Filtering Surface Profiles

1
Filtering Surface Profiles

• Bala Muralikrishnan
• Dept. of MEES
• UNCC

2
(No Transcript)
3
Primary output from filtering
Another output
4
Filtering

• Appears to be some sort of a smoothing
  operation
• We took in a very rough profile with lot of
  high frequency information and produced a
  smoother version of the same
• It is just smooth enough to not see the small
  wiggles, but not smooth enough to miss the long
  waves

5
Averaging

• Have a bunch of numbers that vary over time
  stock market fluctuations, may be
• How to tag a price to it by averaging
• We do the same here, we try to do some sort of
  average to the rough profile

6
Moving Average
7
Moving Average

• Two inputs
• Pprofile
• Window function (or the filter)
• The window function has two pieces of info
• The weights to be used for averaging this is
  automatically determined by specifying a certain
  shape
• The width how large a window this is what we
  can decide this is related to cutoff, larger
  the width means greater averaging or more
  smoothing

8
1/N
0.8 mm (cutoff)
-0.8 mm
If there are N points, the weights are all equal
to 1/N. The sum of all weights in a window is 1
9
a
0.8 mm (cutoff)
-0.8 mm
Triangular window (area under triangle 1.
Thisgives value for a)
10
-0.8
0.8
Gaussian weighting function
11
profile
window
12
Convolution

• Moving average is also called convolution
• If input profile has length n, window has length
  m, output has length nm-1
• In surface metrology, the input is the Pprofile
  and a filter (say, Gaussian), then output is the
  waviness profile W

13
Convolution

• In our examples, the Pprofile will have, say 8000
  points spaced at 0.001 mm between each point
• Filter is typically 1601 points long, also spaced
  at 0.001 mm
• Then, output waviness profile has 80001601-1
  points.
• Then, from this output, we extract the central
  portion of the 8000 points, by throwing away the
  first and last 800 points.

14
Matlab

• Matlab has a conv function to do convolution
• help conv

15
How to specify filter

• Gaussian filter
• Input spacing between points
• Specify filter shape filter equation does this
• Specify width - cutoff

16
Gaussian Filter

• cutoff 0.8 spc 0.001
• x (-cutoffspccutoff)
• const cutoff sqrt(log(2)/pi)
• b exp(-pi(x/const).2) / const
• b b/sum(b)
• plot(x,b)

17
Putting it all together

• load a data file
• RawProfile load('PaperProfile.dat')
• n length(RawProfile)
• spacing 0.001 spacing is in mm
• X (01n-1)spacing
• plot(X,RawProfile)
• PProfile detrend(RawProfile)
• plot(X,PProfile,'r')
• Generate the filter
• cutoff 0.8 spc 0.001
• x (-cutoffspccutoff)
• const cutoff sqrt(log(2)/pi)
• b exp(-pi(x/const).2) / const
• b b/sum(b)
• plot(x,b)

• perform filtering
• m (length(b)-1)/2
• n length(PProfile )
• W conv(PProfile ,b)
• W W(m1mn)
• X (01n-1)spacing
• plot(X, PProfile ,r,X,W,b)
• R Pprofile W
• plot(X,R)

18
Exercise

• Re-write this using functions this will be very
  useful for the first - computational lab
• Function 1 to load and detrend data
• input argument file name and spacing
• Output argument X and Pprofile
• Function 2 to generate filter
• Input arguments are spacing and cutoff
• Output argument is filter points b
• Function 3 to perform convolution
• Inputs are Pprofile and filter b
• Output is waviness profile W and roughness R

19
How to call your functions

• X, Pprofile myLoadData(paperprofile.dat,
  0.001)
• B myGaussian(0.8,0.001)
• R,W myFilter(Pprofile,B)

20

• Load the profile paperprofile.dat
• The spacing between points is 0.001 mm
• Filter using the functions you have just written
  select two different cutoffs as inputs 0.25
  mm and 0.8 mm and observe the roughness and
  waviness plots

21
Summary

• Filtering is the process of smoothing the profile
• We do this by performing a moving average
• For this, we need a weight array called the
  filter window, the filter or the weighting
  function
• The moving average is also called convolution

22
Some questions

• A surface is typically viewed as comprising a
  range of surface wavelengths
• Roughness comprises high frequency or small
  wavelengths
• Waviness has larger wavelengths
• Form has even longer wavelengths
• So, in what we have done does far we did use a
  cutoff term for the filter that sort of relates
  to wavelengths
• But we never quite explicitly looked at a
  profile as being composed of different sinusoids
  that can separated into bands!

23

• So, Instead of looking at filtering as simply an
  averaging process, we will look at a surface as
  being composed of many sinusoids that can be
  filtered into high frequency, medium frequency
  and small frequency bands
• Welcome to the world of the Fourier Transform

24

• Step 1 Preprocessing
• Step 2 Filtering
• Still some unfinished business here!
• Step 3 Parameters