3D Geometry for Computer Graphics. 2 The plan today Least squares approach General / Polynomial...

3D Geometry forComputer Graphics

2

The plan today

Least squares approach General / Polynomial fitting Linear systems of equations Local polynomial surface fitting

3

y = f (x)

Motivation

Given data points, fit a function that is “close” to the points

y

x

Pi = (xi, yi)

4

Motivation

Local surface fitting to 3D points

5

Line fitting

Orthogonal offsets minimization – we already learned (PCA)

x

y

6

Line fitting

y-offsets minimization

x

y

Pi = (xi, yi)

7

Line fitting

Find a line y = ax + b that minimizes

E(a,b) is quadratic in the unknown parameters a, b

Another option would be, for example:

But – it is not differentiable, harder to minimize…

2

1

( , ) [ ( )]n

i ii

a b ay bE x

n

iii baxybaAbsErr

1

)(),(

8

Line fitting – LS minimization

To find optimal a, b we differentiate E(a, b):

E(a, b) = (–2xi)[yi – (axi + b)] = 0

E(a, b) = (–2)[yi – (axi + b)] = 0

a

n

i 1

n

i 1b

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We get two linear equations for a, b:

(–2xi)[yi – (axi + b)] = 0

(–2)[yi – (axi + b)] = 0

n

i 1

n

i 1

10



[xiyi – axi2 – bxi] = 0

[yi – axi – b] = 0

n

i 1

)1(

n

i 1

)2(

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( xi2) a + ( xi) b = xiyi

( xi) a + ( 1) b = yi

n

i 1

n

i 1

n

i 1

n

i 1

n

i 1

n

i 1

12


Solve for a, b using e.g. Gauss elimination

Question: why the solution is the minimum for the error function?

E(a, b) = [yi – (axi + b)]2

n

i 1

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Fitting polynomials

y

x

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Decide on the degree of the polynomial, k Want to fit f (x) = akx

k + ak-1xk-1 + … + a1x+ a0

Minimize:

E(a0, a1, …, ak) = [yi – (akxik+ak-1xi

k-1+ …+a1xi+a0)]2

E(a0,…,ak) = (– 2xm)[yi – (akxik+ak-1xi

k-1+…+ a0)] = 0

Fitting polynomials

n

i 1

n

i 1ma

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Fitting polynomials

We get a linear system of k+1 in k+1 variables

1 1 1 1

2 1

1 1 1 1

1 2

1 1

0

1 1

1

1 1n n n n

ki i i

i i i i

n n n nk

i i i i ii i i i

n n n nk k k k

i i i i ii i i

ki

x x y

x x x x

a

a

a

y

x x x x y

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General parametric fitting

We can use this approach to fit any function f(x) Specified by parameters a, b, c, … The expression f(x) linearly depends on the

parameters a, b, c, …

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Want to fit function fabc…(x) to data points (xi, yi)

Define E(a,b,c,…) = [yi – fabc…(xi)]2

Solve the linear system

...1

...1

( , , , ) ( 2 ( ))[ ( )] 0

( , , , ) ( 2 ( ))[ ( )] 0

n

abc i i ii

n

abc i i ii

E a b c f x y f xa a

E a b c f x y f xb b

n

i 1

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It can even be some crazy function like

Or in general:

2

222

71 3

1( ) sinx

f x x x e

1 2, , 1..., 1 2 2( ) ( ) ( ) ... ( )k kkf x f x f x f x

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Solving linear systems in LS sense

Let’s look at the problem a little differently: We have data points (xi, yi)

We want the function f(x) to go through the points:

i =1, …, n: yi = f(xi)

Strict interpolation is in general not possible In polynomials: n+1 points define a unique interpolation

polynomial of degree n. So, if we have 1000 points and want a cubic polynomial, we

probably won’t find it…

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We have an over-determined linear system nk:

f(x1) = 1 f1(x1) + 2 f2(x1) + … + k fk(x1) = y1

f(x2) = 1 f1(x2) + 2 f2(x2) + … + k fk(x2) = y2

…

…

…

f(xn) = 1 f1(xn) + 2 f2(xn) + … + k fk(xn) = yn

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In matrix form:

1 1 2 1 1 1

1 2 2 2 2 2

1 2

( ) ( ) ... ( )

( ) ( ) ... ( )

...

( ) ( ) ... ( )

k

k

n n k n n

f x f x f x y

f x f x f x y

f x f x f x y

1

2

k

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In matrix form:

Av = y

.

1 2

1 2

where ( ) is a rectangular matr

( , ,...,

ix, n k

( , ,..., )

)

j j

T

i i

n

k

T

A f x n k

y y y

y

v

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More constrains than variables – no exact solutions generally exist

We want to find something that is an “approximate solution”:

2arg min A

vv v y

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Finding the LS solution

v Rk

Av Rn As we vary v, Av varies over the linear

subspace of Rn spanned by the columns of A:

Av = A2A1 Ak

1

2

.

. k

= 1A 1 A 2 A k + 2 +…+ k

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We want to find the closest Av to y:2

min A v

v y

Subspace spannedSubspace spannedby columns of by columns of AA

yy

Rn

Avclosest to y

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The vector Av closest to y satisfies:

(Av – y) {subspace of A’s columns}

column Ai, <Ai, Av – y> = 0

i, AiT(Av – y) = 0

AT(Av – y) = 0

(ATA)v = ATy

These arecalled the

normal equations

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We got a square symmetric system (ATA)v = ATy (kk)

If A has full rank (the columns of A are linearly independent) then (ATA) is invertible.

2

1

min

( )T T

A

A A A

vv y

v y

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Weighted least squares

Sometimes the problem also has weights to the constraints:

1 21 2

2, ,..., i, ,. ,

ii

..1

min [ ( )] , and doesn't depend on

min ( ) ( ), where is a diagonal matrix

( ) this is a

0

s

W

quare system

kk

n

i i i

Ti

T

ii

v

T

w w

W

y f x

A A

A A v A y

w

W W

v y v y

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Normals? Lighting? Upsampling?

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Locally approximate

a polynomial surface

from points

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Fitting local polynomial

X Y

ZReference plane

Fit a local polynomial around a point P

P

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Fitting local polynomial surface

Compute a reference plane that fits the points close to P Use the local basis defined by the normal to the plane!

z

x y

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Fit polynomial z = p(x,y) = ax2 + bxy + cy2 + dx + ey + f

z

x y

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z

x

y

35



z

x y

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Again, solve the system in LS sense:

ax12 + bx1y1 + cy1

2 + dx1 + ey1 + f = z1

ax22 + bx2y2 + cy2

2 + dx2 + ey2 + f = z1

. . .

axn2 + bxnyn + cyn

2 + dxn + eyn + f = zn

Minimize ||zi – p(xi, yi)||2

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Also possible (and better) to add weights:

wi ||zi – p(xi, yi)||2, wi > 0

The weights get smaller as the distance from the origin point grows.

See you next time

3D Geometry for Computer Graphics. 2 The plan today Least squares approach General / Polynomial...

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Transcript of 3D Geometry for Computer Graphics. 2 The plan today Least squares approach General / Polynomial...