Elements of Computational Metrology

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Elements of Computational Metrology

• Vijay Srinivasan
• IBM Columbia U.

DIMACS Workshop on CAD/CAM, Rutgers U., October
7, 2003.
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A Very Old Problem

• How tall is the pyramid of Cheops?
• Measure the length of the pyramids 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
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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 Conformance
• Is the manufactured object within
  designer-specified tolerances?
• Process 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 of manufacturing imprecision
• All manufacturing processes are inherently
  imprecise and produce parts that vary.
• Axiom 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 statementof their uncertainty.
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Fitting
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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
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A flatness assessment Good old way
Part under inspection
Inspection plate
Dial indicator
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Other form tolerances

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
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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
• Of Sets

and their discrete versions.
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Gaussian Filter (Mean-line Filter)
 
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Morphological Operations

• Primary operations
• Dilation
• Erosion
• Secondary operations
• Opening
• Closing

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
• Width and Convex Hulls
• Non-linear Least Squares
• Circular Elements and Proximity Diagrams
• More Chebyshev Fits
• Geometry of Engineered Surfaces
• Integral Transforms and Convolutions
• 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.