data handling with two independent variables and the bezier filter

P. Venkataraman

Mechanical Engineering

P. Venkataraman Rochester Institute of Technology

DETC2009 – 86330 Data Handling with Two Independent Variables and the Bezier Filter

29th CIE, San Diego, California

DATA HANDLING WITH TWO INDEPENDENT VARIABLES

ANDTHE BEZIER FILTER

P. Venkataraman

P. Venkataraman





2What is a Bezier Function ?

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0.5

1

1.5

2

2.5

3

3.5

4

x: independent variable

y: d

ep

end

en

t va

ria

ble

s

[a1,b1][a1,b1]

[a2,b2]

[a3,b3]

[a4,b4]

[a5,b5]

Convex hullBezier VerticesBezier Curve: order 4

,0

( ) ( ) ( ) , 0 1

n

i n ii

Bx p y p J p p

1, ( ) ( )i n in i

nJ p p p

i

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)

P. Venkataraman





3Matrix Description of Bezier Function (2D)

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0.5

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1.5

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2.5

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3.5

4

x: independent variable

y: d

ep

end

en

t va

ria

ble

s

[a1,b1][a1,b1]

[a2,b2]

[a3,b3]

[a4,b4]

[a5,b5]

Convex hullBezier VerticesBezier Curve: order 4

[ ( ) ( )] [ ][ ][ ]x p y p P N B

4 3 2[ ] [ 1];

1 -4 6 -4 1 0 0

-4 12 -12 4 0 1 3

[ ] 6 -12 6 0 0 [ ] 2 1

-4 4 0 0 0 3 2

1 0 0 0 0 5

P p p p p

N B

0

This allows the use of Array Processing for shorter computer time

P. Venkataraman





4

For a selected order of the Bezier function (n) Given a set of (m) vector data ya,i , or [Y], find the coefficient matrix, [B] so that the corresponding data set yb,i , [YB ] produces the least sum of the squared error

2

, ,

m

a i b ii

E y y

T T

B B A AE Y Y Y Y Y P NB Y P NB

0E

B

1[ ] [ ] [ ]T TA A AB P P P Y

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

P. Venkataraman





5Two Dimensional Example

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1.25

1.3

1.35

1.4

1.45x 10

4

Ori

gin

al D

ata,

Fit

ted

Dat

a

points

Closing DJIA between Aug and Dec 2007

0 50 100 150 200 2501.2

1.25

1.3

1.35

1.4

1.45x 10

4

Ori

gin

al D

ata,

Fit

ted

Dat

a

points

DJIA - Adjust Close 17 Sep - Dec 18

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

P. Venkataraman





6Bezier Function in 3D

A 3D Bezier function will be a surface in 2D. Bezier surface can be described as a vector-valued function of two parameters r and s

[ ( , ) ( , ) ( , )]; 0 , 1x r s y r s u r s r s

, ,

, , ,0 0

( ) (1 ) ; 0 1; ( ) (1 ) ; 0 1

[ ( , ) ( , ) ( , )] ( , ) ( ) ( )

i m i j n j

m i n j

m n

i j m i n ji j

m nJ r r r r K s s s s

i j

x r s y r s u r s Q r s B J r K s

2 2

4 3 2

3 4 4

3 2 3

3 2 2 3 2

( , ) -1 3 3

( , ) - 4 8 - 6 8

( , ) (-1 48 - 78 12 )

(4 -112 180 - 24 )

(-6 60 - 90 ) (-8 12 8)

0 1 0 1

x r s r r r

y r s s s s s

u r s r r r s

r r r s

r r s r r s

r s

P. Venkataraman





7Matrix Form of Bezier Function in 3D

[ ( , ) ( , ) ( , )]; 0 , 1x r s y r s u r s r s

[ ( , ) ( , ) ( , )] ( , )T

x r s y r s u r s Q r s R M B N S

4 3 21[ ] [ ... ] 1TTn nS s s s s s s s

1 3 2 1[ ...1] 1m mR r r r r r

1 0

1 2

1 0

0

11 1 ... 1

0 0 1 1

11 1 ... 0

0 1 1 2

[ ] . . . 0

11 1 ... 0

0 1 1 0

1 0 ... 00 0

m m

m m

m m m m m m m

m m m m

m m m m

m m

M

m m m m

m m

[ ]M

-1 3 -3 1

3 -6 3 0

-3 3 0 0

1 0 0 0

[ ]N

1 -4 6 -4 1

-4 12 -12 4 0

6 -12 6 0 0

-4 4 0 0 0

1 0 0 0 0

[ ]B

[0 0 0] [0 2 2] [0 3 3] [0 5 4] [0 6 5]

[1 0 0] [1 2 2] [1 3 3] [1 5 2] [1 6 1]

[3 0 0] [3 2 3] [3 3 0] [3 5 3] [3 6 5]

[5 0 0] [5 2 3] [5 3 0] [5 5 3] [5 6 5]

P. Venkataraman





8

Minimize

FOC:

Bezier Filter for 3D 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

Given a set of array data [U], assuming an order for each dimension (m, n),

find the Bezier function coefficient matrix, [BU] so that the corresponding approximate data [UB] generates the least value for the sum of the squared error over the data array

2B

i j

E U U TTB A U A A U AU R M B N S F B G

0U

E

B

1T TU A A A A AB G IF F G IF U[ ] [ ] [ ]

P. Venkataraman





93 Dimensional Bezier Function – Smooth Datay

x

Original Data

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5

10

15

20

25

30

35

40

45

50

-6

-4

-2

0

2

4

6

Original Data about 2600 points based on MATLAB Peaks function

3D View of the Data

010

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0

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60-8

-6

-4

-2

0

2

4

6

8

10

x

Original Data

y

Ori

gin

al

-6

-4

-2

0

2

4

6

8

y

x

m =12, n =15 ,Least Sum of Absolute Error :179.8217

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5

10

15

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-6

-4

-2

0

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6

Using the Bezier Filter

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60-8

-6

-4

-2

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8

x

Bezier Data

y

Bez

ier

-6

-4

-2

0

2

4

6 Contour Plot

3D Plot

original Bezier

mean 0.317 0.312

std. dev. 1.116 1.086

maximum 8.042 7.360

minimum -6.521 -6.405

average error: 6.91e-02

P. Venkataraman





103 Dimensional Bezier Function – Rough Data

Same peaks function but randomly perturbed on both sides

y

x

Original Data

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-6

-4

-2

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6

Less dominant peaks diffused3D plot

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60-8

-6

-4

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x

Original Data

y

Ori

gin

al

-6

-4

-2

0

2

4

6

8 Bezier FilterContour plot

y

x

m =12, n =12 ,Least Sum of Absolute Error :1702.726

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3D plot

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x

Bezier Data

y

Bez

ier

-6

-4

-2

0

2

4

6

average error: 6.54e-01

original Bezier

mean 0.322 0.325

std. dev. 0.859 1.035

maximum 8.253 7.481

minimum -7.651 -6.565

P. Venkataraman





11

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 (whole field representation)

Gradient and derivative information of the data are easy to obtain

P. Venkataraman





12Current Investigations

Using Bezier functions to solve inverse problems in ODE and PDE

Using Bezier functions in irregular domains

Using Bezier functions in image filtering

P. Venkataraman





13Bezier Function in Four Quadrants

Original Image 671 KB

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Four quads

Bezier function representation

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Function order 20 x 20

Coefficient storage = 4*11 KB (3 color streams) = 44 KB

P. Venkataraman





14

Questions ?

data handling with two independent variables and the bezier filter

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