Computational VisionU. Minn. Psy 5036Daniel KerstenLecture 5GoalsLast time‡Ideal Observer Analysis: Essential ideaIdeal observerModel the data (image) generation processDefine the inference taskDetermine optimal performanceCompare human performance to the idealIdeal normalizes for information availableExplain discrepancies in terms of:functional adaptationmechanismPsychophysical tasks & techniques (from the previous lecture)The Receiver Operating Characteristic (ROC)Although we can't directly measure the internal distributions of a human observer's decision variable, we've seen that we can measure hit and false alarm rates, and thus d'. But one can do more, and use ROC measurements to see if an observer's decisions are consistent with Gaussian distribu-tions with equal variance. If the criterion is varied, we can obtain a set of n data point. To get meaningful and equal d's for each pair of hit and false alarm rates assumes that the underlying relative separation of the signal and noise distributions remain unchanged and that the distributions are Gaussian, with equal standard deviation. We might know this is true (or true to a good approximation) for the ideal, but we have no guarantee for the human observer. Is there a way to check? Suppose the signal and noise distributions look like:If we plot the hit rate vs. false alarm rate data on a graph as the criterion xc varies, we get something that looks like:One can show that the area under the ROC curve is equal to the proportion correct in a two-alternative forced-choice experiment (Green and Swets). Sometimes, sensitivity is operationally defined as this area. This provides a single summary number, even if the standard definition of d' is inappropriate, for example because the variances are not equal.We also saw that one can test the gaussian equal variance assumption by re-plotting the ROC curve in terms of the z-scores of the hit and false alarm rates.Applications of ROC to neural measures2 5_Psychophysics.nbApplications of ROC to neural measuresThe area under the ROC curve provides a useful measure of sensitivity even if the additive gaussian model isn't known to be correct. It can also be thought of as a measure of how much information about signal vs. no signal can be extracted from the data. ROC curves can be used characterize the sensitivity of single neurons, as well as gross overall measures of activity such as comes from brain imaging data. In the figure below, the gray lines represent a behavioral response by a human observer--i.e. when the signal is high, the observer is indicating subjective "detection". The red lines represent a measured brain signal. How well does the brain signal predict what the observer is reporting? The 2AFC (two-alternative forced-choice) methodUsually rather than manipulating the criterion, we would rather do the experiment in such a way that it does not change. Is there a way to reduce the of a fluctuating criterion?‡See the "mini-experiment" from "Statistical efficiency: competing with the ideal observer in a 2AFC task" in the last lecture‡Relating performance (proportion correct) to signal-to-noise ratio, d'. In psychophysics, the most common way to minimize the problem of a varying criterion is to use a two-alternative forced-choice procedure (2AFC). In a 2AFC task the observer is presented on each trial a pair of stimuli. One stimulus has the signal (e.g. high flash), and the other the noise (e.g. low flash). The order, however, is randomized. So if they are pre-sented temporally, the signal or the noise might come first, but the observer doesn't know which from trial to trial. In the spatial version, the signal could be on the left of the computer screen with the noise on the right, or vice versa. One can show that for 2AFC:(1)d'= - 2 z Hproportion correctL5_Psychophysics.nb 3‡Calculating the Pattern Ideal's d' for a two-alternative forced-choice experiment from a z-score of the proportion correct. (see Homework Assignment #1)For our 2AFC experiment, the observer gets two images to compare. One has the signal plus noise, and the other just noise. But the observer doesn't know which one is which. This strategy will result in a single measureable number, the proportion correct, Pc. d' for a 2AFC task is given by the formula:TodayReview some probability and statisticsPattern detection: The signal-known-exactly (SKE) ideal observerDemo of 2AFC for pattern detection in noiseWhat does the eye see best?Make the question precise by asking: For what patterns does the human visual system have the highest detection efficiencies relative to an ideal observer?4 5_Psychophysics.nb‡Animals, or particularly dangerous ones?‡Faces, or a particular face?‡Or something simple, like a spot?‡Or something complex, like a "frozen" noise image?‡Or some pattern motivated by neurophysiology? E.g. the kinds of spatial patterns preferred by single neurons in the primary visual cortex ...5_Psychophysics.nb 5‡Or some pattern motivated by neurophysiology? E.g. the kinds of spatial patterns preferred by single neurons in the primary visual cortex ...Answering this question requires one to first devise a generative model that describes the variations in both the signal and the non-signal conditions. In general this is hard to do, but we can do it for simple cases such as when the signal image is constant, and the data is either "white gaussian noise" or the signal added to white gaussian noise.‡Some intuition: Measures of pattern similarityThe fundamental problem of pattern recognition is deciding whether an input pattern x matches a stored representation s. This decision requires some measure of comparison between the input and the stored "template" s. Given two patterns represented by vectors x and s, how can we measure how close or similar they are? Some possibilities are: Abs[x-s], Cos[x,s], or Dot[x,s]. We will see below that the ideal strategy is to compute the cross - correlation decision variable for each image (i.e. the dot product between each image data vector, say x, and an exact template of the signal, s, one is looking for), and pick the image which gives the larger cross - correlation.6 5_Psychophysics.nbProbability OverviewFor terminology, a fairly comprehensive outline, and overview, see notebook:ProbabilityOverview.nbin the syllabus web page, and for a general introduction, Griffiths and Yuille (2008).For the section below, we'll use the properties of independence. Here is