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  • Notation can be confusing “ The evaluation of relative position of A w.r.t. R is given by a function mu_alpha( R ) ( x ) and mu_A(x) for all x in S” – Bloch “ The fuzzy landscape mu_alpha( R ) at point P is then defined as: mu_alpha( R )( P ) = f(beta_min(P)).” - Bloch
  • Incidentally R=2, O=1 in direction Pi yields 1.000 across the board
  • H=255 G=183 S=7.332% with black counted S2=2.933 S3=4.399
  • Base prob = 1/C Bias = b/100 Exclude A = 1/(C-1) Rest = (100-b)/100 * 1/(C-1) = (100-b)/100*(C-1)
  • G=2 H=4
  • Phi = 126 Gamma = 42
  • GBS_Final.ppt

    1. 1. Mining Fuzzy Spatial Association Rules from Image Data G. Brannon Smith Mississippi State University 6 June 2001
    2. 2. Contents <ul><li>Introduction </li></ul><ul><li>Motivation </li></ul><ul><li>Brief Background </li></ul><ul><ul><li>Fuzzy Relative Position </li></ul></ul><ul><ul><li>Object Co-occurence </li></ul></ul><ul><li>Theory </li></ul><ul><ul><li>Aggregate </li></ul></ul><ul><ul><li>Traditional </li></ul></ul><ul><li>Experiments </li></ul><ul><li>Conclusion </li></ul><ul><li>Future Work </li></ul><ul><li>Selected References </li></ul><ul><li>Acknowledgements </li></ul>
    3. 3. Introduction <ul><li>Operating on raster image data = image space </li></ul><ul><li>Images can be partitioned into regions or objects (groups of like pixels) </li></ul><ul><li>Like objects compose classes </li></ul><ul><li>Would like to know general spatial, i.e., directional, arrangement of these </li></ul>Depends on DSK
    4. 4. Introduction (cont.) <ul><li>Association Rules seem appropriate – but not made for raster data, so… </li></ul><ul><li>Need an approach for finding generalized fuzzy association rules on object spatial relations pulled from image space </li></ul>
    5. 5. Motivation <ul><li>GENERAL: Periodic collection of vast amounts of data = tedious for human to analyze </li></ul><ul><li>SPECIFIC: OKEANOS project sponsored by NAVO collects many seafloor images </li></ul><ul><li>Data Mining/Knowledge Discovery helps </li></ul>
    6. 6. Background <ul><li>Fuzzy Set Theory (Zadeh) </li></ul><ul><li>Fuzzy Relative Position (Bloch) </li></ul><ul><li>Association Rules (Agrawal et al.) </li></ul><ul><li>Fuzzy Association Rules (Kuok, Fu & Wong) </li></ul><ul><li>Spatial Data Mining (Koperski & Han) </li></ul><ul><li>Object co-occurrence rules (Ordo ñ ez & Omiecinski) </li></ul>
    7. 7. Fuzzy Spatial Relations <ul><li>I. Bloch applies fuzzy sets to spatial relations </li></ul><ul><li>Fuzzy concepts of position: right of is fuzzy </li></ul><ul><li>Morphology (shape & size) has effect… </li></ul>A R A R
    8. 8. Fuzzy Spatial Relations (cont.) <ul><li>Objects described as fuzzy sets ( crisp OK ) </li></ul><ul><li> Ex.  A (x) and  R (x) , x  S </li></ul><ul><li>Landscape :   (R)( x) is whole image S in relation to R in direction  </li></ul><ul><li>Relation : want  A (x) and   (R)( x) overlap </li></ul>
    9. 9. Fuzzy Landscape (single) Test Image Landscape RO#2,  =0 Reference Object #2 Background; Empty Space OO#4 H G F
    10. 10. Membership Interval <ul><li>Bloch algorithm on all points in objects </li></ul><ul><li>Result: 3 stats per relation, M  [N,  ]: </li></ul>Captures imprecision
    11. 11. Fuzzy Relation Stats N=0.9959, M=0.9999,  = 1.0000 N=0.7557, M=0.9079,  = 1.0000 R A R A
    12. 12. Image Data Mining <ul><li>Ordo ñ ez and Omiecinski have done preliminary work in image space </li></ul><ul><li>Used Blobworld to convert images to transactions, objects to item meta-data </li></ul><ul><li>ARM to find simple co-occurrence rules </li></ul>
    13. 13. Hypothesis <ul><li>Unified system of above can be made </li></ul><ul><li>Raster Image data input (K&H) </li></ul><ul><li>Fuzzy Spatial Relation metadata (Bloch) </li></ul><ul><li>Fuzzy Assoc Rule mining (Agrawal et al., KFW) </li></ul><ul><li>Result : useful fuzzy rules describing generalities of object spatial relations </li></ul>
    14. 14. Main Problem <ul><li>How to get from Fuzzy Relation metadata tuples (Bloch) to useful rules? </li></ul><ul><li>What are rule forms? </li></ul><ul><li>What are Support and Confidence or analogs thereof? </li></ul><ul><li>Time? Space? Usefulness? </li></ul>
    15. 15. Theory <ul><li>A pre-emptive approach </li></ul><ul><li>By aggregating objects into classes first, can do pseudo-mining right away </li></ul><ul><li>PRO: Few landscapes, small, quick, no mining per se </li></ul><ul><li>CON: lost info (e.g., no more indiv objs) </li></ul>
    16. 16. Fuzzy Landscape (multi) Test Image Landscape RO#7,  =0 Reference Object #7 Background; Empty Space
    17. 17. Theory (cont.) <ul><li>“ Class-class” or “Pixel-Pixel” rule form: </li></ul><ul><li>S & C </li></ul>For any pixel x of class A and any given pixel y of class B, it is implied that y is in direction  of x, with some degree of confidence supported by some portion of the (meta) database.
    18. 18. Theory (cont.) <ul><li>Prev. ex.: </li></ul>1.0000 0.9999 0.9959 G H 0.0  M N OC# RC# alpha
    19. 19. Theory (cont.) <ul><li>More traditional … (aggreg loses obj id) </li></ul><ul><li>Given: relations for all obj pairs in 4 dirs </li></ul><ul><li>1. </li></ul>For any object x of class A, there exists some object y of class B, such that that y is in direction  of x, with some degree of confidence supported by some portion of the (meta) database.
    20. 20. Theory (cont.) <ul><li>Prev. ex. (same source objs): </li></ul>0.6019 0.5434 0.4842 G 5 H 3 0.0 0.8233 0.7590 0.6887 G 4 H 3 0.0 0.8242 0.7603 0.6904 G 5 H 2 0.0 1.0000 0.9999 0.9959 G 4 H 2 0.0 1.0000 0.9999 0.9959 G 5 H 1 0.0 0.8233 0.7590 0.6887 G 4 H 1 0.0  M N OC# OO# RC# RO# alpha
    21. 21. Theory (cont.) Object based
    22. 22. Theory (cont.) <ul><li>2. </li></ul>
    23. 23. Theory (cont.) object
    24. 24. Time  Parallel <ul><li>Landscape generation/Relation extraction independent for given RO,  </li></ul><ul><li>“ Embarrassingly Parallel” </li></ul><ul><li>mpiShell by Wooley shortens development time, allows user to exploit parallel </li></ul><ul><li>Not linear: 16CPU  4  ; BUT very useful considering min implementation effort… </li></ul>
    25. 25. Simple Hand Constructed 3 classes
    26. 26. Hand graph
    27. 27. Experiments: Synthetic Data Sample Graphs Scatter plots of rules mined from synthetic images with a fuzzy spatial relation extractor, using Obj-Obj rules
    28. 28. Synthetic Data <ul><li>S ynthetic D ata G enerator to produce images with bias – “loaded” images </li></ul><ul><li>Can we extract rules that reflect the bias? </li></ul><ul><li>Regular </li></ul><ul><li>Extended </li></ul><ul><li>Half </li></ul>
    29. 29. Side Effects <ul><li>Edge Effect – image edges </li></ul><ul><li>Counterbias – wrong direction </li></ul><ul><li>“ Spillover” - other classes benefit </li></ul><ul><li>Probability – bias is NOT a guarantee </li></ul>
    30. 30. Sample random 2 (6 classes)
    31. 31. R2 graph
    32. 32. Sample 4 R=G, A=H of 6 classes, bias=90% Extended,  =0
    33. 33. 4 graph
    34. 34. Sample 2 R=G, A=H of 6 classes, bias=80% Extended,  =0
    35. 35. 2 graph
    36. 36. Sample 10 R=G, A=H of 6 classes, bias=95%,  =0
    37. 37. 10 graph
    38. 38. Half A=I of 6 classes, bias=85% Half,  =0 R=H, A=I R=J, A=I
    39. 39. Half graph
    40. 40. Seafloor
    41. 41. Seafloor graph
    42. 42. Seafloor Rule #148  
    43. 43. Conclusions <ul><li>Fairly recent discovery of Association Rules (1993) has enjoyed much growth. (Agrawal) </li></ul><ul><li>Expansion into categorical, fuzzy , etc. (Srikant, Kuok/Fu/Wong, et al.) </li></ul><ul><li>Many have done work with Spatial Databases – in Object Space (Koperski & Han) </li></ul><ul><li>BUT… </li></ul>
    44. 44. Conclusions <ul><li>Preliminary investigation on image object co-occurrence rules by Ordonez and Omiecinski aside… </li></ul><ul><li>Very little work done in Association Rule Mining in (raster) Image Space, esp. fuzzy </li></ul><ul><li>We have endeavored to fill this gap </li></ul>
    45. 45. Conclusions <ul><li>Used Bloch Fuzzy Spatial Relations as tool for meta-data generation </li></ul><ul><li>Used techniques inspired by (not implemented) Kuok, Fu & Wong </li></ul><ul><li>Showed that we can find interesting and useful rules – both “loaded” and unknown </li></ul>
    46. 46. Future Work <ul><li>Better exploitation of fuzzy membership interval </li></ul><ul><li>Application of thresholding typical to most AR to prune low fuzzy values </li></ul><ul><li>Addition of a distance measure attribute </li></ul><ul><li>Exploration of different kinds of rules such as Spatial Relation Co-occurence </li></ul>
    47. 47. Summary <ul><li>Introduction </li></ul><ul><li>Motivation </li></ul><ul><li>Brief Background </li></ul><ul><ul><li>Fuzzy Relative Position </li></ul></ul><ul><ul><li>Object Co-occurence </li></ul></ul><ul><li>Theory </li></ul><ul><ul><li>Aggregate </li></ul></ul><ul><ul><li>Traditional </li></ul></ul><ul><li>Experiments </li></ul><ul><li>Conclusion </li></ul><ul><li>Future Work </li></ul><ul><li>Selected References </li></ul><ul><li>Acknowledgements </li></ul>
    48. 48. Selected References <ul><ul><li>Agrawal, R., T. Imielinski, and A. Swami. 1993. Mining associations between sets of items in massive databases. In Proceedings of the 1993 ACM SIGMOD Int’l Conferences on Management of Data held in Washington, DC, May 26-28, 1993, 207-216. New York: ACM Press. </li></ul></ul><ul><ul><li>Bloch, I. 1999. Fuzzy relative position between objects in image processing: A morphological approach. IEEE Transactions on Pattern Analysis and Machine Intelligence 21(7):657-664. </li></ul></ul><ul><ul><li>Fayyad, U. M., G. Piatetsky-Shapiro, P. Smyth, and R. Uthurusamy (Eds.). 1996. Advances in knowledge discovery and data mining . Menlo Parks, CA: AAAI/MIT Press. </li></ul></ul><ul><ul><li>Knorr, E. M., and R. T. Ng. 1996. Finding aggregate proximity relationships and commonalities in spatial data mining. IEEE Transactions on Knowledge and Data Engineering 8(6):884-897. </li></ul></ul>
    49. 49. Selected References (cont.) <ul><ul><li>Koperski, K., J. Adhikary, and J. Han. 1996. Knowledge discovery in spatial databases: Progress and challenges. In Proceedings of the 1996 ACM SIGMOD Workshop on Research Issues on Data Mining and Knowledge Discovery (DMKD’96) held in Montr éal, June 2, 1996 , 55-70. IRIS/Precarn. </li></ul></ul><ul><ul><li>Kuok, C. M., A. W.-C. Fu, and M. H. Wong. 1998. Mining fuzzy association rules in databases. SIGMOD Record 27(1):41-46. </li></ul></ul><ul><ul><li>Luo, J. and S. M. Bridges. 2000. Mining fuzzy association rules and frequency episodes for intrusion detection. International Journal of Intelligent Systems 15(8):687-703. </li></ul></ul><ul><ul><li>Ordonez, C. and E. Omiecinski. 1999. Discovering association rules based on image content. Proceedings of the 1999 IEEE Forum on Research and Technology Advances in Digital Libraries held in Baltimore, MD, May 19-21, 1999 , 38-49. IEEE. </li></ul></ul>
    50. 50. Selected References (cont.) <ul><ul><li>Wooley, B. 2000. mpiShell Documentation . (Accessed 02 May 2001}. </li></ul></ul><ul><ul><li>Zadeh, L.A. 1965. Fuzzy sets. Information and Control 8(3):338-353. </li></ul></ul><ul><ul><li>Zimmerman, H.-J. 1996. Fuzzy set theory – and its applications (3 rd ed.). Boston: Kluwer Academic Publishers. </li></ul></ul>
    51. 51. Acknowledgements <ul><li>Thanks to… </li></ul><ul><li>Dr. Susan Bridges (Major Professor) for being a great editor of a very long document </li></ul><ul><li>Bruce Wooley for creating mpiShell </li></ul><ul><li>Sean Taylor for code review </li></ul>
    52. 52. Acknowledgements <ul><li>Grants from NAVO Research group based at Stennis Space Center in Bay St. Louis, MS </li></ul><ul><ul><li>National Science Foundation Grant #9818489 </li></ul></ul><ul><ul><li>ONR EPSCoR Grant N00014-96-1-1276 </li></ul></ul><ul><ul><li>Naval Oceanographic Office via NASA Stennis NAS1398033 DO92 </li></ul></ul>
    53. 53. URL for Thesis Materials <ul><li> </li></ul><ul><li>Includes this presentation, previous presentations (proposal, seminar, etc.), proposal text and thesis text in PostScript and PDF formats </li></ul>
    54. 54. Questions and Comments?
    55. 58. Fuzzy Set Theory <ul><li>Classical/Crisp set membership is TOTAL or NULL </li></ul><ul><li>Can describe with characteristic function - map universe onto {0,1}, a set itself </li></ul><ul><li>OK, for definite sets, e.g. Turing winners </li></ul>
    56. 59. Fuzzy Set Theory (cont.) <ul><li>PROBLEM: imprecise sets such as TALL </li></ul><ul><li>Where is NOT TALL/TALL boundary? </li></ul><ul><li>Zadeh proposed set membership function </li></ul><ul><li> A (x) mapping to [0,1] (interval), so 0.7 OK </li></ul><ul><li>Exact membership function at user discretion – domain specific </li></ul>
    57. 60. Fuzzy Set Theory (cont.) <ul><li>Classical operator analogs: complement, cardinality, etc. </li></ul><ul><li>Union & Intersection typically max & min respectively ( there are others ) </li></ul><ul><li>Still give proper results for crisp sets </li></ul>
    58. 62. Association Rules <ul><li>Rules of Agrawal et al. usually of form </li></ul><ul><li>antecedent X  consequent Y (s,c) </li></ul><ul><li>XY is set of items in a transaction and </li></ul><ul><li>X  Y =  i.e., disjoint </li></ul><ul><li>Ex. Beer  Chips (support:3%,conf:87%) </li></ul>
    59. 63. Association Rules (cont.) <ul><li>Notions of support and confidence </li></ul><ul><li>Support = % of ALL transactions with both X & Y - high support = “ large ” </li></ul><ul><ul><li>Measures importance (freq) in database </li></ul></ul><ul><li>Confidence = % of X transactions with Y </li></ul><ul><ul><li>Strength of relationship between X and Y </li></ul></ul>
    60. 64. Association Rules (cont.) <ul><li>Rules use binary/boolean attributes </li></ul><ul><li>Ex. “Transaction includes chips/Trans. does NOT include chips” </li></ul><ul><li>Classical Set Theory </li></ul><ul><li>But what about range data (e.g., Price or Age)? </li></ul>
    61. 65. Association Rules (cont.) <ul><li>Srikant & Agrawal offer mapping to Quantitative Rules to use range </li></ul><ul><li>Can map values from range to booleans </li></ul><ul><li>Ex. Price = 700/Price  700 </li></ul><ul><li>Price  [500,999]/Price  [500,999] </li></ul><ul><li>Still use boolean algorithms </li></ul>
    62. 66. Association Rules (cont.) <ul><li>Kuok, Fu & Wong complaint: interval boundaries (like TALL vs. NOT TALL) </li></ul><ul><li>SOLUTION: Use fuzzy set intervals </li></ul><ul><li>[20,25] becomes Young Adult </li></ul><ul><li>Attribute values have degrees of membership in several fuzzy sets </li></ul>
    63. 67. Association Rules (cont.) <ul><li>KFW rule: X is A  Y is B (s,c) </li></ul><ul><li>A and B are sets of fuzzy sets for attribs </li></ul><ul><li>s,c are fuzzy analogs of supp and conf: </li></ul><ul><li>Significance and Certainty </li></ul><ul><li>Weighted by fuzzy vals </li></ul>
    64. 68. Spatial Data Mining <ul><li>Koperski & Han leaders adapting General DM to Spatial Data specifically Spatial DB </li></ul><ul><li>Spatial DB stores spatial data, object attribs, does spatial ops, e.g., Spatial Join </li></ul><ul><li>Object Space </li></ul><ul><li>This work does NOT use a Spatial DB </li></ul>
    65. 69. Spatial Data Mining (cont.) <ul><li>But Koperski & Han do acknowledge Raster Image Data (not in Spatial DB) </li></ul><ul><li>Kind of bridge between strict SDM and Image Processing </li></ul>