Otolith Shape Otolith Shape AnalysisAnalysisusing using

Wavelet TransfomsWavelet Transfomsandand

Curvature Scale Curvature Scale SpaceSpace

Vicenç Parisi Baradad, Joan Cabestany, Jaume Piera

Emili Garcia-Ladona, Toni Lombarte

• Contour codingContour coding• Wavelet TransformWavelet Transform• Curvature Scale Space Curvature Scale Space representationrepresentation• Fourier TransformFourier Transform• Data base retrievalData base retrieval

INTRODUCTION – Contour – Wavelet – CSS – Fourier – Matching

IntroductionIntroduction

• Coordinates (x,y)Coordinates (x,y)

Introduction – CONTOUR – Wavelet – CSS – Fourier– Matching

• Equiangle coordinatesEquiangle coordinates

Introduction – CONTOUR – Wavelet – CSS – Fourier – Matching

• Chain codeChain code

Introduction – CONTOUR – Wavelet – CSS – Fourier – Matching

Wavelet TransformWavelet Transform

)(*)( 22 xfxfW jj

0)( dxx

s

x

sxs 1)(

22

2

12

21 |)(||)(||)(| xfSxfWxfS Jj

J

j

Introduction – Contour – WAVELET – CSS – Fourier – Matching

fS 22

fW 12

fS 32

fW 22

fS 12

fW 32

Introduction – Contour – WAVELET – CSS – Fourier – Matching

• Mother wavelet: smoothing function second derivativeMother wavelet: smoothing function second derivative

22

2

2)(

dx

dx

j

j

))(*(2)(

22

22

2xf

dx

dxfW jj

j

Introduction – Contour – WAVELET – CSS – Fourier – Matching

2

32.2.

......

)),(),((

),(),(),(),(),(

uYuX

uYuXuYuXuk

Curvature Scale SpaceCurvature Scale Space

•Invariance to image translation, scale and rotation changes

•Robust to shear

•Good performance against high frequency noise

Introduction – Contour – Wavelet – CSS – Fourier– Matching

Contour SmoothingContour Smoothing

Introduction – Contour – Wavelet – CSS – Fourier– Matching

Curvature Inflection PointsCurvature Inflection Points

Introduction – Contour – Wavelet – CSS – Fourier– Matching

Sampling InvarianceSampling Invariance150 samples 512 samples

CSS normalized

Introduction – Contour – Wavelet – CSS – Fourier– Matching

Noise inmunityNoise inmunity

Low scales elimination increases noise inmunity

Introduction – Contour – Wavelet – CSS – Fourier– Matching

Scaling invarianceScaling invariance

Introduction – Contour – Wavelet – CSS – Fourier– Matching

Rotation invarianceRotation invariance

Rotation = Maxima translation

Introduction – Contour – Wavelet – CSS – Fourier– Matching

Shear "invariance"Shear "invariance"

Shear produces slight changes

Introduction – Contour – Wavelet – CSS – Fourier– Matching

Fourier TransformFourier Transform

Cosinus Sinus

• Integral covers whole contour

• Singularities not located

Introduction – Contour – Wavelet – CSS – FOURIER – Matching

Data BaseData Base107 otoliths Wavelet

Fourier CSS

Wavelet MatchingWavelet Matching• Energy conservationEnergy conservation

j

xfWf j2

2

2 |)(|||

• Wavelet distanceWavelet distance

j

gWfW (x)||||f(x) – g jj2

22

2 ||||

• Zero Crossing distanceZero Crossing distance

j

xgZxfZZgZfd jj2

22

2 ||)()(||),(

Introduction – Contour – Wavelet – CSS – Fourier – MATCHING

Introduction – Contour – Wavelet – CSS – Fourier – MATCHING

Introduction – Contour – Wavelet – CSS – Fourier – MATCHING

Introduction – Contour – Wavelet – CSS – Fourier – MATCHING

Introduction – Contour – Wavelet – CSS – Fourier – MATCHING

CSS MatchingCSS Matching

Introduction – Contour – Wavelet – CSS – Fourier – MATCHING

Image Model

Noise, Rotation and Shear

Introduction – Contour – Wavelet – CSS – Fourier – MATCHING

Fourier MatchingFourier Matching|)()(|),(

1

n

jjCgjCfCgCfd

Introduction – Contour – Wavelet – CSS – Fourier– MATCHING

Fourier Matching under Rotation

Introduction – Contour – Wavelet – CSS – Fourier – MATCHING

Fourier Matching under Shear

Introduction – Contour – Wavelet – CSS – Fourier – MATCHING

Fourier Matching under Noise

Introduction – Contour – Wavelet – CSS – Fourier– MATCHING

ConclusionsConclusions

• (Wavelet + chain code) and CSS robust under (Wavelet + chain code) and CSS robust under affine transformations and Shearaffine transformations and Shear

• Wavelet and CSS locate SingularitiesWavelet and CSS locate Singularities• Wavelet + CSS allow database compressionWavelet + CSS allow database compression• Wavelet allows perfect contour reconstructionWavelet allows perfect contour reconstruction

Introduction – Contour – Wavelet – CSS – Fourier – Matching