Abstract

A method for testing the linearity of cone combination of chromatic detection mechanisms is applied to S-cone detection. This approach uses the concept of mechanism noise, the noise as seen by a postreceptoral neural mechanism, to represent the effects of superposing chromatic noise components in elevating thresholds and leads to a parameter-free prediction for a linear mechanism. The method also provides a test for the presence of multiple linear detectors and off-axis looking. No evidence for multiple linear mechanisms was found when using either S-cone increment or decrement tests. The results for both S-cone test polarities demonstrate that these mechanisms combine their cone inputs nonlinearly.

© 2007 Optical Society of America

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2006 (2)

T. Hansen and K. R. Gegenfurtner, "Higher level chromatic mechanisms for image segmentation," J. Vision 6, 239-259 (2006).
[CrossRef]

M. Sakurai and K. T. Mullen, "Cone weights for the two cone-opponent systems in peripheral vision and asymmetries of cone contrast sensitivity," Vision Res. 46, 4346-4354 (2006).
[CrossRef] [PubMed]

2005 (3)

G. D. Horwitz, E. J. Chichilnisky, and T. D. Albright, "Blue-yellow signals are enhanced by spatiotemporal luminance contrast in macaque V1," J. Neurophysiol. 93, 2263-2278 (2005).
[CrossRef]

S. M. Wuerger, P. Atkinson, and S. Cropper, "The cone inputs to the unique-hue mechanisms," Vision Res. 45, 3210-3223 (2005).
[CrossRef] [PubMed]

S. G. Solomon and P. Lennie, "Chromatic gain controls in visual cortical neurons," J. Neurosci. 25, 4779-4792 (2005).
[CrossRef] [PubMed]

2004 (2)

E. N. Johnson, M. J. Hawken, and R. Shapley, "Cone inputs in macaque primary visual cortex," J. Neurophysiol. 91, 2501-2514 (2004).
[CrossRef] [PubMed]

D. T. Lindsey and A. M. Brown, "Masking of grating detection in the isoluminant plane of DKL color space," Visual Neurosci. 21, 269-273 (2004).
[CrossRef]

2003 (5)

J. R. Newton and R. T. Eskew, Jr., "Chromatic detection and discrimination in the periphery: a postreceptoral loss of color sensitivity," Visual Neurosci. 20, 511-521 (2003).
[CrossRef]

D. M. Dacey and O. S. Packer, "Colour coding in the primate retina: diverse cell types and cone-specific circuitry," Curr. Opin. Neurobiol. 13, 421-427 (2003).
[CrossRef] [PubMed]

D. M. Dacey, B. B. Peterson, F. R. Robinson, and P. D. Gamlin, "Fireworks in the primate retina: in vitro photodynamics reveals diverse LGN-projecting ganglion cell types," Neuron 37, 15-27 (2003).
[CrossRef] [PubMed]

K. Klug, S. Herr, I. T. Ngo, P. Sterling, and S. Schein, "Macaque retina contains an S-cone OFF midget pathway," J. Neurosci. 23, 9881-9887 (2003).
[PubMed]

S. Chatterjee and E. M. Callaway, "Parallel colour-opponent pathways to primary visual cortex," Nature 426, 668-671 (2003).
[CrossRef] [PubMed]

2002 (1)

C. E. Landisman and D. Y. Ts'o, "Color processing in macaque striate cortex: electrophysiological properties," J. Neurophysiol. 87, 3138-3151 (2002).
[PubMed]

2001 (3)

B. R. Conway, "Spatial structure of cone inputs to color cells in alert macaque primary visual cortex (V-1)," J. Neurosci. 21, 2768-2783 (2001).
[PubMed]

R. T. Eskew, Jr., J. R. Newton, and F. Giulianini, "Chromatic detection and discrimination analyzed by a Bayesian classifier," Vision Res. 41, 893-909 (2001).
[CrossRef] [PubMed]

M. J. Sankeralli and K. T. Mullen, "Bipolar or rectified chromatic detection mechanisms?," Visual Neurosci. 18, 127-135 (2001).
[CrossRef]

2000 (1)

J. S. McLellan and R. T. Eskew, Jr., "ON and OFF S-cone pathways have different long-wave cone inputs," Vision Res. 40, 2449-2465 (2000).
[CrossRef] [PubMed]

1999 (3)

E. J. Chichilnisky and D. A. Baylor, "Receptive-field microstructure of blue-yellow ganglion cells in primate retina," Nat. Neurosci. 2, 889-893 (1999).
[CrossRef] [PubMed]

C. F. Stromeyer 3rd, R. Thabet, A. Chaparro, and R. E. Kronauer, "Spatial masking does not reveal mechanisms selective to combined luminance and red-green color," Vision Res. 39, 2099-2112 (1999).
[CrossRef] [PubMed]

K. Shinomori, L. Spillmann, and J. S. Werner, "S-cone signals to temporal OFF-channels: asymmetrical connections to postreceptoral chromatic mechanisms," Vision Res. 39, 39-49 (1999).
[CrossRef] [PubMed]

1998 (4)

N. P. Cottaris and R. L. De Valois, "Temporal dynamics of chromatic tuning in macaque primary visual cortex," Nature 395, 896-900 (1998).
[CrossRef] [PubMed]

M. D'Zmura and K. Knoblauch, "Spectral bandwidths for the detection of color," Vision Res. 38, 3117-3128 (1998).
[CrossRef]

F. Giulianini and R. T. Eskew, Jr., "Chromatic masking in the (ΔL/L,ΔM/M) plane of cone-contrast space reveals only two detection mechanisms," Vision Res. 38, 3913-3926 (1998).
[CrossRef]

C. F. Stromeyer 3rd, A. Chaparro, C. Rodriguez, D. Chen, E. Hu, and R. E. Kronauer, "Short-wave cone signal in the red-green detection mechanism," Vision Res. 38, 813-826 (1998).
[CrossRef] [PubMed]

1997 (1)

1996 (1)

1994 (1)

D. M. Dacey and B. B. Lee, "The 'blue-on' opponent pathway in primate retina originates from a distinct bistratified ganglion cell type," Nature 367, 731-735 (1994).
[CrossRef] [PubMed]

1993 (2)

A. Chaparro, C. F. Stromeyer 3rd, E. P. Huang, R. E. Kronauer, and R. T. Eskew, Jr., "Colour is what the eye sees best," Nature 361, 348-350 (1993).
[CrossRef] [PubMed]

G. R. Cole, T. Hine, and W. McIlhagga, "Detection mechanisms in L-, M-, and S-cone contrast space," J. Opt. Soc. Am. A 10, 38-51 (1993).
[CrossRef] [PubMed]

1992 (3)

K. R. Gegenfurtner and D. C. Kiper, "Contrast detection in luminance and chromatic noise," J. Opt. Soc. Am. A 9, 1880-1888 (1992).
[CrossRef] [PubMed]

Q. Zaidi, A. Shapiro, and D. Hood, "The effect of adaptation on the differential sensitivity of the S-cone color system," Vision Res. 32, 1297-1318 (1992).
[CrossRef] [PubMed]

C. F. Stromeyer 3rd, J. Lee, and R. T. Eskew, Jr., "Peripheral chromatic sensitivity for flashes: a post-receptoral red-green asymmetry," Vision Res. 32, 1865-1873 (1992).
[CrossRef] [PubMed]

1991 (2)

D. G. Pelli and L. Zhang, "Accurate control of contrast on microcomputer displays," Vision Res. 31, 1337-1350 (1991).
[CrossRef] [PubMed]

A. Stockman, D. I. A. MacLeod, and D. D. DePriest, "The temporal properties of the human short-wave photoreceptors and their associated pathways," Vision Res. 31, 189-208 (1991).
[CrossRef] [PubMed]

1990 (1)

1988 (1)

C. F. Stromeyer 3rd and J. Lee, "Adaptational effects of short wave cone signals on red-green chromatic detection," Vision Res. 28, 931-940 (1988).
[CrossRef] [PubMed]

1987 (1)

1985 (1)

C. F. Stromeyer 3rd, G. R. Cole, and R. E. Kronauer, "Second-site adaptation in the red-green chromatic pathways," Vision Res. 25, 219-237 (1985).
[CrossRef] [PubMed]

1984 (2)

A. M. Derrington, J. Krauskopf, and P. Lennie, "Chromatic mechanisms in lateral geniculate nucleus of macaque," J. Physiol. (London) 357, 241-265 (1984).

C. H. Elzinga and C. M. M. de Weert, "Nonlinear codes for the yellow/blue mechanism," Vision Res. 24, 911-922 (1984).
[CrossRef] [PubMed]

1983 (4)

C. Noorlander and J. J. Koenderink, "Spatial and temporal discrimination ellipsoids in color space," J. Opt. Soc. Am. 73, 1533-1543 (1983).
[CrossRef] [PubMed]

J. J. Wisowaty, "An action spectrum for the production of transient tritanopia," Vision Res. 23, 769-774 (1983).
[CrossRef] [PubMed]

R. M. Boynton, A. L. Nagy, and C. X. Olson, "A flaw in equations for predicting chromatic differences," Color Res. Appl. 8, 69-74 (1983).
[CrossRef]

A. Burgess and H. B. Barlow, "The precision of numerosity discrimination in arrays of random dots," Vision Res. 23, 811-820 (1983).
[CrossRef] [PubMed]

1982 (1)

J. Krauskopf, D. R. Williams, and D. W. Heeley, "Cardinal directions of color space," Vision Res. 22, 1123-1131 (1982).
[CrossRef] [PubMed]

1981 (1)

1980 (1)

P. G. Polden and J. D. Mollon, "Reversed effect of adapting stimuli on visual sensitivity," Proc. R. Soc. London, Ser. B 210, 235-272 (1980).
[CrossRef]

1979 (3)

A. B. Watson, "Probability summation over time," Vision Res. 19, 515-522 (1979).
[CrossRef] [PubMed]

J. S. Werner and B. R. Wooten, "Opponent chromatic mechanisms: relation to photopigments and hue naming," J. Opt. Soc. Am. 69, 422-434 (1979).
[CrossRef] [PubMed]

E. N. Pugh, Jr. and J. D. Mollon, "A theory of the π1 and π3 color mechanisms of Stiles," Vision Res. 19, 293-312 (1979).
[CrossRef] [PubMed]

1975 (1)

J. Larimer, D. H. Krantz, and C. M. Cicerone, "Opponent process additivity. II. Yellow/blue equilibria and nonlinear models," Vision Res. 15, 723-731 (1975).
[CrossRef] [PubMed]

Albright, T. D.

G. D. Horwitz, E. J. Chichilnisky, and T. D. Albright, "Blue-yellow signals are enhanced by spatiotemporal luminance contrast in macaque V1," J. Neurophysiol. 93, 2263-2278 (2005).
[CrossRef]

Atkinson, P.

S. M. Wuerger, P. Atkinson, and S. Cropper, "The cone inputs to the unique-hue mechanisms," Vision Res. 45, 3210-3223 (2005).
[CrossRef] [PubMed]

Barlow, H. B.

A. Burgess and H. B. Barlow, "The precision of numerosity discrimination in arrays of random dots," Vision Res. 23, 811-820 (1983).
[CrossRef] [PubMed]

Baylor, D. A.

E. J. Chichilnisky and D. A. Baylor, "Receptive-field microstructure of blue-yellow ganglion cells in primate retina," Nat. Neurosci. 2, 889-893 (1999).
[CrossRef] [PubMed]

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, 1969).

Boynton, R. M.

R. M. Boynton, A. L. Nagy, and C. X. Olson, "A flaw in equations for predicting chromatic differences," Color Res. Appl. 8, 69-74 (1983).
[CrossRef]

Brainard, D. H.

D. H. Brainard, "Cone contrast and opponent modulation color spaces," in Human Color Vision, 2nd ed., P.K.Kaiser and R.M.Boynton, eds. (Optical Society of America,1996).

Brown, A. M.

D. T. Lindsey and A. M. Brown, "Masking of grating detection in the isoluminant plane of DKL color space," Visual Neurosci. 21, 269-273 (2004).
[CrossRef]

Burgess, A.

A. Burgess and H. B. Barlow, "The precision of numerosity discrimination in arrays of random dots," Vision Res. 23, 811-820 (1983).
[CrossRef] [PubMed]

Burgess, A. E.

Callaway, E. M.

S. Chatterjee and E. M. Callaway, "Parallel colour-opponent pathways to primary visual cortex," Nature 426, 668-671 (2003).
[CrossRef] [PubMed]

Chaparro, A.

C. F. Stromeyer 3rd, R. Thabet, A. Chaparro, and R. E. Kronauer, "Spatial masking does not reveal mechanisms selective to combined luminance and red-green color," Vision Res. 39, 2099-2112 (1999).
[CrossRef] [PubMed]

C. F. Stromeyer 3rd, A. Chaparro, C. Rodriguez, D. Chen, E. Hu, and R. E. Kronauer, "Short-wave cone signal in the red-green detection mechanism," Vision Res. 38, 813-826 (1998).
[CrossRef] [PubMed]

A. Chaparro, C. F. Stromeyer 3rd, E. P. Huang, R. E. Kronauer, and R. T. Eskew, Jr., "Colour is what the eye sees best," Nature 361, 348-350 (1993).
[CrossRef] [PubMed]

Chatterjee, S.

S. Chatterjee and E. M. Callaway, "Parallel colour-opponent pathways to primary visual cortex," Nature 426, 668-671 (2003).
[CrossRef] [PubMed]

Chen, D.

C. F. Stromeyer 3rd, A. Chaparro, C. Rodriguez, D. Chen, E. Hu, and R. E. Kronauer, "Short-wave cone signal in the red-green detection mechanism," Vision Res. 38, 813-826 (1998).
[CrossRef] [PubMed]

Chichilnisky, E. J.

G. D. Horwitz, E. J. Chichilnisky, and T. D. Albright, "Blue-yellow signals are enhanced by spatiotemporal luminance contrast in macaque V1," J. Neurophysiol. 93, 2263-2278 (2005).
[CrossRef]

E. J. Chichilnisky and D. A. Baylor, "Receptive-field microstructure of blue-yellow ganglion cells in primate retina," Nat. Neurosci. 2, 889-893 (1999).
[CrossRef] [PubMed]

Cicerone, C. M.

J. Larimer, D. H. Krantz, and C. M. Cicerone, "Opponent process additivity. II. Yellow/blue equilibria and nonlinear models," Vision Res. 15, 723-731 (1975).
[CrossRef] [PubMed]

Cole, G. R.

Conway, B. R.

B. R. Conway, "Spatial structure of cone inputs to color cells in alert macaque primary visual cortex (V-1)," J. Neurosci. 21, 2768-2783 (2001).
[PubMed]

Cottaris, N. P.

N. P. Cottaris and R. L. De Valois, "Temporal dynamics of chromatic tuning in macaque primary visual cortex," Nature 395, 896-900 (1998).
[CrossRef] [PubMed]

Cropper, S.

S. M. Wuerger, P. Atkinson, and S. Cropper, "The cone inputs to the unique-hue mechanisms," Vision Res. 45, 3210-3223 (2005).
[CrossRef] [PubMed]

Dacey, D. M.

D. M. Dacey, B. B. Peterson, F. R. Robinson, and P. D. Gamlin, "Fireworks in the primate retina: in vitro photodynamics reveals diverse LGN-projecting ganglion cell types," Neuron 37, 15-27 (2003).
[CrossRef] [PubMed]

D. M. Dacey and O. S. Packer, "Colour coding in the primate retina: diverse cell types and cone-specific circuitry," Curr. Opin. Neurobiol. 13, 421-427 (2003).
[CrossRef] [PubMed]

D. M. Dacey and B. B. Lee, "The 'blue-on' opponent pathway in primate retina originates from a distinct bistratified ganglion cell type," Nature 367, 731-735 (1994).
[CrossRef] [PubMed]

De Valois, R. L.

N. P. Cottaris and R. L. De Valois, "Temporal dynamics of chromatic tuning in macaque primary visual cortex," Nature 395, 896-900 (1998).
[CrossRef] [PubMed]

de Weert, C. M. M.

C. H. Elzinga and C. M. M. de Weert, "Nonlinear codes for the yellow/blue mechanism," Vision Res. 24, 911-922 (1984).
[CrossRef] [PubMed]

DePriest, D. D.

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J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

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Neuron (1)

D. M. Dacey, B. B. Peterson, F. R. Robinson, and P. D. Gamlin, "Fireworks in the primate retina: in vitro photodynamics reveals diverse LGN-projecting ganglion cell types," Neuron 37, 15-27 (2003).
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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

J. Krauskopf, D. R. Williams, and D. W. Heeley, "Cardinal directions of color space," Vision Res. 22, 1123-1131 (1982).
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J. S. McLellan and R. T. Eskew, Jr., "ON and OFF S-cone pathways have different long-wave cone inputs," Vision Res. 40, 2449-2465 (2000).
[CrossRef] [PubMed]

J. J. Wisowaty, "An action spectrum for the production of transient tritanopia," Vision Res. 23, 769-774 (1983).
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[CrossRef] [PubMed]

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[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

(a) Test stimulus vector v in the first quadrant (QI) of the ( l , m ) plane of cone contrast space, with contrast κ; (b) compound, L M noise vector in the same plane. The two dots represent the chromaticities of the binary noise. The noise may be decomposed into its L- and M-cone contrast components as shown. (c) Model observer used in the study.

Fig. 2
Fig. 2

Noises used in the main study. The inset in panel (b) represents the appearance of the noise rings. The three panels show the ( l , m ) , ( l , s ) , and ( m , s ) planes. The arrows show the compound noises; the cone components of each of these compounds were also used (i.e., three times as many noise stimuli were used than shown in each panel). In subsequent figures these bipolar noises are represented by their polar angles in quadrants I or II in the plane, with 0 ° being the horizontal rightward axis, and other angles are given in a counterclockwise direction. In the ( l , s ) and ( m , s ) planes, the chosen noise angles were close to the S-cone axis because of the eightfold to tenfold lower S-cone contrast sensitivity [9, 11]. In a given plane, the naive observers (JDP and WL) were run with somewhat fewer noise angles than was FG.

Fig. 3
Fig. 3

Test of linearity for a G-detected equiluminant green stimulus in the ( l , m ) plane for three observers. The filled symbols show the elevations produced by compound noises at the indicated angles. The open circles and triangles are the predictions made by linear opponent and nonopponent models, respectively. Three thresholds are represented at each angle (from two-cone-component and one-compound noises). Standard error bars are plotted wherever they are larger than the symbols. For panel (c) (RTE), a different apparatus was used (note the different vertical scale). A Sony monitor, controlled by a video card with 10-bit DACs, was freely viewed, monocularly, with a mean luminance of 50 cd m 2 . Because no background fields were added, the noise powers were about four times as high as those for the other observers.

Fig. 4
Fig. 4

Elevations for S-cone increments for the two main observers. The left column shows data from observer FG; the right column shows observer JDP. Triangles and circles are as in Fig. 3. The rows represent compound noises in the ( l , m ) , ( l , s ) , and ( m , s ) planes at the angles indicated. The open squares show the predictions of nonlinear model 1 (see Discussion).

Fig. 5
Fig. 5

Elevations for S-cone increments produced by noises in the ( l , m ) plane for observer WL. Triangles and circles are as in Fig. 4.

Fig. 6
Fig. 6

Elevations for decrement S-cone tests and predictions of the linear models for observers FG (left column) and JDP (right column). Rows represent compound noises in the ( l , m ) , ( l , s ) , and ( m , s ) planes. Symbols are as in Fig. 4.

Fig. 7
Fig. 7

Simulations of multiple linear mechanisms and off-axis looking. The solid points represent simulated threshold elevations produced by a compound noise at the indicated angle. The two curves show the opponent and nonopponent linear predictions, which cross at 90 ° (see discussion of Fig. 3). The area between the two linear predictions, where all the actual data in the main experiment fall, is colored gray. In each panel, the legend indicates the plane of cone space used, the set of mechanism directions in that plane in curly brackets, and the percentage of simulated compound elevations that lie between the two linear predictions. In panels (a) and (b), noises of constant power are simulated for illustration. In the other panels, noises of realistic powers were used.

Fig. 8
Fig. 8

Compound ( x y ) noise and its x and y components in one quadrant of cone space. Noise vectors are represented by heavy lines with open circles at the ends. The open-headed arrow represents a mechanism vector. Lengths of vectors are shown in italic font, aligned with the vector. The projection of the x noise onto the mechanism vector, with length B f Cos ( β ) Cos ( ϕ ) , is illustrated [cf. Eq. (B.2)]; the projections of the y and x y noises onto the mechanism vector are not shown.

Tables (2)

Tables Icon

Table 1 Nonlinear Models Tested

Tables Icon

Table 2 Parameter Estimates for Model 1: a s + sign ( b l + c m ) b l + c m α

Equations (56)

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κ = ( l , m , s ) = l 2 + m 2 + s 2 .
M e c h = f ( l , m , s ) = f ( v ) .
M e c h ( v ) = w L l + w M m + w S s = ( w L w M w S ) ( l m s ) = f v .
E = d 2 e ( N + N e q ) = p N + E 0 ,
E [ M e c h ( v ) ] = E 0 + p E [ M e c h ( N ) ] .
M e c h ( v ) = f v = ( w L l + w M m + w S s ) ,
( τ L M τ 0 ) 2 = 1 + [ ( τ L τ 0 ) 2 1 ± ( τ M τ 0 ) 2 1 ] 2 ,
( w L w M ) = ± ( n M n L ) 2 τ L 2 τ 0 2 τ M 2 τ 0 2 .
( τ ( L M S ) τ 0 ) 2 = 1 + [ ( τ L τ 0 ) 2 1 ± ( τ M τ 0 ) 2 1 ± ( τ S τ 0 ) 2 1 ] 2 .
( τ n o i s e τ 0 ) = 1 + E [ M e c h ( n o i s e ) ] τ 0 2 .
v = ( l , m , s ) T q ( x , y , t ) ,
v = κ u q ( x , y , t ) ,
E [ M e c h ( v ) ] = f [ l ( x , y , t ) , m ( x , y , t ) , s ( x , y , t ) ] 2 d x d y d t = ( f v ) 2 q [ l ( x , y , t ) , m ( x , y , t ) , s ( x , y , t ) ] 2 d x d y d t = κ 2 ( f u ) 2 Q ,
V = κ ( 1 4 ( f n p ) 2 + 1 4 ( f n n ) 2 ) = κ 2 ( f n ) 2 ,
E [ M e c h ( n o i s e ) ] = κ 2 ( f n ) 2 Q n ,
E [ M e c h ( v ) ] = ( f v ) 2 Q = E 0 .
( f v ) 2 Q = E L ,
( f v ) 2 Q = E M ,
( f v ) 2 Q = E L M .
h τ L 2 = E L ,
h τ M 2 = E M ,
h τ L M 2 = E L M .
E [ M e c h ( L noise ) ] = ( f ± n L ) 2 Q n = ( f n L ) 2 Q n .
E [ M e c h ( M noise ) ] = ( f ± n M ) 2 Q n = ( f n M ) 2 Q n .
E [ M e c h ( L M noise ) ] = [ f ( ± n L , ± n M , 0 ) T ] 2 Q n = [ f ± ( n L , n M , 0 ) T ] 2 Q n = [ f ( n L , n M , 0 ) T ] 2 Q n .
E [ M e c h ( L M noise ) ] = [ f ( ± n L , n M , 0 ) T ] 2 Q n = [ f ± ( n L , n M , 0 ) T ] 2 Q n = [ f ( n L , n M , 0 ) T ] 2 Q n .
E [ M e c h ( v ) ] = E 0 + p E [ M e c h ( L noise ) ] = E 0 + p Q n [ f ( n L , 0 , 0 ) T ] 2 = E L ,
E [ M e c h ( v ) ] = E 0 + p E [ M e c h ( M noise ) ] = E 0 + p Q n [ f ( 0 , n M , 0 ) T ] 2 = E M .
E [ M e c h ( v ) ] = E 0 + p E [ M e c h ( L M noise ) ] = E 0 + p Q n [ f ( n L , n M , 0 ) T ] 2 = E ( L M ) .
E [ M e c h ( v ) ] = E 0 + p E [ M e c h ( L M noise ) ] = E 0 + p Q n [ f ( n L , n M , 0 ) T ] 2 = E ( L M ) .
E L E 0 = h ( τ L 2 τ 0 2 ) = p Q n [ f ( n L , 0 , 0 ) T ] 2
E M E 0 = h ( τ M 2 τ 0 2 ) = p Q n [ f ( 0 , n M , 0 ) T ] 2 ,
E L M E 0 = h ( τ L M 2 τ 0 2 ) = p Q n [ f ( n L , n M , 0 ) T ] 2 ,
E L M E 0 = h ( τ L M 2 τ 0 2 ) = p Q n [ f ( n L , n M , 0 ) T ] 2 .
τ L 2 τ 0 2 τ 0 2 = p Q n [ f ( n L , 0 , 0 ) T ] 2 h τ 0 2 ,
τ M 2 τ 0 2 τ 0 2 = p Q n [ f ( 0 , n M , 0 ) T ] 2 h τ 0 2 ,
τ L M 2 τ 0 2 τ 0 2 = p Q n [ f ( n L , n M , 0 ) T ] 2 h τ 0 2 ,
τ L M 2 τ 0 2 τ 0 2 = p Q n [ f ( n L , n M , 0 ) T ] 2 h τ 0 2 .
τ L M 2 τ 0 2 τ 0 2 = p Q n [ f ( n L , n M , 0 ) T ] 2 h τ 0 2 = p Q n [ f ( n L , 0 , 0 ) T ± f ( 0 , n M , 0 ) T ] 2 h τ 0 2 = [ p Q n h f ( n L , 0 , 0 ) T τ 0 + ang ( L M ) p Q n h f ( 0 , n M , 0 ) T τ 0 ] 2 ,
ang ( L M ) = { + 1 , L M in quadrants I III 1 , L M in quadrants II IV .
p Q n h f ( n L , 0 , 0 ) T τ 0 = τ L 2 τ 0 2 τ 0 2 .
p Q n h f ( 0 , n M , 0 ) T τ 0 = τ M 2 τ 0 2 τ 0 2 .
p Q n h f ( n L , 0 , 0 ) T τ 0 = sgn ( w L ) τ L 2 τ 0 2 τ 0 2 ,
p Q n h f ( 0 , n M , 0 ) T τ 0 = sgn ( w L ) sgn ( w L w M ) τ M 2 τ 0 2 τ 0 2 ,
sgn ( x ) = { + 1 x 0 1 x < 0 .
τ L M 2 τ 0 2 τ 0 2 = [ sgn ( w L ) τ L 2 τ 0 2 τ 0 2 + sgn ( w L ) sgn ( w L w M ) ang ( L M ) τ M 2 τ 0 2 τ 0 2 ] 2 = [ τ L 2 τ 0 2 τ 0 2 + sgn ( w L w M ) ang ( L M ) τ M 2 τ 0 2 τ 0 2 ] 2 ,
τ L 2 τ 0 2 τ M 2 τ 0 2 = [ f ( n L , 0 , 0 ) T f ( 0 , n m , 0 ) T ] 2 = [ w L n L w M n M ] 2
( w L w M ) = ± ( n M n L ) 2 τ L 2 τ 0 2 τ M 2 τ 0 2 .
h w l M e c h = ρ ( w L l + w M m + w S s ) = ρ ( ( w L w M w S ) ( l m s ) ) ,
ρ ( x ) = { x x > 0 0 x 0 .
V = { ( f n ) 2 2 , f n > 0 0 , f n 0 .
E [ M e c h ( x y ) ] = B 2 f 2 Cos 2 ( ϕ β ) Q n .
E [ M e c h ( x ) ] = B 2 f 2 Cos 2 ( β ) Cos 2 ( ϕ ) Q n ,
E [ M e c h ( y ) ] = B 2 f 2 Sin 2 ( β ) Sin 2 ( ϕ ) Q n .
E [ M e c h ( s i g n a l ) ] = κ 2 f 2 Sin 2 ( ϕ ) Q .
E [ M e c h ( s i g n a l ) ] E [ M e c h ( y ) ] = κ 2 f 2 Sin 2 ( ϕ ) Q B 2 f 2 Sin 2 ( β ) Sin 2 ( ϕ ) Q n = κ 2 Q B 2 Sin 2 ( β ) Q n ,

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