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 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

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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 MeasurementC.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.

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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distance between two parallel planes = 0.04 units

Some Form Tolerances …

Ø2 ± 0.1

0.05 0.04

radial separationbetween two coaxialcylinders = 0.05 units

Syntax

Semantics

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A flatness assessment …Good old way

Inspection plate

Dial indicator

Part under inspection

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Other form tolerances …

Type Characteristic Symbol

Form

Straightness —

Flatness

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

(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.

Input: set of points XOutput: a point “x0” on the plane and a direction vector “a” normal to it.

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A sample code for TLS Plane

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);

Input: set of points XOutput: a point “x0” on the plane and a direction vector “a” normal to it.

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Minimax PlaneA 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 ProblemDeterministic 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 ?

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

dssxKszxy )()()(

},:{ ByAxyxBA

… and their discrete versions.

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Gaussian Filter (Mean-line Filter)

 

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Morphological Operations

BABAD ),(

BABAE ),(

)),,((),(

BBAEDBAO

)),,((),(

BBADEBAC

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

Input profile Output profile

All dimensions are in micrometers

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All dimensions are in micrometers; disk radius 50 micrometer

Closing Filter (Envelope Filter)

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

Fitting lines and planes

Filtering

Other fits

Introduce fractals

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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.