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Mathematical Morphology A Geometric Approach to Image Processing and Analysis

Mathematical Morphology A Geometric Approach to Image Processing and Analysis. John Goutsias Department of Electrical and Computer Engineering Image Analysis and Communications Laboratory The Johns Hopkins University Baltimore, MD 21218. Question. What is Mathematical Morphology ?.

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Mathematical Morphology A Geometric Approach to Image Processing and Analysis

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  1. Mathematical MorphologyA Geometric Approach to Image Processing and Analysis John Goutsias Department of Electrical and Computer Engineering Image Analysis and Communications Laboratory The Johns Hopkins University Baltimore, MD 21218

  2. Question • What is Mathematical Morphology ?

  3. A Commercial Answer • Mathematical Morphology is FAST !! • Mathematical Morphology is CHEAP !!

  4. An (imprecise) Mathematical Answer • A mathematical tool for investigating geometric structure in binary and grayscale images.

  5. Shape Processing and Analysis • Visual perception requires transformation of images so as to make explicit particular shape information. • Goal: Distinguish meaningful shape information from irrelevant one. • The vast majority of shape processing and analysis techniques are based on designing a shape operator which satisfies desirable properties.

  6. Grayscale Images Binary Images Example • Image analysis consists of obtaining measurements characteristic to images under consideration. • Geometric measurements (e.g., object location, orientation, area, length of perimeter)

  7. Set Union (overlapping objects): • Set Intersection (occluded objects): Morphological Shape Operators • Objects are opaque and shape information is not additive !! • Shapes are usually combined by means of:

  8. Morphological Dilation Morphological Erosion Morphological Shape Operators • Shape operators should distribute over set-unions and set-intersections (a type of “linearity”) !

  9. Morphological Operators • Erosions and dilations are the most elementary operators of mathematical morphology. • More complicated morphological operators can be designed by means of combining erosions and dilations.

  10. Question • What is Mathematical Morphology ?

  11. Applications Image Processing and Analysis A (precise) Mathematical Answer • A mathematical tool that studies operators on complete lattices Algebra Complete Lattices Topology Hit-or-Miss Mathematical Morphology Operators Erosions-Dilations Geometry Convexity - Connectivity Distance

  12. Some History • George Matheron (1975)Random Sets and Integral Geometry, John Wiley. • Jean Serra (1982)Image Analysis and Mathematical Morphology, Academic Press. • Petros Maragos (1985)A Unified Theory of Translations-Invariant Systems with Applications to Morphological Analysis and Coding of Images, Doctoral Thesis, Georgia Tech.

  13. Translation Invariant Operators

  14. “LINEARITY” TRANSLATION INVARIANCE Morphological Erosion

  15. Morphological Erosion

  16. Morphological Erosion Structuring Element Pablo Picasso, Pass with the Cape, 1960

  17. “LINEARITY” TRANSLATION INVARIANCE Morphological Dilation

  18. Morphological Dilation

  19. Structuring Element Morphological Dilation Pablo Picasso, Pass with the Cape, 1960

  20. Morphological Opening

  21. Structuring Element Morphological Opening Pablo Picasso, Pass with the Cape, 1960

  22. Morphological Opening • Is a smoothing filter ! • Amount and type of smoothing is determined by the shape and size of the structuring element. • Approximates a shape from below, since

  23. DEGRADED ORIGINAL Henri Matisse, Woman with Amphora and Pomegranates, 1952 FILTERED Filtering Example

  24. Increasing Operator + Translation Invariant Operator !! An Important Result

  25. Main Idea • Examine the geometrical structure of an image by matching it with small patterns at various locations. • By varying the size and shape of the matching patterns, called structuring elements, one can extract useful information about the shape of the different parts of the image and their interrelations. • Results in image operators which are well suited for the analysis of the geometrical and topological structure of an image.

  26. Question • What about gray-scale images ?

  27. TRANSLATION INVARIANCE “LINEARITY” MINIMUM Grayscale Erosion

  28. TRANSLATION INVARIANCE “LINEARITY” MAXIMUM Grayscale Dilation

  29. Flat Erosion Flat Dilation Remark

  30. ORIGINAL EROSION DILATION OPENING Grayscale Morphology

  31. Structuring Element Grayscale Opening

  32. Henri Matisse, Woman with Amphora and Pomegranates, 1952 Question • Can we automatically extract the largest connected component (the woman’s body) in this image ?

  33. ORIGINAL MARKER MARKER EROSION MARKER MARKER Answer This is amorphological operatorthat filters out connected image components of a certain size and shape CONNECTED OPERATORS !!

  34. DATA Targets MORPHOLOGICAL RECONSTRUCTION OPENING MARKER An Application - Target Detection

  35. DATA MORPHOLOGICAL RECONSTRUCTION CLOSING MARKER An Application: Target Detection

  36. DATA FINAL RESULT THRESHOLDING Incorrectly detected target Correctly detected targets An Application: Target Detection

  37. The End

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