Two eyes: ffiffiffi 2 p better than one? William A. Simpson a, * , Velitchko Manahilov b , Uma Shahani b a School of Psychology, University of Plymouth, Drake Circus, Plymouth PL4 8AA, UK b Department of Vision Sciences, Glasgow Caledonian University, Cowcaddens Road, Glasgow G4 0BA, UK article info Article history: Received 15 March 2008 Received in revised form 14 March 2009 Accepted 16 March 2009 Available online 29 April 2009 PsycINFO classification: 2323 Keywords: Stereo vision Binocular summation Signal detection theory abstract Classical data on the detection of simple patterns show that two eyes are more sensitive than one eye. The degree of binocular summation is important for inferences about the underlying combination mech- anism. In a signal detection theory framework, sensitivity is limited by internal noise. If noise is added centrally after binocular combination, binocular sensitivity is expected to be twice as good as monocular. If the noise is added peripherally at each eye prior to combination, binocular sensitivity will be ffiffiffi 2 p higher than monocular. In a large sample of observers (51), we measured contrast sensitivity for detection of gratings at several spatial frequencies using left, right, or both eyes. Estimates of binocular summation using both binocular summation ratios and Minkowski coefficients show a summation ratio with means in the range of 1.5–1.6. The 95% confidence interval overlaps with the value of ffiffiffi 2 p predicted by the peripheral noise model and does not overlap with the value of 2 predicted by the central noise model. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction Having two eyes confers many advantages. Binocular stereopsis is the most obvious benefit of having two eyes. But another benefit is that having two eyes allows the viewer to detect faint patterns better. Exactly how such binocular summation in the detection of luminance patterns is performed in the brain is unknown. In an ef- fort to find the mechanism, many studies have been done. Detection of a faint pattern is a problem of detecting a signal in noise (Green & Swets, 1988). Besides noise contained in the stim- ulus delivered to the observer (either deliberately generated or due to imperfect electronics, for example), there is also noise inside the observer’s visual system (Burgess, Wagner, Jennings, & Barlow, 1981; Legge, Kersten, & Burgess, 1987; Pelli, 1990; Pelli & Farrell, 1999; Simpson, Falkenberg, & Manahilov, 2003). In the case of bin- ocular detection of signals, there are two possible ways in which noise might be introduced in the visual system. In the central noise model, the outputs of left and right eyes are combined at some cen- tral site (binocular simple cells in V1 for example), and noise is introduced at that stage (Blake, Sloane, & Fox, 1981, p. 274). In the peripheral noise model, noise is introduced peripherally at each eye prior to binocular combination. These models make dif- ferent predictions about binocular summation. The models predict the performance of an ideal observer who knows the signal exactly, including such things as its spatial frequency, phase, and whether it is binocularly or monocularly presented to the left or right eye. First let us consider monocular detection, and then we can com- pare binocular detection to monocular. A stimulus composed of a contrast signal c(x, y, t) embedded in Gaussian noise with variance r 2 is delivered to the observer. The ideal observer cross-correlates the noisy stimulus with a stored representation of the signal. Cross-correlation means that the observer multiplies the stimulus point-by-point with the signal and sums. Because of the cross-cor- relation operation, the observer’s performance depends on the sig- nal energy; if the stimulus matches the signal, the product at each point amounts to squaring, and the sum gives the energy. The en- ergy E is proportional to RRR c 2 ðx; y; tÞdxdydt. The detectability d 0 of a signal having energy E and having added noise with variance r 2 is d 0 ¼ ffiffiffiffiffi E r 2 r (Whalen, 1971, pp. 159–163). In many experiments there is little or no added noise in the stimulus. Therefore, the assumption is made that the noise is added internally. At threshold d 0 = 1. By squaring both sides and solving for the threshold energy, the result is r 2 . En- ergy is proportional to the sum of contrast squared, so the monoc- ular contrast threshold is r. Now suppose that two eyes view the same stimulus, and that the decision is based on the central combination of the outputs of the two eyes. Then the contrast signal is 2c(x,y,t) and so the energy is 4 RRR c 2 ðx; y; tÞdxdydt, four times the monocular signal 0001-6918/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.actpsy.2009.03.006 * Corresponding author. Tel.: +44 (0)1752 584 856; fax: +44 (0)1752 233 362. E-mail address: william.simpson@plymouth.ac.uk (W.A. Simpson). Acta Psychologica 131 (2009) 93–98 Contents lists available at ScienceDirect Acta Psychologica journal homepage: www.elsevier.com/locate/actpsy