Symbols in Order of Appearance1 Historical Introduction2 Template Estimation for Linear Classification of Two Signals in Additive White Gaussian Noise3 Measuring the Internal Noise4 Discussion and ConclusionsAppendixAcknowledgmentsReferencesJournal of Vision (2002) 2, 121-131 http://journalofvision.org/2/1/8/ 121 Classification image weights and internal noise level estimation Albert J. Ahumada, Jr. NASA Ames Research Center, Moffett Field, CA, USA For the linear discrimination of two stimuli in white Gaussian noise in the presence of internal noise, a method is described for estimating linear classification weights from the sum of noise images segregated by stimulus and response. The recommended method for combining the two response images for the same stimulus is to difference the average images. Weights are derived for combining images over stimuli and observers. Methods for estimating the level of internal noise are described with emphasis on the case of repeated presentations of the same noise sample. Simple tests for particular hypotheses about the weights are shown based on observer agreement with a noiseless version of the hypothesis. Keywords: discrimination, detection, vision, noise Symbols in Order of Appearance β 0, H random shifted bias of the human observer model m the number of image components γ 2 variance of β 0, H s s s = 0, 1; 1 by m signal vectors α2 proportion of external noise in the classification variable, 1/(1+ γ 2) p s s = 0, 1; probability of signal s s n 1 by m noise vector with components n(i), i = 1, m d H' sensitivity of the human observer model β H performance bias of the human observer model g 1 by m trial stimulus vector with components g(i), i = 1, m φ(·) standard normal distribution density function E[·] averaging or expectation operator Var[·] variance computing operator n s, R a random noise n conditional on signal s s and detection response R σ2 variance of n(i) w 1 by m classification vector with components w(i), i = 1, m a s, R the average of N s, R noises n s, R v s, R the expected value of n s, R when m = 1. β bias of linear classifier x, y, z standard normal variables R the observer's response, 0 or 1 U an orthonormal m by m transformation T matrix transpose operator I the m by m identity transformation ||·|| vector length, ||w|| = (w w T) 1/2 z i standard normal variables Pr{} probability of enclosed event N s, R the number of presentations of stimulus s that led to response R p s, R probability of response R given signal s s , Pr{R|s s} N s the number of presentations of stimulus s, N s = N s, 0 + N s, 1 Φ(·) cumulative standard normal distribution function e the decision contribution from the external noise, replacing w n T d 0' sensitivity of linear classifier β 0 shifted bias of linear classifier, β – w s 0 T M R R = 0, 1; the event that an internal-noise-free model made response R Z(·) functional inverse of the cumulative standard normal distribution function, Φ -1(·) p M, s, 0 the probability of event M 0 given that the signal was s s w I classification vector w of the ideal observer β M, s the signal-dependent, internal-noise-free model criterion, β 0 if s 0 , or β 0 − d 0' if s 1 d I' sensitivity of the ideal observer ρ 2 the sampling efficiency of w, ρ = w w IT DOI 10:1167/2.1.8 Received November 20, 2001; published March 22, 2002 ISSN 1534-7362 © 2002 ARVOAhumada 122 1 Historical Introduction In 1965, a frustrated graduate student in physiological psychology was looking for a thesis topic in the auditory research laboratory of E. C. Carterette and M. P. Friedman, the editors to be of the Handbook of Perception. They recommended that he tape record the stimulus of the traditional tone-in-noise yes-no detection experiment and analyze the sounds in the four different types of trials to determine whether correlates could be found in the stimuli relating to the observer responses. The noise masker was continuous wide-band noise, and marker tones were recorded on a second track to keep track of the signals presented. The tapes were digitized and analyzed, but the signal-to-noise level at threshold was so low that no trace of the signals could be found in the digitized records. To ensure earning a degree in the foreseeable future, the student made several changes in the experiment. To improve the signal-to-noise ratio on the tape, the noise bandwidth was narrowed, and the noise was turned on only during the short interval when the signal might be present. To reduce the effects of observer noise, the tape was repeatedly presented to the observer to get average ratings of signal presence. To minimize degrees of freedom in the stimulus measurement, the stimulus was reduced to the energy passed by a filter tuned to the signal tone frequency. This combination of changes allowed the student to find that on signal trials, very narrow filter outputs correlated best with observer ratings, whereas on noise trials, wider filter outputs correlated best, contradicting the prediction of single linear filter models for auditory tone detection (Ahumada, 1967). To gain better control of the masking noise and avoid the limitations of tape recording, Ahumada and Lovell (1971) used computer-generated tones and noises defined by their Fourier component amplitudes and reported linear regressions on the component energies with average observer ratings. These results were essentially auditory classification images that again demonstrated results contrary to simple linear filter theory: frequency components were weighted differently on signal trials from noise-only trials and negative weights were frequently observed. The results of both experiments seemed to be consistent with models with multiple linear channels that were being nonlinearly combined. Ahumada, Marken, and Sandusky (1975) extended the experiment to the combined time and frequency domains with similar results. Our first visual classification images (Ahumada, 1996) were done to see whether the method we had used in audition could be used to elucidate the features used by observers to accomplish a vernier acuity task. Figure 1 shows a raw classification image and the same image smoothed and quantized so only weights that are significantly different from zero are colored differently from the gray background. The ideal observer would have only weights on the right side, the side of the line that was either