Symmetry as an Intrinsically Dynamic Feature
Abstract
:1. Introduction
2. Symmetry Constancy
2.1. Dynamics Through Erosion
- When going from a regular pattern to a modified version of it, any perturbation gives a signature. Moreover, experiments about ratios of pattern size over defect size, or about the number of defects, allow precision assessment.
- Curve variations follow, at least in a qualitative manner, tendencies of symmetry variations with the angle: constancy, monotony, smoothness etc.
2.2. Dynamics Through Multi-Resolution
- direct: building the pyramid and running the symmetry operator S layer by layer;
- indirect: running S at the bottom (image) and then building the pyramid from these values;
- hierarchical: recursively running S to build the pyramid.
3. Capturing Symmetry
- a model is made explicit to support optimality claims;
- this model puts forward:
- the invariance to the transform again in relying on explicit comparison between the pattern and a transformed version of it;
- the distance that could evolve into approximate comparison for similarity;
- pattern inclusion introduces set operations (such as Minkowski’s), associating as a result logic and geometry.
3.1. Optimal Symmetry Detection
3.2. Symmetry Measures
3.3. Results
- sensitivity of the symmetry detection to the centre position
- validity of λ the measure (degree) of symmetry in comparing patterns to their kernel through the elongation η and then the kernel evolutions with IOT
- quality of the correlation kernel wrt the kernel
- validity of the symmetry axis from correlation wrt the best axis over shifts
4. Symmetry Detection and Face Expressions
4.1. An Approach to Face Expression Recognition Based on Broken Symmetry Detection
- Human face is by nature mostly symmetrical. More so in the so-called Neutral expression (see FACS [19]).
- Any expression different from Neutral is obtained by stretching a different subset of face muscles (the so-called action units [20]).
- Such stretching is rarely completely symmetrical; as such, the more marked those changes in expression are, the more breakage of symmetry is introduced in different parts of the face.
- Collecting and measuring those differences in symmetry from different portions of the face allows us to compile a typical signature for each expression. These signatures are then vectorized and feed to a classifier.
4.2. Method
4.2.1. Dataset
4.2.2. Procedure
- 1)
- parameters (e.g., axis position and angle) for which max_included and min_including are obtained are the same, and
- 2)
- the measure given above is invariant for translation, rotation, and scaling.
4.3. Experimental Results and Discussion
5. Conclusions
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Image | λg | αg | λc | αc | OST | αOST |
---|---|---|---|---|---|---|
1a | 0.76 | 135.00° | 0.76 | 11.25° | 0.86 | 112.50° |
2a | 0.74 | 90.00° | 0.79 | 33.75° | 0.93 | 101.00° |
3a | 0.82 | 157.50° | 0.76 | 22.50° | 0.87 | 56.25° |
4a | 0.76 | 0.00° | 0.80 | 0.00° | 0.80 | 0.00° |
1b | 0.80 | 90.00° | 0.80 | 90.00° | 0.72 | 90.00° |
2b | 0.70 | 90.00° | 0.89 | 90.00° | 0.92 | 45.00° |
1c | 0.99 | 90.00° | 0.99 | 135.00° | 0.90 | 90.00° |
2c | 0.99 | 0.00° | 0.99 | 90.00° | 0.96 | 0.00° |
Image | ρ | α |
---|---|---|
1a | 0.67 | 101.25° |
2a | 0.67 | 112.50° |
3a | 0.58 | 112.50° |
4a | 0.55 | 157.50° |
1b | 0.80 | 90.00° |
2b | 0.94 | 90.00° |
1c | 0.99 | 90.00° |
2c | 0.98 | 0.00° |
Neutral | Anger | Disgust | Fear | Happiness | |
---|---|---|---|---|---|
Neutral | 0,88 | 0,00 | 0,02 | 0,05 | 0,05 |
Anger | 0,07 | 0,61 | 0,12 | 0,13 | 0,07 |
Disgust | 0,02 | 0,00 | 0,93 | 0,00 | 0,05 |
Fear | 0,07 | 0,07 | 0,05 | 0,76 | 0,05 |
Happiness | 0,15 | 0,07 | 0,12 | 0,05 | 0,61 |
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Di Gesu, V.; Tabacchi, M.E.; Zavidovique, B. Symmetry as an Intrinsically Dynamic Feature. Symmetry 2010, 2, 554-581. https://doi.org/10.3390/sym2020554
Di Gesu V, Tabacchi ME, Zavidovique B. Symmetry as an Intrinsically Dynamic Feature. Symmetry. 2010; 2(2):554-581. https://doi.org/10.3390/sym2020554
Chicago/Turabian StyleDi Gesu, Vito, Marco E. Tabacchi, and Bertrand Zavidovique. 2010. "Symmetry as an Intrinsically Dynamic Feature" Symmetry 2, no. 2: 554-581. https://doi.org/10.3390/sym2020554