Elements of Computational Metrology Vijay Srinivasan IBM & Columbia U. DIMACS Workshop on CAD/CAM, Rutgers U., October 7, 2003.

2 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 A.D.

3 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.

4 The Big Picture … Metrology Dimensional (Geometric) Metrology Coordinate and Surface Metrology Computational Metrology - Fitting and Filtering

5 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)

6 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

7 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.

8 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 )

9 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.

Fitting

11 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?

12 distance between two parallel planes = 0.04 units Some Form Tolerances … Ø2 ± radial separation between two coaxial cylinders = 0.05 units Syntax Semantics

13 A flatness assessment … Good old way Inspection plate Dial indicator Part under inspection

14 Other form tolerances … TypeCharacteristicSymbol Form Straightness — Flatness Roundness Cylindricity … and many more types of tolerances.

15 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 L 2 norm.  Chebyshev Fit – when the objective function uses L  or other norm. … and report the uncertainty in Y if you know uncertainties in X.

16 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 X Output: a point “ x0 ” on the plane and a direction vector “ a ” normal to it.

17 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 X Output: a point “ x0 ” on the plane and a direction vector “ a ” normal to it.

18 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.

19 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?

20 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 ?

Filtering

22 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!

23 Filtering as Convolution …  Of Functions  Of Sets … and their discrete versions.

24 Gaussian Filter (Mean-line Filter)

25 Morphological Operations  Primary operations  Dilation  Erosion  Secondary operations  Opening  Closing … and alternating sequence operations.

26 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).

27 Erosion Filter Input profile Output profile All dimensions are in micrometers

28 All dimensions are in micrometers; disk radius 50 micrometer Closing Filter (Envelope Filter) Input profile Output profile

29 Alternating Sequence Filter

30 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

31 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.