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# lecture16 - = 29 29 29 29 29 03 21 12 30 11 2 03 21 2 12 30...

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Shape Descriptors

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Desired Properties ° Uniqueness ° Invariance ° Size ° Rotations ° Translations ° Noise
Moments ( 29 °° = x y q p pq y x B y x m , Discrete Form

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Uniqueness Theorem ° The double moment sequence {m pq } is uniquely determined by B(x,y) and conversely B(x,y) is uniquely determined by {m pq }
Characteristic Function/ Moment Generating Function Characteristic Function: ( 29 ( 29 ( 29 ± ± - - + = dxdy y x B vy ux v u M , exp , Moment Generating Function: ( 29 ( 29 ( 29 ± ± - - + = dxdy y x B ivy iux v u , exp , φ

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Characteristic Function/ Moment Generating Function Characteristic Function: Moment Generating Function: ( 29 ( 29 ( 29 °° = = = 0 0 ! ! , p q q p pq q iv p iu m v u φ ( 29 °° = = = 0 0 ! ! , p q q p pq q v p u m v u M
Central Moments Translation Invariant

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Central Moments
Hu Moments 02 20 1 u u v + = ( 29 2 11 2 02 20 2 4 u u u v + - = ( 29 ( 29 2 03 12 2 12 30 3 3 3 u u u u v - + - = ( 29 ( 29 2 03 21 2 12 30 4 u u u u v + + + = Translation, Rotation, Scaling Invariant

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Hu Moments ( ( ( 29 ( 29 [ ] ( 29 ( 29 ( 29 ( 29 [ ] 2 03 21 2 12 30 03 21 03 21 2 03 21 2 12 30 12 30 12 30 5 3 3 3 3 u u u u u u u u u u u u u u u u v + - + + - + + - + + - = ( 29 ( 29 ( 29 [ ( 29 ( 29 ]

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Unformatted text preview: = ( 29 ( 29 ( 29 [ ( 29 ( 29 ] 03 21 12 30 11 2 03 21 2 12 30 02 20 6 4 u u u u u u u u u u u v + + +--+-= ( 29 ( 29 ( 29 ( 29 [ ] ( 29 ( 29 ( 29 ( 29 [ ] 2 03 21 2 12 30 03 21 12 30 2 12 30 2 12 30 12 30 03 21 7 3 3 3 3 u u u u u u u u u u u u u u u u v +-+ +-+ +-+ +-= Hu Moments Medial Axis Transform Medial Axis Transform Medial Axis Transform Medial Axis Transform Medial Axis Transform Medial Axis Transform Suggested Reading & M-K. Hu., “Visual pattern recognition by moment invariants,” Computer methods in image analysis. & H. Blum, “A transformation for extracting new descriptors of shape,” Computer methods in image analysis. & Chapter 4, Mubarak Shah, “Fundamentals of Computer Vision”....
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