Compact description of 3- -D image D image Compact description of - - PowerPoint PPT Presentation

Compact description of 3 Compact description of 3-

• D image

D image gamut by r gamut by r-

• image method

image method

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• Jounal

Jounal of Electronic Image, vol.12,

• f Electronic Image, vol.12,
• pp. 660
• pp. 660-
• 668, Oct. 2003

668, Oct. 2003 School of Electrical Engineering and Computer Science Kyungpook National Univ.

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Flow chart Flow chart 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 Abstract 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 Introduction 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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3-D image gamut shell by r-image 3 3-

• D image gamut shell by r

D image gamut shell by r-

• image

image

Extraction of 3-D image gamut shell

– Image color center

where P is number of pixel in input image

(1) ) ( 1 , ) ( 1 ), ( 1 ] , , [ r

1 1 * 1 * * * * *

      = =

∑ ∑ ∑

= = = P i P i i P i i i

b P a P L P b a L

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– An arbitrary pixel is represent by ] , , [

* * * i i i i

b a L c =

i

r

(2) i 1 P r c r

i i

≤ ≤ − = (3) π θ θ 2 tan

* * * * 1

≤ ≤         − − =

− i i i i

a a b b

(4) π ϕ π ϕ ≤ ≤         − + − − + =

− i i i i i

b b a a L L

2 1 2 * * 2 * * * * 1

} ) ( ) {( tan ) 2 / (

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

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– Radial matrix r-gamut and segmentation

• M, N are segmentation factors

(5) 1 1 and ( for r r

• r

N k N M j M k k j i j gamut

i i jk

≤ ≤ = ∆ ≤ ≤ = ∆ ∆ ≤ ≤ ∆ − ∆ ≤ ≤ ∆ − = = / / 2 ) 1 ( ) 1 } max{ ] [ π ϕ π θ ϕ ϕ ϕ θ θ θ

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(a) image “wool” (b) color map (c) radial vectors (d) surface colors (e) wire frame (f) polygon gamut surface Fig.2. Extraction of image gamut of maximum radial vector.

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(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.

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(a) color chip map (b) radial vectors (c) surface colors (d) wire frame (e) polygon gamut surface (f) out of gamut r vectors Fig.3. Extraction of device gamut by maximum radial vectors (Epson PM800C inkjet printer).

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(a) color distribution of chips (b) radial vectors (c) gamut shell in wire frame (d) gamut shell surface Fig.3-1. Extraction of device gamut by maximum radial vectors (Epson PM800C inkjet printer 1331 chips).

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r-image as 3-D GBD

– Definition of the r-image (M x N rectangular matrix r) – The radial vector magnitude arranged in a discrete integer ( j , k ) address

[ ]

(6) ) ( ) ( ) ( r

2 1 2 * * jk 2 * * jk 2 * * jk

b b a a L L

jk

− + − + − =

(7) ] r r N k M, j

jk

. 1 1 [ ≤ ≤ ≤ ≤ =

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– Approximate reconstruction from r image (8) r r r ϕ ϕ θ ϕ θ ∆ − − ≅ + ∆ − ∆ − ≅ + ∆ − ∆ − ≅ ) 5 . cos( ˆ ) 5 . sin( ) 5 . sin( ˆ ) 5 . sin( ) 5 . cos( ˆ

* * * * * *

k L L b k j b a k j a

jk jk jk jk jk jk

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(a) image “wool” (b) 3D maximum r vectors

*

a

*

L

*

b

ϕ

θ

(c) color chip map (d) radial vectors (e) surface colors Fig.4. GBD of image wool by 2-D r-image and its surface colors.

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Compact GBD by compression of r image Compact GBD by compression of r image Compact GBD by compression of r image

Discrete Cosine Transform

(10) 1,2,..., 2,..., for 1 for M j M k M k j M k M a a A rA A R R

jk jk t jk DCT

= =              − − = = = = = 2 ) 1 )( 1 2 ( cos 2 1 ] [ ] [ π

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– Being concentrated spatial frequency energy in low frequency – Compression of r image

(11)

t DCT A

AR r =

[ ]

     > ≤ = = ≅ (12) for for m k j m k j R R R R A AR r

jk m jk m jk m t m

, , , , ˆ

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Compression of r-image by SVD

where U, V : the eigenvectors of and : the diagonal matrix containing the singular values of r

(14)

1 1

              = = Λ

M trV

U λ λ λ M K K K O M K K K r ) 13 ( ] [ t V U jk r Λ = =

r r ′ r r ′ Λ

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– Compression by m(<M) in singular values matrix

and eigen vectors matrix U,V

Λ

(15) V U ] [

t m m mΛ

≅ =

jk

r r ˆ ˆ

(16)               = Λ

m m

λ λ λ M K K K O M K K K

1 1

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              =

Mm M M m m

U U U U U U U U U U L M M L L

2 1 2 22 21 1 12 11 m

(17)               =

mN m m N N m

V V V V V V V V V V L M M L L

2 1 2 22 21 1 12 11

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– Compression rate

where If M m, C is given by

(20)

2

) 1 2 ( M m M C + =

values singular and vectors row vectors colomn m M m V m M U

m m

× × : :

〉〉

M m C / 2 ≅

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Compression of r image by Wavelet

– Applied a Daubechies filter as the discrete wavelet scaling function

(18) rW W R R

t jk DWT

= = ] [

(19) r

t DWT W

WR =

• Fig. 5. 2-D discrete wavelet transform of r-image wool.

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Experimental results Experimental results Experimental results

Quantization error in r image by segmentation

• Fig. 6. Quantization error on the gamut shell surface
• f r image by segmentation.

Saturated to 0.57 and 0.90 each other

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Image gamut reconstruction from reduced

DCT and SVD parameters

(a) original “wool” r – image gamut shell

• Fig. 7. Reconstructed r image and gamut shell from reduced

DCT and SVD coefficients.( continuous )

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(b) DCT r-imge (c) DCT gamut shell dimension 4 x 4 8 x 8 16 x 16

• Fig. 7. Reconstructed r image and gamut shell from reduced

DCT and SVD coefficients .( continuous )

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(d) SVD r-imge (c) SVD gamut shell dimension 4 x 4 8 x 8 16 x 16

• Fig. 7. Reconstructed r image and gamut shell from reduced

DCT and SVD coefficients .( continuous )

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SVD parameters for r image reconstruction

– Selection of the first four eigenvectors and singular values

(a) distribution of eigen values

• Fig. 8. SVD parameters .( continuous )

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(b) first four eigen vectors in matrix U (c) first four eigen vectors in matrix V

• Fig. 8. SVD parameters.

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Gamut reconstruction error from reduced

DCT and SVD coefficients

• Fig. 9. Gamut surface reconstruction error from reduced

DCT and SVD coefficients.

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Compression of gamut shell shape

(a) image “bride” (b) image “wool”

• Fig. 10. Reconstruction error of gamut shell by compression for

48 x 48 segmentations in r-image.

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• riginal wavelet=1/6 wavelet=1/10 JPEG=1/5 SVD=1/6

“bride” (384 byte) (230 byte) (460 byte) (388 byte) Δ E94=2.6 Δ E94=3.7 Δ E94=4.0 Δ E94=4.5

• riginal wavelet=1/6 wavelet=1/10 JPEG=1/5 SVD=1/6

“wool” (384 byte) (230 byte) (460 byte) (485 byte) Δ E94=4.0 Δ E94=5.1 Δ E94=5.3 Δ E94=5.1

• Fig. 11. Comparison of gamut shell shape compressed by JEPG, SVD, and wavelet.

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Conclusions Conclusions Conclusions

Compact description of 3-D image gamut

– Description of 3-D I-D GMA by compact r-image

• A set of radial vectors
• Segmented polar angles

– Detection to the out of gamut segments easily without comparing all of the pixels