Compact description of 3-D image gamut by r-image

Compact description of 3-D image gamut by r-image method Hiroaki Kotera and Ryoichi Saito Jounal of Electronic Image, vol.12, pp. 660-668, Oct. 2003 School of Electrical Engineering and Computer Science Kyungpook National Univ.

Flow chart Input image 3D image gamut by 2D monochorometic image (r-image) Compact Gamut Boundary Descriptor by compression Reconstruction of the surface colors

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Abstract ‹ 3-D

image to device gamut mapping by

r-image – Each pixel in the r-image is denoted the maximum radial vector magnitude in CIELAB color space – Segmentation by using the discrete polar angle color space

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Introduction ‹ 2-D

LC (lightness-chroma) Gamut Mapping Algorithm – Device to device concept – Advance toward 2-D into 3-D

‹ 3-D

I-D gamut mapping algorithm

– Simple and compact image GBD – Easily performed by the pixel to pixel direction comparison between r-image of device and image

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‹ Key

factor

– To extract the 3-D image gamut shell from the random color distributions – To describe its boundary surface with a small number of data

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33-D -D image gamut shell by rr-image -image ‹ Extraction

of 3-D image gamut shell

– Image color center 1 P * 1 P * 1 P *  r0 = [ L , a , b ] =  ∑ ( Li ), ∑ (ai ), ∑ (bi ) P i =1 P i =1  P i =1  * 0

* 0

* 0

(1)

where P is number of pixel in input image

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– An arbitrary pixel ri = ci − r0

ci = [ L*i , ai* , bi* ] is represent by ri

1≤ i ≤ P

 * * −1 bi − b0  θi = tan  ai* − a0* 

(2)

0 ≤ θi ≤ 2π

  ϕi = (π / 2) + tan * * 2 * * 2 1 2 {(ai − a0) + (bi − b0 ) }  

−1

L*i − L*0

(3)

0 ≤ ϕi ≤ π

(4)

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Fig.1. Maximum radial vector in segmented polar angle space.

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– Radial matrix r-gamut and segmentation r - gamut = [r jk ] = max{ri } for ( j − 1)∆θ ≤ θi ≤ j∆θ and (k − 1)∆ϕ ≤ ϕi ≤ k∆ϕ ∆θ = 2π / M

1≤ j ≤ M

∆ϕ = π / N

1≤ k ≤ N

(5)

• M, N are segmentation factors

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(a) image “wool”

(d) surface colors

(b) color map

(e) wire frame

(c) radial vectors

(f) polygon gamut surface

Fig.2. Extraction of image gamut of maximum radial vector. 10 / 33

(a) image “bride

(b) color map in CIELAB

(c) radial vector to gamut surface (d) gamut shell Fig.2-1. Extraction of image gamut of maximum radial vector. 11 / 33

(a) color chip map

(d) wire frame

(b) radial vectors

(c) surface colors

(e) polygon gamut surface (f) out of gamut r vectors

Fig.3. Extraction of device gamut by maximum radial vectors (Epson PM800C inkjet printer). 12 / 33

(a) color distribution of chips

(c) gamut shell in wire frame

(b) radial vectors

(d) gamut shell surface

Fig.3-1. Extraction of device gamut by maximum radial vectors (Epson PM800C inkjet printer 1331 chips). 13 / 33

‹ r-image

as 3-D GBD

– Definition of the r-image (M x N rectangular matrix r)

[

r jk = ( L*jk − L*0 ) 2 + ( a *jk − a 0* ) 2 + ( b *jk − b 0* ) 2 r = [ r jk ]

1 ≤ j ≤ M,

]

1 2

1≤ k ≤ N.

(6) (7)

– The radial vector magnitude arranged in a discrete integer ( j , k ) address

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–

Approximate reconstruction from r image aˆ *jk ≅ r jk cos( j − 0.5) ∆θ sin( k − 0.5) ∆ϕ + a0* bˆ*jk ≅ r jk sin( j − 0.5) ∆θ sin( k − 0.5) ∆ϕ + b0* Lˆ*jk ≅ L*0 − r jk cos( k − 0.5) ∆ϕ

(8)

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L*

b*

ϕ

(a) image “wool”

(c) color chip map

θ a*

(b) 3D maximum r vectors

(d) radial vectors

(e) surface colors

Fig.4. GBD of image wool by 2-D r-image and its surface colors. 16 / 33

Compact GBD by compression of r image ‹

Discrete Cosine Transform R DCT = [ R jk ] = At rA A = [ a jk ]    a jk =    

1

for k = 1

M  ( 2 j − 1)( k − 1)π  cos   2M   M

2

for k = 2,..., M

j = 1,2,..., M

(10)

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– Being concentrated spatial frequency energy in low frequency – Compression of r image r = ARDCT At m t

rˆ ≅ AR A ,

(11) R

m

[ ]

= Rm jk ,

 R jk m  R jk =  0

for j , k ≤ m for j , k > m

(12)

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‹

Compression of r-image by SVD r = [r jk ] = UΛV t where

(13)

U, V : the eigenvectors of r r ′and r ′r

Λ : the diagonal matrix containing the singular values of r

λ1 0 K K 0    0 0 0 K λ 1  Λ = U t rV =  M M  O    0 K K K λM 

(14)

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– Compression by m(