Elements of
Computational Metrology
Vijay Srinivasan
IBM & Columbia U.
DIMACS Workshop on CAD/CAM, Rutgers U., October 7, 2003.
A Very Old Problem …
How tall is the pyramid of Cheops?
 “Measure
the length of the pyramid’s
shadow when your own shadow exactly
equals your height.”
- Thales, ca. 600 B.C.
 “Add
the measured heights of each of the
203 steps. Its uncertainty is 14 times the
uncertainty in measuring a single step.”
- Fourier, ca. 1800 A.D.
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First, Some Definitions …

Metrology is the art and science of
measurements.
 Measurement is the association of one or
more numerical values to physical objects
and characteristics.
 Our focus today in on geometric
measurements and computations on them.
 Specifically, our focus is on fitting and
filtering discrete geometric data.
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The Big Picture …
Metrology
Dimensional (Geometric) Metrology
Coordinate and Surface Metrology
Computational Metrology
- Fitting and Filtering
4
In our context …
 Fitting
 Optimization
 Continuous
optimization (e.g., least
squares fitting)
 Combinatorial optimization (e.g., minimax
fitting)
 Filtering
 Convolution
 Convolutions
of functions (e.g., Gaussian
filters)
 Convolutions of sets (e.g., envelope filters
using Minkowski sums)
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Industrial Setting …
Why do we bother?
 Product

Is the manufactured object within
designer-specified tolerances?
 Process


Conformance
Characterization
What is the capability of the manufacturing
process?
Is it under control over time?
These are major questions that arise in
computer-aided design and manufacture
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Two Basic Axioms …
 Axiom

All manufacturing processes are inherently
imprecise and produce parts that vary.
 Axiom

of manufacturing imprecision:
of measurement uncertainty:
No measurement can be absolutely
accurate and with every measurement
there is some finite uncertainty about the
measured value or measured attribute.
These are independent axioms and
both should be considered operative in any real situation.
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Imprecision in Manufacturing

No man-made artifact has ideal geometric
form.


No manufactured object can be perfectly planar, or
cylindrical, and so on.
There is experimental evidence that the
geometry of an engineered surface is more
like a fractal.

Over the range of engineering scales - from a
nanometer to a kilometer. (A dynamic range of
1012)
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Uncertainty in Measurement
C.H.Meyer (NIST) reporting on his measurement of the
heat capacity of ammonia (circa 1970):
“We think our reported value is good to 1 part in
10,000: we are willing to bet our own money
at even odds that it is correct to 2 parts in
10,000. Furthermore, if by chance our value
is shown to be in error by more than 1 part in
1000, we are prepared to eat the apparatus
and drink the ammonia”
Results of our computations should be accompanied by a statement
of their uncertainty.
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Fitting
Fitting …What is it and Why
do we care?
Associating ideal geometric form(s) to a
discrete set of points sampled on a
manufactured surface.
 Datum establishment – for relative
positioning geometric objects.
 Deviation assessment – how far has a
part deviated from its intended ideal
form?
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Some Form Tolerances …
Syntax
Ø2 ± 0.1
0.05
0.04
Semantics
radial separation
between two coaxial
cylinders = 0.05 units
distance between two
parallel planes = 0.04 units
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A flatness assessment
…Good old way
Part under inspection
Inspection plate
Dial indicator
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Other form tolerances …
Type
Characteristic
Straightness
Symbol
—
Flatness
Form
Roundness
Cylindricity
… and many more types of tolerances.
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Fitting as an optimization problem …
Given a set of points X, fit ideal geometric
element(s) Y that minimize an objective
function involving distances between X and Y,
subject to certain constraints.
Two popular fits:
 Least Squares Fit – when the objective
function uses L2 norm.
 Chebyshev Fit – when the objective function
uses L or other norm.
… and report the uncertainty in Y
if you know uncertainties in X.
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TLS Plane Problem
Input: set of points X
Output: a point “x0” on the plane and
a direction vector “a” normal to it.
(Total Least Squares plane minimizes the sum of
the squares of the perpendicular distances of the
points from the plane.)
Solution:
• X0 is the centroid of the input set X.
• a is the singular vector associated with the
smallest singular value of the central
coordinate matrix of the input set X.
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A sample code for TLS Plane
Input: set of points X
Output: a point “x0” on the plane and
a direction vector “a” normal to it.
function [x0, a] = lsplane(X)
x0 = mean(X)';
A = [(X(:, 1) - x0(1)) (X(:, 2) - x0(2)) (X(:, 3) - x0(3))];
[U, S, V] = svd(A, 0);
[s, i] = min(diag(S));
a = V(:, i);
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Minimax Plane
 A plane
that minimizes the maximum
(perpendicular) distance of the input set
of points.
 Equivalent to the “width of a set”
problem.
 A good example of combinatorial
optimization.
 Implementation
is more challenging than
the TLS plane.
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Some Interesting Questions …

Given uncertainties in the input data points,
what is the uncertainty of the computed
“width”?
 Can the TLS fit give us a statistical estimate
of the “out-of-flatness”?


RMS deviation from the TLS plane?
Can TLS plane or Minimax plane help us to
establish a planar datum?

Supporting plane that minimizes the sum of the
distances of the input points from that plane?
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Soft Gaging
Set Containment Problem
 Deterministic Version: Given two
sets A and B, is there a rigid motion
r such that A  rB (subject to some
constraints)?
 Probabilistic Version: If A is given
with some uncertainty, what is the
probability that A  rB ?
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Filtering
Filtering … What is it and Why
do We Care?
 Geometry
of engineered surface is
more like a fractal, in the engineering
range of scale.
 Engineering function is scale dependent
– rough versus smooth surfaces.
 Main purpose of filtering is to extract
scale dependent information
– and not compression of data!
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Filtering as Convolution …
 Of
Functions

y ( x) 
 z (s) K ( x  s)ds

 Of
Sets
A  B  {x  y : x  A, y  B}
… and their discrete versions.
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Gaussian Filter (Mean-line
Filter)
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Morphological Operations

Primary operations


Dilation
D( A, B)  A  B

E ( A, B)  A B
 Secondary operations

Erosion



Opening
Closing
O( A, B)  D( E ( A, B), B)

C ( A, B)  E ( D( A, B), B)
… and alternating sequence operations.
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Types of Morphological Filters

Same as morphological operations






Dilation filters
Erosion filters
Opening filters
Closing filters
… and alternating sequence filters
Most commonly used structuring elements
are disks (balls) and line-segments (flats).
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Erosion Filter
All dimensions are in micrometers
Input profile
Output profile
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Closing Filter (Envelope Filter)
All dimensions are in micrometers;
disk radius 50 micrometer
Input profile
Output profile
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Alternating Sequence Filter
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Columbia Lectures on
Elements of Computational Metrology

Introduction
 A Brief History of Engineering Metrology
 Linear and Orthogonal Regression
Fitting lines and planes
 Width and Convex Hulls
 Non-linear Least Squares
Other fits
 Circular Elements and Proximity Diagrams
 More Chebyshev Fits
 Geometry of Engineered Surfaces
Introduce fractals
 Integral Transforms and Convolutions
Filtering
 Wavelets
 Morphological Transforms
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Summary

Computational Metrology - A discipline in its
own right.
 Seemingly different practices are being
consolidated under optimization (fitting) and
convolution (filtering).


We can now provide better scientific basis.
Industrial need is the driver.

Several problems still remain open, especially
involving measurement uncertainty.
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