Recovering Circles and Spheres From LADAR Data Christoph Witzgall - - PowerPoint PPT Presentation

Recovering Circles and Spheres From LADAR Data

Christoph Witzgall Geraldine Cheok Anthony Kearsley National Institute of Standards and Technology

May 23, 2006

DISCLAIMER

Certain company products are shown in this presentation. In no case does this imply recommendation or endorsement by the National Institute

• f Standards and Technology, nor

does it imply that the products are necessarily the best available for the purpose.

LADAR

LADAR – Laser Detection and Ranging

Point of impact (x, y, z) LADAR Light beam Object LADAR Facility at NIST

LADAR Imaging This is not a photograph!

Point Cloud & Sphere

Point cloud Fitted sphere

Fitting

• One way of recovering spheres from point

clouds is by “fitting”

– Select a “gauge” function = measure of deviation – Find the sphere that minimizes that gauge function

• Fitting spheres comes in two flavors:

– Fixed radius – Variable radius

Gauge Functions

• Define a gauge function

– Specify deviation concept: – Select norm * for vector of deviation

• Common norms

– – –

( )

surface from point i

i

∆ = ∆

n

∆ ∆ ∆ = , , , function gauge

2 1

• Chebychev"

" ) ( Max

∞

∆ L

i

squares" Least " ) (

2 2

L

i

∆

absolutes" Least " ) (

1

L

i

∆

How well does fitting work?

• r rather

What gauge function is appropriate?

Locating an I-Beam

For pick-up by an automated crane

B

The spheres are targets for “registration”.

A C

Registration

Those coordinates are based on the instrument. They need to be registered to the real world coordinates. The coordinates of the sphere centers, when determined both in real and instrument coordinates. “Rosetta Stone” Relating the two different coordinate systems.

A Red Flag

Repeated measurements (n ~ 90)

• Trouble at Sphere C

7.6 7.6 True (cm) 0.3 0.1 (cm) 6.9 5.4 Average (cm) Geometric Fitting Algebraic Fitting Radius

Circles vs. Spheres

Fitting methods for spheres are Analogous to Fitting methods for circles Discussion of fitting methods will be in terms of circles

Algebraic Fitting

• Measure-of-deviation is compliance with equation
• Use least squares for gauge function

Ordinary linear regression Unique solution

2 2

= + + + + C By Ax y x

( )

• +

+ + +

i 2 2 2

Minimize C By Ax y x

i i i i

( )

C By Ax y x

i i i i

+ + + − ~

2 2

* * *

, , C B A

Completing the Square

2 2 : THEOREM

* 2 * 2 *

≥ −

• +
• C

B A

( ) ( ) ( ) ( ) ( )

radius

• ptimal

center,

• ptimal

, becomes this 2 2 , 2 , 2 with 2 2 2 2 as written be can

* * * 2 * 2 * 2 * * 2 * 2 * 2 * * * * * * * 2 * 2 * 2 * * * * 2 2

= =

• =

− − + −

• −
• +
• =

− = − = =

• −
• +
• −
• +

+

• +

= + + + + r y x r y y x x C B A r B y A x C B A B y A x C y B x A y x

How Good Is It?

• 0.05
• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.02 0.03 0.04 20 40 60 80 100 120 140 Data Point Residual (m) Radius Residual Algebraic Residual

Example of an algebraic fit

Plots actual distance of data from fitted sphere

Geometric Fitting

The gauge function is based on the orthogonal distances of data points to a circle (sphere)

• Does the minimum always exist?
• Is the minimum uniquely defined?

(xo, yo) (xi , yi) r

( ) ( )

r y i y x i x − − + − 2 2

( ) ( )

( )

• −

− + −

• i

r y x r y i y x i x , , for 2 2 2

Non-Uniqueness

No Solution

What is a Reasonable Data Set?

#1 “Full” #2 “Upper” #2 “Lower”

Geometric Fitting Results

103.66

• 72.61
• 196.82
• 6258.61

Lower

102.36

• 83.02
• 196.37
• 6258.27

Upper

98.41

• 78.85
• 196.51
• 6254.99

Full R (mm) Z (mm) Y (mm) X (mm)

VARIABLE RADIUS

101.6

• 73.98
• 196.77
• 6256.59

Lower

101.6

• 82.55
• 196.36
• 6257.52

Upper

101.6

• 78.87
• 196.58
• 6259.19

Full R (mm) Z (mm) Y (mm) X (mm)

FIXED RADIUS

Actual Error

How about minimizing deviation in scan direction?

Actual error (in scan direction) Orthogonal distance error

Data point (xi yi zi)

LADAR

(xo yo zo)

Assumption: distance error is dominant

Scan ray

Fitting in Scan Direction

101.60

• 79.12
• 198.01
• 6259.38

Lower

101.22

• 78.90
• 198.15
• 6259.06

Upper

101.29

• 79.18
• 198.07
• 6258.98

Full r Z Y X Gauge function = Least squares of scan errors

103.66

• 72.61
• 196.82
• 6258.61

Lower

102.36

• 83.02
• 196.37
• 6258.27

Upper

98.41

• 78.85
• 196.51
• 6254.99

Full r Z Y X

Compare to Geometric Fit

Scan Ray Geometry

Actual error (in scan direction)

Data point (xi yi zi)

LADAR

xo, yo zo

i

l

ai si bi

i i i i i i i i i i

s l a error l z z l y y l x x a − − − − − − − − = = = = + + + + + + + + = = = =

How to Compute - 1

• Difficulty fitting in scan direction

– Errors incurred only if the Tentative sphere is actually hit by the scan ray – Gauge function is minimized by simply moving it out of the way

• Remedy

– Define deviation error for scan rays missing the sphere

Orthogonal error Scan ray

How to Compute - 2

• Difficulty with proposed remedy

– Appended gauge function not differentiable – Orthogonal errors cause distortion

• Solution to non-differentiability problem

– Use optimizer which handles

• Non-differentiability
• Multiply local minima

– Kearsley’s modification of the BFGS algorithm

• BFGS = Broyden Fletcher Goldfarb Shanno method
• Hybrid algorithm combines aspects of BFGS and Nelder-

Mead type approach improving on both

How to Compute - 3

• Solution to distortion problem

– Iterative procedure

• Solve with appended orthogonal error

Sphere

• Temporarily delete data points missing the

sphere

• Repeat until set of misses stabilizes
• Actually delete points missing the sphere

– Final optimization

• To determine a fragile local minimum

Plane Fitting

Message

When fitting a curve or a surface, it may not be sufficient to provide the data coordinates only. If there are directions in which the individual data points have been obtained, then those directions need to be taken into account.

Not But