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Automated analysis of Spatial Grids

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  1. Automated analysis of Spatial Grids Part of the short course on Artificial Intelligence at the American Meteorological Society Annual Meeting, Jan. 2006, Atlanta.

  2. Information from grids • Observed data are often on grids • Satellite imagery • Ground-based radar • Other data can be gridded • You may need to extract information from these grids • Is there a boundary? • Where are the troughs?

  3. Spatial grids as images • Our spatial grids are often two dimensional • Automated extraction of information from images is a long-established field in machine intelligence • We can treat spatial grids as images • And apply image processing techniques to our data.

  4. Image processing terminology

  5. Other terms for this field • There are subtle differences between image processing and • Data mining • Relationships between identified signatures • Knowledge discovery • A human draws the inferences • Computer Vision • Pattern recognition when it doesn’t work

  6. Limitations of image processing • Scenarios where pattern recognition succeeds: • Highly controlled environments • Assembly lines, robotics • Photography • Red, green, blue channels can be assumed independent • IR channels can not! • Objects have sharp boundaries • Storms, etc. do not • The most mature sub-fields • Filters (smoothing, edge finding) • Segmentation

  7. Workflow of AI applications • Artificial Intelligence applications that operate on spatial data usually follow these steps: • Gridding: take spatial data and make it a locally uniform resolution grid • Filtering: smoothing, pattern matching, etc. • Edge finding: find areas of drastic spatial change • Segmentation: find contiguous areas of constant value • Feature extraction: find features based on edges/regions • Classification: classify features using a rule base, neural network, support vector machine, etc. • Normally done with all the features from all your data fields • This will be covered by the other speakers

  8. A note on convention • Image processing uses matrix notation • (0,0) is the top-left corner of the image • (0,1) is the first row, second column. • As long as you don’t think of the first index as “x” and the second index as “y”, you will be okay • In what direction does the first index increase? Answer: South or downwards

  9. Image processing assumptions • Traditionally image processing assumes a Markov random process • Two pixels are correlated if they are adjacent to each other • If you have pixels: a b c • a and b are correlated as are b and c • But the correlation between a and c is captured solely by the correlation between a-b and b-c • No “second-order” correlation term. • A simplifying assumption that usually causes no problems. • If your field is random (no correlation at all) or if the correlation is second-order image processing techniques may not work

  10. Image processing assumptions • An assumption that could cause problems: • The field has uniform resolution • Projected spatial grids are not uniform resolution. • Locally, most projected grids can be assumed uniform. • Pixels 10 km apart don’t vary that much in size. • Be careful with global operators on projected grids • Segmentation is a global operation • Filtering, edge finding, etc. are fine. • Filters are local operations.

  11. Radar data • Radar data are NOT uniform resolution • Size of a “pixel” varies with range • Dramatic changes • Two options: • Use varying-size windows for local operators • Hard to keep track! • Convert to a Cartesian grid before applying image processing techniques • Will cause problems for local operators at far ranges where data are “repeated” • This is an issue with most projected data as well

  12. Missing Data • Some times, you will have spatial data where some pixels are missing • Can you treat this data as some value? • Say zero? • Will simplify your task considerably • Can dramatically speed up operations • If not, you can vary the formulae to use only those pixels for which you have data • Filtering results may have higher quality • Operations will be much slower

  13. Summary: Spatial grids • To treat spatial grids as images, • Operate on the raw, un-projected data • Less chance of repeated values • For “gridded” data, operate on grids at the resolution closest to half the spacing between observations. • Change “missing” data to background value if available. • Use local operators (e.g: filters) freely • Be careful with global operations (e.g: segmentation) • Convert computed features (such as area) to projected systems

  14. Gridding From insitu observations to grids

  15. In-situ observations • Some environmental data are measured by in-situ instruments • Not remotely sensed • These measurements are at geographically dispersed points • Need to be converted into grids

  16. Gridding observations • In-situ observations have to be interpolated on to grids to process as images • What is the pixel resolution? • What interpolation techniques are available?

  17. Pixel resolution • If the chosen pixel resolution is too coarse, lose some of the observations • If the chosen pixel resolution is too fine, strong gradients may not be apparent • A good rule of thumb: • Choose pixel resolution to be half the mean distance between observation points

  18. Interpolation methods • Interpolation methods: • Cressman analysis • Barnes analysis • Kriging

  19. Cressman analysis • The value at each pixel of grid: • Every observation gets weighted based on its distance from grid point • R is the “radius of influence” • Higher the R, the more distant points are considered

  20. Barnes analysis • The problem with a Cressman analysis: • Even if a pixel is collocated with an observation, the final value of pixel will be different • Due to (small) effect of pixels further away • Barnes analysis tries to account for this

  21. Barnes analysis equation • Barnes also changed the Cressman interpolation function to be exponential • Smoother than simple polynomials

  22. Barnes analysis: the technique • Perform Cressman analysis on obs • Compute errors at observation points • Perform Cressman analysis on errors • Give the error field a weight • Add the two fields • One pass of Barnes analysis • Can repeat Barnes analysis • N-pass Barnes analysis • Closer and closer to the value of the observations at the observation points.

  23. Kriging • Kriging is a principled method of interpolation • Compute correlation between observations • Use variogram to determine scaling parameter • Works only if the observations are very close to each other • M. A. Oliver and R. Webster "Kriging: a method of interpolation for geographical information system", Int’l. J. Geographical Information Systems, 1990, VOL. 4, No. 3, 313-332

  24. Convolution Filters A surprisingly powerful local operation

  25. Smoothing: an intuitive look • How would you smooth the image on the top right? • So that the edges aren’t quite so ragged? • To get an image like bottom right?

  26. Averaging is a smoothing operation • The most obvious way to smooth is to replace each pixel by a local average: • I used k=3 in the WDSS-II filter “average” • • The dataset I’m using for demonstration is available online. See: •

  27. How it works • What does this equation mean? • For every pixel x,y, look in a 2D neighborhood, add up all the values and divide by the number of pixels. • (2k+1) is the neighborhood size • k is the half-size • Because the neighborhood size needs to be odd, most software will ask you for the half-size, k

  28. Convolution • Another way to think about smoothing: • At each pixel, you assign weights to the neighboring pixels. • And compute a weighted average. • W is a (2k+1)x(2k+1) matrix shown on right. • Called the convolution matrix or kernel • Technically, this is cross-correlation, but if you are using symmetric kernels (as we almost always will be), cross-correlation and convolution are the same thing.

  29. Smoothing kernels • The “boxcar” kernel • Poor characteristics in transition areas. • More spatial relocation of maxima • Smudges more • A better convolution kernel is where the kernel function drops off with distance. • A Gaussian convolution kernel offers the best space/frequency trade off • Frequency = raggedness • Space = location • You can smooth a lot more effectively because the features don’t move or smudge as much. Gauss:9 Average:3

  30. Gaussian Kernel • x and y range from –k to k. • Since the Gaussian has infinite range, you need to choose k appropriately. • Choosing k to be 3 * largest sigma is a reasonable choice. • The sigmas are the scale • Divide by total weight (The W above doesn’t add up to 1)

  31. Convolution kernels • By changing the weights in the convolution kernel, you can extract many features.

  32. Matching filters • In general, make your convolution kernel be the feature you are trying to extract. • The above kernel returns large magnitudes for thin vertical lines • But not just thin vertical lines • Need to normalize by smoothed value also • Can use the matched filter idea to find objects

  33. Scale and rotation • Convolution filters are: • Scale dependent • What if the gradient is 5 pixels apart? • Need a matching filter with columns 5 apart! • Orientation dependent • What if the boundary is horizontal? • Need a matching filter with that orientation. • Remember typical usage? • Do you know the size of a machine part before hand? • How about a squall line?

  34. Scale and orientation • If you specify your convolution filter as Gaussian functions, it is easy to specify rotation and scale. • x and y range from –k to k. • Since the Gaussian has infinite range, you need to choose k appropriately. • Choosing k to be 3 * largest sigma is a reasonable choice. • Theta is the orientation from x-axis (anti-clockwise from south) • The sigmas are the scale (use same sigma for circular kernels; different sigmas for elliptical ones) • Divide by total weight (The W above doesn’t add up to 1)

  35. Speed • The larger the scale of interest, the larger the half-size of the kernel. • Will take a long time to compute. • Since we need to compute weighted average at every pixel. • Two speed ups possible • Convolution can be performed in Fourier Transform space. • Convolution can be performed using separable filters.

  36. Convolution with Fourier Transforms • To convolve with Fourier Transforms: • Compute 2D FFT of original image • Can not have missing data • Fill with background value somehow • Compute 2D FFT of convolution kernel • The convolution kernel has to be same size as image (pad with zeros) • Multiply the FFT values pixel-by-pixel • Take inverse 2D FFT of result. • The larger your kernel, the more dramatic the speed up. • Can be 100-200 times faster. • Seconds instead of hours! • Resources • fftw is a free, nice Fourier Transform library. • Lakshmanan, V: 2000: Speeding up a large scale filter. J. of Oc. and Atm. Tech., 17, 468-473

  37. Convolution with separable filters • What if you have missing data? • Can not use FFT method • Problem with radar data in particular. • If the convolution filter is “separable”: • Compute weighted average row-wise on original image. • Compute weighted average on the row-wise result column-by-column. • If your filter was 129x129, 60 times faster than doing the standard way.

  38. What is a separable filter • A filter is separable if • For example, if the convolution filter is a simple Gaussian: • If you want to vary orientation, see: • Lakshmanan, V 2004: A separable filter for directional smoothing. IEEE Geosc. and Remote Sensing Letters, 1, 192-195

  39. Edge finding • The “line-finding” filter is not a robust way to find edges • Similar problems to using boxcar for smoothing • Use Laplacian of a Gaussian (Canny) • Differentiate the smoothing kernel twice. • Convolve image with that kernel • Look for zero crossings in the convolution result. • J. Canny, ``A computational approach to edge detection", IEEE Trans. on Pattern Analysis and Machine Intelligence, 8(6),pp679-698 (1986)

  40. Multi scale or orientation analysis • Use filter banks: • To perform multi scale (multi resolution) or orientation analysis • A bunch of filters at different scales and orientations • Combine their results by a weighted average or maximum • Drawbacks: • Repeated convolutions on original image. • Results of the filters are not related. • Can get very expensive • Simplifications possible: • We can steer the results based on a few filters • The resulting features are related • The squall line contains storm cells • So far, we have only considered image-to-image • The resulting images are related. • Use wavelets

  41. Wavelets • Wavelet analysis • breaks up images • smoother and smoother results on the right (s0,s1) • The lost “detail” information on the left • 3 images of the size of the corresponding s • Can not use any old filter • So that s1 is half the size of s0 • Convolution filter has to be a wavelet function • Haar function (boxcar) • Daubechies p-functions • Many others • Fast Wavelet transformations exist • Use wavelets only if you need this decomposition • Otherwise, use simple multi scale analysis with lots of Gaussian filters. • Gaussian filters have lots of nice properties. • Plus everybody knows what they are!

  42. Texture • In convolution, we compute weighted average of the original image’s pixels • Can also do other operations inside neighborhood • Texture operators • Operators based on relative ordering

  43. Texture operators • Texture is a good way to distinguish different objects • Not just on data value, but on distribution of data values • Lakshmanan, V., V. DeBrunner, and R. Rabin, 2000: Texture-based segmentation of satellite weather imagery. Int'l Conference on Image Processing, Vancouver, 732-735. • 2nd and 3rd order statistics • Variance, standard deviation • Homogeneity • Entropy • Shannon’s measure of information content • Bin the data values and form a histogram • P is the fraction of pixels in each bin

  44. Percent filters • Operators based on sorting • Take the median value • Excellent smoothing filter • Optimal when you have speck noise • Take the maximum value • Effect of dilating the field spatially • Minimum erodes the field spatially • Use percentiles • 50% is the median (maximum noise reduction) • 0% is the min (heavily noise dependent) • 100% is the max • Intermediate values trade off on noise reduction

  45. Dilation and Erosion filters Using the second-highest or second-lowest value in the neighborhood is an excellent dilation/erosion filter. Implemented this way in WDSSII. dilate erode orig

  46. Filter Banks and Neural Networks • Classification using NN • Use filter banks • Convolution filters • Texture operators • Percentile operators • Each pixel provides a pattern • May have to select pixels – not use all of them • Since a 1000x1000 image has 1 million patterns. • But is only data case

  47. NN random error Effect of post processing QCNN • Features computed in local neighborhoods of radar gates • Classified by NN • Returns [0,1] • Is it perfect? • If we use this directly as a mask, there will be holes in the result! • Lakshmanan, V., A. Fritz, T. Smith, K. Hondl, and G. J. Stumpf, 2005: An automated technique to quality control radar reflectivity data. J. Applied Meteorology, subm. Original Effect of Pre-processing (don’t care) QC’ed

  48. NN random error Effect of post processing Actual QC technique Original Effect of Pre-processing (don’t care) QC’ed

  49. Images to Objects • So far, we have looked at ways of processing images • Result is still image • Bunch of pixels • Need to group pixels together into objects • Classify, track, etc. these objects if possible.

  50. Segmentation From images to objects