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1 NAIST-IS-MT

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3 4 Fast ICA, NAIST-IS- MT8511, i

4 ,,, Fast ICA, ii

5 Organization of Concentric Receptive Fields in the Primary Visual Cortex by Using Independent Component Analysis Takayoshi Aoki Abstract Previous experimental studies showed that neurons in the primary visual cortex (V1) have four firing properties to represent visual information. Four firing properties are named luminance gabor, oriented double-opponent, concentric single-opponent, and concentric double-opponent receptive fields. Previous theoretical studies demonstrated that redundancy compressions like independent component analysis (ICA) and sparse coding can derive basic functions corresponding to luminance gabor and oriented double-opponent receptive fields from natural images. On the other hand, emergences of basic functions corresponding to two concentric receptive fields via redundancy compression of natural images have not been reported. In this study, I propose a low pass filtering function of neurons in lateral geniculate nucleus (LGN) is important to derive concentric basic functions in addition to redundancy compression. Neurons in LGN project their axons to neurons in V1. Therefore, it is plausible that LGN contributes to form receptive fields of V1. I show that the concentric double-opponent basis function can be obtained by applying the low pass filtering and Fast ICA to natural images in a computer simulation. However, the concentric single-opponent basis function cannot be not obtained. This reason is thought that the concentric doubleopponent basis function functionally includes the concentric single-opponent basis Master s Thesis, Department of Bioinformatics and Genomics, Graduate School of Information Science, Nara Institute of Science and Technology, NAIST-IS-MT8511, February 5, 21. iii

Four firing properties are named luminance gabor, oriented double-opponent, concentric single-opponent, and concentric double-opponent receptive fields.

6 function from the view point of redundancy compression. I also clarify how shapes of basic functions depend on ones of learning images, and how color components of basic functions depend on a color distribution of learning images. Keywords: Concentric Double-Opponent, Single-Opponent, Redundancy compressions, Fast ICA, Low-pass filter iv

how color components of basic functions depend on a color distribution of learning

7 PCA PCA Fast ICA v

4............................... 22 2.5 PCA................. 23 2.6 PCA.................. 26 2.7 Fast ICA............................ 27 2.8...................... 28 2.

8 A. 61 B. 62 C. 63 D. 63 E. 64 vi

57 58 61 A. 61 B. 62 C. 63 D. 63 E. 64 vi

9 Hz Hz Hz Hz Hz Hz vii

......................... 22 15......... 24 16............. 34 17 22.8Hz........ 35 18 11.4Hz........ 36 19 5.7Hz......... 37 2 35.

10 25 5.7Hz Hz Hz Hz Hz viii

.... 53 1................. 6 2...................... 15 3.

11 (Photon) (Retina) (Cone) ( ) ( ) ( ) ( ) ( ) ( ) (Retinal Ganglion Cells RGCs) ( 1) L M Photon S Retina Cone Retina Luminance R-G B-Y RGCs 1 (Photon) (Retina) (Cone) (Retinal Ganglion Cells RGCs) L M S (Luminance) (R-G) (B-Y) 3 1

12 2 1 L nm M nm S nm [5] 3 [5] S ( ) nm( ) M ( ) nm( ) L ( ) nm( ) ( 2) [2]

13 A -(L+M+S) B +(L+M+S) -M +L C -S + : Excitatory - : Inhibitory +(L+M) D E F +(L+M+S) -(L+M+S) -M +L +(L+M) -S Luminance Channel Red-Green Opponent Channel Blur-Yellow Opponent Channel 3 (A) (C) (D) (F) (A) (D) (B) (E) (C) (F) L M S + (Excitatory) (Inhibitory) (Luminance) (R-G) (B-Y) 3 ( 3) 3A D 3A 3

Channel Blur-Yellow Opponent Channel 3 (A) (C) (D) (F) (A) (D) (B) (E) (C)

14 ( 1) S 3B E 3B ( ) L M 3C F L M (L+M) [2] (Lateral Geniculate nucleus LGN) [2] [9] (Primary Visual Cortex V1) 4

15 A B -(L+M+S) +(L+M+S) -(L+M+S) M- L+ L+ M- C D M+ L- M- L+ M+ L+ L- M- M+ L+ L+ M- M- M+ L- 4 (A) (B) (C) (D) [1] ( 4) 4A B C D 4A C 4B D 2 [6] 2 5

16 1 (Single-Opponent) (Concentric Double-Opponent) (Retina) (LGN) (Primary Visual Cortex V1) (Luminance Gabor) (Oriented Double-Opponent) Luminance Gabor Single-Opponent Oriented Double-Opponent Concentric Double-Opponent Retina LGN V1 ( 1) [2] 6

Double-Opponent) Luminance Gabor Single-Opponent Oriented

17 ON ON OFF OFF Single-Opponent Eye Lateral geniculate nucleus ON ON ON Single-Opponent OFF OFFOFF Primary visual cortex Luminance Gabor 5 ON OFF ( 5) ( ) 7

18 ON OFF ON OFF Single-Opponent Eye Lateral geniculate uncleus OFF ON ON ON OFF OFF Single-Opponent M+ L- Primary visual cortex L+ M- M+ L- L+ M- Oriented Double-Opponent 6 ON OFF ( 6) 8

19 Single-Opponent ON ON OFF OFF ON ON ON Eye Lateral geniculate uncleus OFF OFF OFF ON OFF ON OFF OFF Single-Opponent ON OFF ON OFF OFF ON ON Primary visual cortex L+ M- M+ L- M+ L- L+ M- Concentric Double-Opponent 7 ON OFF ( 7) [3] ( ) 9

Single-Opponent ON OFF ON OFF OFF ON ON Primary visual

20 2 1 [11][18] ( 1) ( ) = ( 1) ( 1) + ( 2) ( 2) +... (1) 1 ( ) Q bit/sec R bit/sec R Q Q R 1 bit/sec R Q 1

21 4 ( 4A) ( 4B) ( 4C) ( 4D) (Independent Component Analysis ICA) ( A) ICA ICA Olshausen ( A C) [15] ( 8 ) ( 8 ) [16] K( f ) 11

22 8 192 (16 16) [15] ( 2) ( ) 4 f K( f ) = High( f )Low( f ) = f exp (2) f f =2(cycles/picture) High( f ) Low( f ) K( f ) 2 12 f

A B K(f) 1 1 fy 1 1 1 1 f fx 9 (A) f K(f) (B)XY f x X f

23 A B K(f) 1 1 fy f fx 9 (A) f K(f) (B)XY f x X f y Y 1 [16] Olshausen Dharmesh ICA [21] C

図 1 先行研究において独立成分分析で得られた基底関数学習にはカラーの自然画像を用い得られた 18 個の基底関数 (6 6 3) の内半分を表示した各画素の色は赤緑青 3 色の組み合わせより表現される赤緑青はそれぞれ実数値を持つ黒色は赤緑青の全てがの値であることを示す + は正の値を持つ領域をは負の値を持つ領域を示すつまり 1 つの基底関数を正

24 図 1 先行研究において独立成分分析で得られた基底関数学習にはカラーの自然画像を用い得られた 18 個の基底関数 (6 6 3) の内半分を表示した各画素の色は赤緑青 3 色の組み合わせより表現される赤緑青はそれぞれ実数値を持つ黒色は赤緑青の全てがの値であることを示す + は正の値を持つ領域をは負の値を持つ領域を示すつまり 1 つの基底関数を正負に分けて表示しているこれらの基底関数は初期視覚野にある白黒ガボール型及び二重反対色ガボール型の受容野特性を持つ図は [21] より引用した I 行 4 列内において + 行の基底関数には輝度を示す白線があるそれに隣接して行も白線があるつまり白黒ガボール型の基底関数であった V 行 3 列の基底関数は赤と緑色が存在し赤緑型の二重反対色ガボール型の基底関数であった IV 行 2 列において青と黄色が存在し青黄型の二重反対色ガボール型の基底関数であった 14

25 2 Olshausen et al.[15] Dharmesh et al.[21] This study Filter Contrast filter Low-pass filter Redundancy compressions Sparse coding ICA Fast ICA Luminance gabor Oriented double-opponent Single-opponent Concentric double-opponent Olshausen Dharmesh ( 2) Dharmesh Olshausen 1.3 Elizabeth ( B) [1] ( 11A) ( ) 15

26 A 6 B 3 Spikes/sec 3 Spikes/sec Spatial freq (c/deg) Spatial freq (c/deg) 11 (A) (B) [1] ( 11B) [4][19] 11A B 1.4 Fast ICA 2 16

27 Fast ICA

28 2. 材料と方法本研究では google 画像検索を用いてランダムに選んだ 2 枚のカラー自然画像を学習サンプルとして用いた (図 12) 電気生理学実験に基づいたローパスフィルターを学習サンプルに掛けた (図 14) またアルゴリズムを簡単化するために画像サンプルの各画素値における平均を分散を 1 とする標準化を行った大きく分散の異なる変数に対してそのまま主成分分析 (Principal Component Analysis PCA) や ICA を適用すればその結果は分散の大きな変数の影響を強く受け変数間の関係を正しく得られない可能性があるからである計算時間と図 12 本研究において学習サンプルとして用いた 2 枚のカラー自然画像 18

29 PCA ( 15) PCA 1 ICA Aapo Hyvarinen Fast ICA [8][1] 2.1 ( ) 3 8bit 13 ( ) ( ) ( ) m m 3 (m m ) X R X G X B X = {X R X G X B } 3 x R 1,1 x R 1,m X R =..... x R m,1 x R m,m (3) X R

30 Purple Red 255 Black (,,) Blue 255 Yellow Aqua White (255,255,255) 255 Green mm 1,Hz 5Hz / Hz 5Hz (1/128) = 3.9Hz (4) Hz ( ) 2

31 5.7Hz(3.9Hz 13) Hz 11.7Hz 2.3 X R X G X B x r = x r E[X R ] s x g = x g E[X G ] s x b = x b E[X B ] s (5) x r x g x b ( 6) E[X] E[X] = 1 x r,g,b (6) N r N N = R + G + B s ( 7) s = 1 N g b (x r,g,b E[X]) 2 (7) r g xr x g x b b 21

32 2.4 ˆx n ( 14) ˆx n ˆX ˆX A B pixel Low-pass filter C Standardization D pixel 1 Red Plane Green Plane Blue Plane E E E (A) (B) (C) (D) (16 16 ) 22

33 2.5 PCA PCA ˆX = {ˆx n } n = 1,..., N ˆx n D PCA M < D ( 15) 1 (M = 1) D u 1 u 1 u T 1 u 1 = 1 T ˆx n u T 1 ˆx n u T 1 x x 1 N N {u T 1 ˆx n u T 1 x}2 = 1 N n=1 = 1 N = 1 N N {u T 1 (ˆx n x)} 2 n=1 N {u T 1 (ˆx n x)}{u T 1 (ˆx n x)} n=1 N {u T 1 (ˆx n x)}{(ˆx n x) T u 1 } n=1 = u T 1 Su 1 (8) S S = 1 N N (ˆx n x)(ˆx n x) T (9) u T 1 x u 1 n=1 u 1 u T 1 u 1 = 1 λ 1 ( D ) E(u 1 ) = u T 1 Su 1 + λ 1 (1 u T 1 u 1) (1) E( ) u 1 23

34 1 Input Data X^ 1 Basis Function 1 1 F , PCA Reduction 15, Eigen Value Decomposition ICA 768 ED 1/2 Restoration M=25 PCAed Data M M=25 M Z* Independent Component s 15 (PCA) (Input Data) (PCAed Data) (ICA) (Independent Component) (ED 1/2 ) (Basis Function) Su 1 = λ 1 u 1 (11) u 1 S u T 1 ut 1 u 1 = 1 24

35 u T 1 Su 1 = λ 1 (12) u 1 λ 1 1 M S M λ 1,..., λ M M u 1,..., u M PCA x S S M M D = 768 PCA 25 (M = 25) M D ˆx n = (ˆx T n u a )u a + (ˆx T n u a )u a (13) a=1 a=m M z n = (ˆx T n u a )u a (14) a=1 z n 1 M M (Z = (z 1... z n ) T ) M=D=768 25

36 2.6 PCA Z = (z 1... z n) T E[z iz j] = δ i j (15) z i 1 I E{z z T } = I 1 PCA n ˆx Z Z = Vˆx (16) V V ˆx V = D 1/2 E T (17) D = diag(d 1 d n ) C x E = (e 1 e n ) C x = E{ˆxˆx T } 1 C x E ˆx PCA ( 9 12) E E T E = EE T = I D C x = EDE T Z E[Z Z T ] = VE[ˆxˆx T ]V T = D 1/2 E T EDE T ED 1/2 = I (18) Z Z (ED 1/2 )Z = ˆx (19) Z ˆx 26

37 2.7 Fast ICA Z = (z 1... z R) T s = (s 1... s R ) T Z = As (2) A z r A s r s = A 1 Z (21) A 1 W s y = (y 1 y 2... y r ) y = WZ (22) y y r [7][17] [7][17] Fast ICA J(y) y H(y) = p(y) log p(y)dy (23) J(y) = H(y gauss ) H(y) (24) y gauss y y J(y) 27

38 y Z [7][17] 2.8 J(y) y y y r ( ) [14] J(y r ) 1 12 E{y3 r} kurt(y r) 2 (25) kurt 4 kurt ( ) y r = E{y 4 r } 3 [ E{y 2 r } ] 2 = E{y 4 r } 3 (26) y r 1 4 y 3 r y4 r G i E{G i (y 3 r)} 2 G 1 G 2 G 1 G 2 J(y r ) k i (E{G 1 (y r )}) 2 + k 2 (E{G 2 (y r )} E{G 2 (ν)}) 2 (27) k 1 k 2 ν 1 y r

39 G 1 (y r ) = y 3 r G 2 (y r ) = y 4 r ( 25) 2 G 2 G J(y r ) [ E{G (y r )} + E{G (µ)} ] 2 (28) G G G [8] G (y r ) = 1 log (cosh y r ) (29) a [8] Fast ICA W w Fast ICA w w T Z J(w T Z ) E{(w T Z ) 2 } = w 2 = 1 ω ( D ) J(w T Z ) = [ E{G ( w T Z ) } + E{G (µ)} ] 2 + ω( w 2 1) (3) w 29

40 J(w T Z ) w = [ E{G(w T Z )} ] 2 + 2E{G(µ)} E{G(wT Z )} + ω wwt w w w = 2E{G(w T Z )}E{Z g(w T Z )} + 2E{G(µ)}E{Z g(w T Z )} + 2ωw = 2 [ E{G(w T z)} + E{G(µ)} ] E{Z g(w T Z )} + 2ωw = 2γE{Z g(w T Z )} + 2ωw (31) w = w/ w (32) γ E{G(w T z)} + E{G(µ)} 32 w w T Z w w g G( 29) γ 33 w = E{Z g(w T Z )} (33) 33 [8] 29 α w (1 + α)w = E{Z g(w T Z )} + αw (34) E{(w T Z ) 2 } = w 2 = 1 E{G(w T Z )} ( D ) F = (1 + α)w = E{Z g(w T Z )} + αw = (35) w α 35 ( E ) w F F w = E{Z Z T g (w T Z )} + αi (36) 3

41 g ( ) g( ) 36 1 E{Z Z T g (w T Z )} E{Z Z T }E{g (w T Z )} = E{g (w T Z )}I (37) F ŵ = w [ E{g (w T Z )} + α ] 1 [ E{Z g(w T Z )} + αw ] (38) 38 E{g (w T Z )} + α ˇw = E{Z g(w T Z )} E{g (w T Z )}w (39) Fast ICA 1 w PCA ED 1/2 M = (F) ( 15) 31

42 Dharmesh 2 ( 16A) ( 16B C) 16B 16C B C Dharmesh ( 1) Hz 11.7Hz ( 17 21) ( ) ( ) A B 22.8Hz 17A B ( 17C D) 32

43 18A B 11.4Hz 17B 18B ( 18C) ( 18D) 19A B 5.7Hz 19B 18B ( ) ( ) ( 19C D) ( ) 2A B 35.1Hz 2B 19B ( ) ( ) ( 2C D) 21A B 11.7Hz 2B 21B ( ) ( ) ( ) ( 21C D) ( 17 21)

44 A B + C - 16 (A) (B) (C) 34

45 A B C + D Hz (A) (C) 16 35

46 A B C + D Hz (A) (C)

47 A B C D Hz (A) (C) 16 37

48 A B C D Hz (A) (C) 16 38

49 A B C + D Hz (A) (C) 16 ( ) ( ) ( ) 39

50 A 22B 22B 22C 22D 22B 5.7Hz 22B 22D 22B 22E 22C 22E 3.4 Fast ICA ( 23) ( 23A) 1( 23B) 2( 23C) 3 B C ( ) ( 1 2) 23A C 2 ( ) ( 23D F) ( 23D)

51 A B C D E 22 (A) (B) (C)B (D) 5.7Hz (E)D 2 1 ( 23E) ( 23D) 2 ( 23F) ( 23D F) ( 23A C) 41

52 A B C D E F 23 (A) (C) (D)A 2 (E)B 1 2 (F)C ( 24 29)

53 1 4 ( ) 3 (13) 3 ( ) 24A ( ) 3 24C ( ) 25C 26C 3 ( ) C 28C 29C C 25C 26C 24C 26C 26B 25C ( 24C 26C) ( 25C) ( )=( ) (Purple) 24C 25C 26C 24C 43

54 24B 25C 25B 25C 26C ( 24C 25C) ( 26C) 13 ( )=( ) (Yellow) 3.2 ( ) 44

55 A +B -R C G +G Blue.5 B +R B -5-5 Green Number Blue -5-5 Green Hz (A) (B)2 (C)B XY 45

56 A +R -G C B +B Red.6 B Number +G R -5-5 Blue 5 5 Red -5-5 Blue Hz (A) (C) 24 46

57 A +G -B C R +R Green.5 B +B G -5-5 Red Number.6 5 Green -5-5 Red Hz (A) (C) 24 47

58 A +G +R C B +B Green 1 B Number -R G -5-5 Blue 5 5 Green -5-5 Blue Hz (A) (C) 24 48

59 A B Number -G 1 4 +B -R +R -B +G Blue C Red Blue -5-5 Red Hz (A) (C) 24 49

60 A B Number -B 1 4 +R -G +G -R +B Red C Green Red -5-5 Green Hz (A) (C) 24 5

61 3.6 ( ) ( 3) 1 ( ) ( ) 3.2 3A (1 2 3) =( ) 6 3B 3 3B ( ) 2 ( ) 3 ( ) (1 2 3) =( ) 3.5 ( 31) 3 ( 31A ) ( 31A C D E F G ) 31A 24C 24C 31A 31A 24C 3B (Aqua) (13)

62 A B Axis (1, 2, 3) Basis Function Axis (1, 2, 3) Basis Function (R, G, B) (B, R, G) (R, B, G) (G, R, B) (B, G, R) (G, B, R) 3 (A) (1 2 3) (R G B) (1 2 3) =(R G B) ( 13 ) (B) 31B ( 13) 52

63 Blue A 1 4 B Red C 5 5 Green D 5 5 Blue Green.6 Blue E 5 Green 5 Red F 5 Red 5 Red Blue -5-5 Green 5 31 R G B (1 2 3)=( ) (1 2 3)=( ) (A) (B) (C) (D) (E) (F) 53

64 Fast ICA ( 18 2) ( 22) ( 23) ( 21) ( 24 29) ( 3 31) PCA ( 14 15) ( ) PCA ED 1/ ( 16 21)

65 3 ( ) (Layer) [1] Layer Single-Opponent (n = 13) Double-Opponent (n = 51) 4/13, 31% 19/51, 37% /13, % 1/51, 2% 1/13, 8% 9/51, 18% /13, % 3/51, 6% 1/13, 8% 2/51, 4% 4/13, 31% 6/51, 12% 3/13, 23% 11/51, 22% ( 3) ( ) 1 4 ( ) 3 7 [12] 55

66 4.2 1 [1] ( ) ( 3) 2 1 ( ) [2] 56

67 5. ( ) ( ) Fast ICA 57

68 58

69 [1] [2] 2., [3] Bevil R. Conway and Margaret S. Livingstone. Spatial and temporal properties of cone signals in alert macaque primary visual cortex. J. Neurosci., 26(42): , 26. [4] A. M. Derrington and P. Lennie. Spatial and temporal contrast sensitivities of neurones in lateral geniculate nucleus of macaque. J. Physiol, 357:219 24, [5] Karl R. Gegenfurtner and Daniel C. Kiper. Color vision. Annu. Rev. Neurosci., 26:181 26, 23. [6] Hubel D. H. and Wiesel T. N. Receptive fields, binocular interaction and functional architecture in the cat s visual cortex. J. Physiol, 195: , [7] Aapo Hyvarinen. Survey on independent component analysis. Neural Computing Surveys, 2:94 128, 1999a. [8] Aapo Hyvarinen. Fast and robust fixed-point algorithms for independent component analysis. IEEE Trans. on Neural Networks, 1(3): , 1999b. [9] L. Iwai and H. Kawasaki. Molecular development of the lateral geniculate nucleus in the absence of retinal waves during the time of retinal axon eye-specific segregation. Neuroscience, 159: , 29. [1] Elizabeth N. Johnson, Michael J. Hawken, and Robert Shapley. The orientation selectivity of color-responsive neurons in macaque v1. J. Neurosci., 28. [11] Avi Karni and Dov Sagi. Where practice makes perfect in texture discrimination: Evidence for primary visual cortex plasticity. Neurobiology, 88: ,

70 [12] Carole E. Landisman and Daniel Y. Color processing in macaque striate cortex: Relationships to ocular dominance, cytochrome oxidase, and orientation. J. Neurophysiol, 87: , 22. [13] D. Luenberger. Optimization by Vector Space Methods, Wiley, [14] Jone M. and Sibson R. What is projection pursuit? J. of the Royal Statistical Society, ser. A, 15:1 36, [15] Bruno A. Olshausen and David J. Field. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381:67 69, [16] Bruno A. Olshausen and David J. Field. Sparse coding with an overcomplete basis set: A strategy employed by v1? Vision Res., 37(23): , [17] Comon P. Independent component analysis - a new concept? Signal Processing, 36: , [18] D. R. Peeples and D. Y. Teller. Colour vision and brightness discrimination in two-month-old human infants. Science, 189: , [19] Shapley R and Lennie P. Spatial frequency analysis in the visual system. Rev. Neurosci., 8: , [2] E. Sernagor, S. Eglen, B. Harris, and R. Wong. Retinal development. 26. [21] Dharmesh R. Tailor, Leif H. Finkel, and Gershon Buchsbaum. Color-opponent receptive fields derived from independent component analysis of natural images. Vision Research, 4: , 2. 6

71 A. I(a, b) a b I(a, b) a b a b a i I(a, b) = a i ϕ i (a, b) (4) i ϕ i (a, b) i 192 n n Î(a, b) = a i ϕ i (a, b) (41) I(a, b) Î(a, b) [ E 1 = ] 2 n I(a, b) Î(a, b) = I(a, b) a i ϕ i (a, b) a,b a,b i i E 1 ϕ i (a, b) a i a i 2 (42) 61

72 a i 42 E 2 n E 2 = I(a, b) a i ϕ i (a, b) a,b i 2 + λ n i [ log 1 + ( ai ) 2 ] σ σ a i λ E 2 ϕ i (a, b) a i ϕ i (a, b) E 2 a i (43) E 2 = ϕ i (a, b)i(a, b) ϕ i (a, b)ϕ j (a, b)a j λ 1 + a i σ 2 a i a,b j a,b 1 + ( ) a 2 (44) i σ ϕ i (a, b) ϕ i (a, b) ϕ i (a, b) = η [ a ] i I(a, b) Î(a, b) (45) η < > a i 44 1 ϕ i (a, b) , B. Elizabeth

73 1 ( ) ( 11A) ( 11B) C. Olshausen , [15] Dharmesh , ICA [21] D. y k J(y k )( 28) min J(y k ), subject to K i (w) =, (i = 1,..., p) (46) 63

74 K i (w) = y k L(y k, λ 1,..., λ k ) = J(y k ) + k λ i H i (y k ) (47) λ 1,..., λ p (47) ( 46) L(y k, λ 1,..., λ k ) y k λ i L(y k, λ 1,..., λ k ) λ i K i (y k ) K i (y k ) = L(w, λ 1,..., λ k ) w J(w) w + k i=1 i=1 λ i H i (w) w = (48) E. ( ) J(w) (2.7 ) J(w) w [ ] T J(w) J(w ) = J(w) + (w w) + 1 w 2 (w w) T 2 J(w) (w w) +... (49) w 2 J(w) w J(w) w w = w 64

75 [ ] T J(w) J(w ) J(w) = w J(w) w 2 wt w (5) w 2 w 5 w (J(w ) J(w)) w = J(w) w J(w) w + w 2 [ ] = J(w) w = J(w) w 2 2 J(w) w w 2 ( ) 2 T J(w) w w J(w) w 2 w (51) 2 J(w) w 2 51 [ ] 2 1 J(w) J(w) w = w 2 w (52) J(w) w [ ] w 2 1 J(w) J(w) = w w 2 w (53) w 53 [13] 2 1 Fast ICA ICA 65

76 1 66

1 (L-cone, M-cone, S-cone) (Luminance, L-M, S-(LM)) CRT ( 2) (a) (b) (c) (d) RGB rgb XYZ LMS DKL (e) 2 RGB ( 3) G 255 B R 3 RGB XYZ xyy

1 (L-cone, M-cone, S-cone) (Luminance, L-M, S-(LM)) CRT ( 2) (a) (b) (c) (d) RGB rgb XYZ LMS DKL (e) 2 RGB ( 3) G 255 B R 3 RGB XYZ xyy DKL 2008 1 13 1) DKL (Luminance, L-M, S-(LM)) RGB DKL 2) Brainard 1996 LMS DKL 3) DKL 1 1.1 (L-cone, M-cone, S-cone) L-, M-cone parvocellular pathway S-cone koniocellular pathway DKL retinotectal pathway