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ter Haar Romeny, FEV Nuclei of fungus cell Paramecium Caudatum Spatial gradient Illumination spectrum -invariant gradient Color RGB original Color-Scale Differential Structure Geusebroek et al, LNCS 1852, 459-464, 1999

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ter Haar Romeny, FEV The color of an object depends on color of the illuminating light illumination intensity sensor sensitivity direction of surface normal surface reflectance properties Assumptions: Scene is uniformly illuminated light source is colored surface has Lambertian reflectance

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ter Haar Romeny, FEV What causes color ? Lamp object color spectral color

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ter Haar Romeny, FEV Object reflectance function for the observed spectrum for a resp. 2500K, 6500K and 10,000K light source: Spectrum reflected from an arbitrary object Emission spectrum of black body radiator

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ter Haar Romeny, FEV Color receptive fields 0 1 2

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ter Haar Romeny, FEV Self-organization: receptive fields from Eigenpatches (12x12 pixels)

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ter Haar Romeny, FEV Colour receptive fields from Eigenpatches

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ter Haar Romeny, FEV Hering basis 0102030405060708090 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Idea Koenderink: Gaussian derivatives of zero, first and second order in the wavelength domain wavelength RF sensitivity How can we measure color?

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ter Haar Romeny, FEV Taylor color model Luminance Blue-yellowness Purple-greenness 300400500600700800900 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 L M S Cone sensitivity

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ter Haar Romeny, FEV Spatial color Color scale-space starts by probing this space. s Energy densities cannot be measured at a point, … … one probes a certain volume

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ter Haar Romeny, FEV Reflectance of light Lamp object color spectra l color What are invariant properties?

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ter Haar Romeny, FEV Reflectance model

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ter Haar Romeny, FEV Transparent materials

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ter Haar Romeny, FEV The reflected spectrum is: v = viewing direction n = surface patch normal s = direction of illumination f = Fresnel front surface reflectance coefficient in v R = body reflectance

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ter Haar Romeny, FEV Because of projection of the energy distribution on the image plane the vectors n, s and v will depend on the position at the imaging plane. So the energy at a point x is then related to: We assume an illumination with a locally constant color:

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ter Haar Romeny, FEV Aim: describe material changes independent of the illumination. Both equations have many common terms

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ter Haar Romeny, FEV The normalized differential determines material changes independent of the viewpoint, surface orientation, illumination direction, illumination intensity and illumination color!

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ter Haar Romeny, FEV The derivative jet to x and forms a complete family of geometric invariants: These are observed properties, so we convolve with Gaussian derivatives

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ter Haar Romeny, FEV Color edges can be defined as the thresholding of the spatial gradient (color-invariant equivalent of L w ): Color invariants

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ter Haar Romeny, FEV Spatial color model and tracing color edges in microscopy Influence of illumination color temperature on edge strength, scale is 3.0 px. Skin tissue section illuminated by a halogen bulb at 4000 K (top) and 2600 K (bottom) color temperature.

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ter Haar Romeny, FEV Color-invariant multi-scale structural operators

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ter Haar Romeny, FEV Total edge strength

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ter Haar Romeny, FEV Some color differential invariants

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ter Haar Romeny, FEV Feulgen stain, red-green edges Paramecium caudatum, Feulgen and Fast green stain Color canny, red-green normalized edges, scale 3

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ter Haar Romeny, FEV Hematoxylin eosin stain Pituitary gland, sheep, adenohypophysis 40x Cell: E 0, scale 1.0 Nuclei: E 0, E +E < 0, scale 3.0 additional constraint added to refine selection

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ter Haar Romeny, FEV Safranin O stain E > 0, E > 0, scale sigma 1.0 Safranin O stain for proteoglycans (mouse knee joint) Courtesy of Koen Gijbels and Paul Stoppie

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ter Haar Romeny, FEV Oil red O stain Oil red O stain of fat emboli in lung E > 0, E > 0, scale 1.5

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ter Haar Romeny, FEV PAS stain L ww > 0, L vv L ww -L vw 2 > 0, E -E > 0, scale sigma 2.0 P.A.S. stain for carbohydrates (goblet cells, gut) carbohydrates stain magenta - elliptic patches

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ter Haar Romeny, FEV Blood smear Blood smear, Giemsa stain, 100x, JPEG compression RBC: E > 0, E +E > 0, scale 0.5 Leucocytes: E < 0, scale 12 Leucocyte nuclei: E 0, scale 3

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ter Haar Romeny, FEV Blue-yellow edges Note the complete absence of detection of black-white edges.

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ter Haar Romeny, FEV Color edges can also be defined as the zero-crossings of the second order derivative in the spatial gradient direction (color-invariant equivalent of L ww ): Second order color invariants

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ter Haar Romeny, FEV Color invariant edge detection Luminance gradient edge detection

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ter Haar Romeny, FEV Conclusions Color ‘scale-space’ compatible with classical luminance scale- space The model enables the design of practical image analysis ‘color reasoning’ solutions, e.g. invariance for illumination The color-scale invariant differential operators are building blocks for a differential geometry on color images

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