University Technology Showcase, February 12, 2009 Reduce and Super Smooth your Scattered Surface Data with the Bezier Filter Mechanical Engineering One Dimensional Example Closing DJIA between Aug and Dec 2007 A Bezier function over all the data Order of function = 20 Mean original data = 13172.432 Mean Bezier data = 13172.423 Avg. Error = 98.34 Maximum Data = 14164.53 Std. Dev (original) = 530.19 Std. Dev. (Bezier) = 514.68 1

3. P. Venkataraman Mechanical Engineering Rochester Institute of TechnologyCEIS University Technology Showcase, February 12, 2009 Reduce and Super Smooth your Scattered Surface Data with the Bezier Filter Mechanical Engineering What is a Bezier Function ? p : parameter Bernstein basis Number of vertices: 5 Order of the function : 4 A Bezier function is a Bezier curve that behaves like a function The Bezier curve is defined using a parameter Instead of y=f(x); both x and y depend on the same parameter value; x = x(p) and y = y(p) 2

4. P. Venkataraman Mechanical Engineering Rochester Institute of TechnologyCEIS University Technology Showcase, February 12, 2009 Reduce and Super Smooth your Scattered Surface Data with the Bezier Filter Mechanical Engineering Matrix Description of Bezier Function This allows the use of Array Processing for shorter computer time 3

5. P. Venkataraman Mechanical Engineering Rochester Institute of TechnologyCEIS University Technology Showcase, February 12, 2009 Reduce and Super Smooth your Scattered Surface Data with the Bezier Filter Mechanical Engineering For a selected order of the Bezier function (n) Given a set of (m) vector data y a,i, or [Y], find the coefficient matrix, [B] so that the corresponding data set y b,i, [Y B ] produces the least sum of the squared error Minimize FOC: The Best Bezier Function to fit the Data Once the coefficient matrix is known, all other information can be generated using array processing For the filter, the best order is chosen on minimum absolute error 4

6. P. Venkataraman Mechanical Engineering Rochester Institute of TechnologyCEIS University Technology Showcase, February 12, 2009 Reduce and Super Smooth your Scattered Surface Data with the Bezier Filter Mechanical Engineering The matrix definition for the Bezier function is It can be recognized as And can be decoupled as Decoupling Independent and Dependent Variables 5

7. P. Venkataraman Mechanical Engineering Rochester Institute of TechnologyCEIS University Technology Showcase, February 12, 2009 Reduce and Super Smooth your Scattered Surface Data with the Bezier Filter Mechanical Engineering Two Dimensional Bezier Function – Smooth Data Original Data about 2600 points based on MATLAB Peaks function 3D View of the Data Using the Bezier Filter Contour Plot 3D Plot originalBezier mean0.3170.312 std. dev.1.1161.086 maximum8.0427.360 minimum-6.521-6.405 average error: 6.91e-02 6

8. P. Venkataraman Mechanical Engineering Rochester Institute of TechnologyCEIS University Technology Showcase, February 12, 2009 Reduce and Super Smooth your Scattered Surface Data with the Bezier Filter Mechanical Engineering Two Dimensional Bezier Function – Rough Data Same peaks function but randomly perturbed on both sides Less dominant peaks diffused 3D plot Bezier Filter Contour plot 3D plot average error: 6.54e-01 originalBezier mean0.3220.325 std. dev.0.8591.035 maximum8.2537.481 minimum-7.651-6.565 7

9. P. Venkataraman Mechanical Engineering Rochester Institute of TechnologyCEIS University Technology Showcase, February 12, 2009 Reduce and Super Smooth your Scattered Surface Data with the Bezier Filter Mechanical Engineering Bezier Function in Image Handling The original image is 960 x 1280 pixels of size 671 KB True image processing in MATLAB Bezier filter applied to Red, Green and Blue color separately and combined Highly nonlinear color distribution 8

10. P. Venkataraman Mechanical Engineering Rochester Institute of TechnologyCEIS University Technology Showcase, February 12, 2009 Reduce and Super Smooth your Scattered Surface Data with the Bezier Filter Mechanical Engineering Single Bezier Functions for the Image Size = 671 KB Bezier function representation Function order 20 x 20 Coefficient storage = 11 KB (3 color streams) Original image 9

11. P. Venkataraman Mechanical Engineering Rochester Institute of TechnologyCEIS University Technology Showcase, February 12, 2009 Reduce and Super Smooth your Scattered Surface Data with the Bezier Filter Mechanical Engineering Bezier Function in Four Quadrants Original Image 671 KB Four quads Bezier function representation Function order 20 x 20 Coefficient storage = 4*11 KB (3 color streams) = 44 KB 10

12. P. Venkataraman Mechanical Engineering Rochester Institute of TechnologyCEIS University Technology Showcase, February 12, 2009 Reduce and Super Smooth your Scattered Surface Data with the Bezier Filter Mechanical Engineering Bezier filter is easy to incorporate and can work for regular, unpredictable data, and images The Bezier functions have excellent blending and smoothing properties High order but well behaved polynomial functions can be useful in capturing the data content and underlying behavior The mean of the Bezier data is the same as the mean of the original data Bezier functions naturally decouples the independent and the dependent variables Conclusions A single continuous function is used to capture all data Gradient and derivative information of the data are easy to obtain 11

13. P. Venkataraman Mechanical Engineering Rochester Institute of TechnologyCEIS University Technology Showcase, February 12, 2009 Reduce and Super Smooth your Scattered Surface Data with the Bezier Filter Mechanical Engineering Questions