Compact description of 3- -D image D image Compact description of 3 gamut by r- -image method image method gamut by r H i r o a k i K o t e r aa a n d R y o i c h i S a i t o H i r o a k i K o t e r a n d R y o i c h i S a i t o Jounal of Electronic Image, vol.12, of Electronic Image, vol.12, Jounal pp. 660- -668, Oct. 2003 668, Oct. 2003 pp. 660 School of Electrical Engineering and Computer Science Kyungpook National Univ.
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 2 / 33
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 3 / 33
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 4 / 33
� 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 5 / 33
3- -D image gamut shell by r D image gamut shell by r- -image image 3 3-D image gamut shell by r-image � Extraction of 3-D image gamut shell – Image color center P P P 1 1 1 ∑ ∑ ∑ = = * * * * * * r [ L , a , b ] ( L ), ( a ) , ( b ) (1) 0 0 0 0 i i i P P P = = = i 1 i 1 i 1 where P is number of pixel in input image 6 / 33
* * * c = r – An arbitrary pixel is represent by [ L , a , b ] i i i i i = − ≤ ≤ 1 i (2) r c r P i i 0 * * − − b b 1 θ = i 0 ≤ θ ≤ π 0 (3) tan 2 i i * * − a a i 0 * * − − L L 1 ϕ = π + i 0 ≤ ϕ ≤ π 0 (4) ( / 2 ) tan i i * − * 2 + * − * 2 1 2 {( a a ) ( b b ) } i 0 i 0 7 / 33
Fig.1. Maximum radial vector in segmented polar angle space. 8 / 33
– Radial matrix r- gamut and segmentation = = r - r r gamut [ ] max{ } jk i − ∆ θ ≤ θ ≤ ∆ θ − ∆ ϕ ≤ ϕ ≤ ∆ ϕ for ( and j 1 ) i j ( k 1 ) k i ∆ θ = π ≤ ≤ 1 2 / M j M ∆ ϕ = π ≤ ≤ 1 (5) / N k N • M, N are segmentation factors 9 / 33
(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. 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 (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). 12 / 33
(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). 13 / 33
� r-image as 3-D GBD – Definition of the r-image ( M x N rectangular matrix r) [ ] = − + − + − 1 2 * * 2 * * 2 * * 2 r ( L L ) ( a a ) ( b b ) (6) jk jk 0 jk 0 jk 0 = ≤ ≤ ≤ ≤ r r ] (7) [ 1 1 . j M, k N jk – The radial vector magnitude arranged in a discrete integer ( j , k ) address 14 / 33
– Approximate reconstruction from r image * * ≅ − ∆ θ − ∆ ϕ + ˆ r a cos( j 0 . 5 ) sin( k 0 . 5 ) a jk jk 0 ˆ * * ≅ − ∆ θ − ∆ ϕ + r b sin( j 0 . 5 ) sin( k 0 . 5 ) b jk jk 0 ˆ * ≅ * − − ∆ ϕ r (8) L L cos( k 0 . 5 ) jk 0 jk 15 / 33
* L * b θ ϕ * a (a) image “wool” (b) 3D maximum r vectors (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. 16 / 33
Compact GBD by compression of r image Compact GBD by compression of r image Compact GBD by compression of r image � Discrete Cosine Transform t = = R [ R ] A rA DCT jk = A [ a ] jk 1 = for 1 k M = a jk − − π 2 ( 2 j 1 )( k 1 ) cos 2 M M = = for 2,..., 1,2,..., (10) k M j M 17 / 33
– Being concentrated spatial frequency energy in low frequency – Compression of r image t r = (11) AR DCT A [ ] ≤ for R j , k m jk m t m m m ≅ = = ˆ r AR A , R R , R jk jk > for (12) 0 j , k m 18 / 33
� Compression of r-image by SVD = = Λ t r [ r ] U V ( 13 ) jk r ′ ′ r r r where U, V : the eigenvectors of and Λ : the diagonal matrix containing the singular values of r λ K K 0 0 1 λ K 0 0 0 t rV 1 Λ = = U (14) M M O λ K K K 0 M 19 / 33
Λ – Compression by m( <M ) in singular values matrix and eigen vectors matrix U,V t = ≅ m Λ ˆ ˆ [ ] U V (15) r r m m jk λ K K 0 0 1 λ K 0 0 0 1 Λ = (16) m M M O λ K K K 0 m 20 / 33
L U U U 11 12 1 m L U U U 21 22 2 m = U m M M L U U U M 1 M 2 Mm L V V V 11 12 1 N L V V V 21 22 2 N = (17) V m M M L V V V m 1 m 2 mN 21 / 33
– Compression rate + ( 2 M 1 ) m = (20) C 2 M where × colomn vectors U : M m m × row vectors V : m M m and singular values m 〉〉 ≅ If M m, C is given by C 2 m / M 22 / 33
� Compression of r image by Wavelet t = = (18) R [ R ] W rW DWT jk t = r (19) WR DWT W – Applied a Daubechies filter as the discrete wavelet scaling function Fig. 5. 2-D discrete wavelet transform of r-image wool. 23 / 33
Experimental results Experimental results Experimental results � Quantization error in r image by segmentation Saturated to 0.57 and 0.90 each other Fig. 6. Quantization error on the gamut shell surface of r image by segmentation. 24 / 33
� 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 ) 25 / 33
(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 ) 26 / 33
(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 ) 27 / 33
� 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 ) 28 / 33
(b) first four eigen vectors in matrix U (c) first four eigen vectors in matrix V Fig. 8. SVD parameters. 29 / 33
� Gamut reconstruction error from reduced DCT and SVD coefficients Fig. 9. Gamut surface reconstruction error from reduced DCT and SVD coefficients. 30 / 33
� 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. 31 / 33
original 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 original 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. 32 / 33
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 33 / 33
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