FACULTY

For each assignment, write an essay (or essays for assignments with multiple part question), approximately 5 pages per assignment, with references and optionally with figures. Each essay must include background, summarizing the relevant material from the lectures and readings, in addition to the specific answer to the question. Submit your essay by email to (as a pdf file) to Prof. Landy.

Assignment 1: Detection/Cue Integration (due 10/16)

Hecht, Schlaer and Pirenne were interested in determining the minimal visible stimulus under the best of conditions. However, they did not run their task as a signal-detection experiment, as was mentioned in class.

(a) Describe a true signal-detection-task version of the experiment. What would be the different stimuli, presented in what order, and blocked how? Please justify your choices.

(b) Describe precisely how you would estimate d' and beta (for standard signal detection theory using unit-variance Gaussians) for each condition in (a). Be specific about which data are used for each such estimate.

(c) However, the underlying model here is that performance is limited by photon statistics. Outline as precisely as you can a procedure for interpreting the data from (a) assuming that photoisomerizations are Poisson (not Gaussian) distributed. Note that photoisomerizations occasionally occur when no stimulus is delivered (which is given the lovely oxymoronic name of "dark light"). You can assume that the noise in the rods due to these spontaneous photoisomerizations is the only source of biological variability in behavioral performance. Your goal is to estimate the "dark light" and the number of photoisomerizations that are needed for threshold performance in detecting a flash of light. You can assume that the display is perfectly calibrated so that you know the mean number of photons emitted for each trial (you don't know the exact number of photons emitted on a given trial, only the average number emitted at any given light level). Furthermore, you can assume that you know the proportion of photons emitted that lead to photoisomerizations (recalling that some of the photons are reflected or absorbed by the rest of the stuff in the eye).

(d) However, it is not realistic to assume that the display is perfectly calibrated nor that you know the proportion of photons emitted that lead to photoisomerizations. So now, without making these assumptions, how would you estimate the "dark light" and the number of photoisomerizations that are needed for threshold performance in detecting a flash of light. Hint: Assume that the "dark light" corresponds to a mean rate of 1 photoisomerization per trial, that a test flash corresponds to a mean rate of 1 photoisomerization per trial and that twice the test flash gives twice the mean rate. What will the internal response (probability density) curves and the ROC curves look like? Next, assume that the "dark light" corresponds to a mean rate of 1 photoisomerization per trial, and that the test flashes yield mean rates of 10 or 20 photoisomerizations. In what way will the internal response and ROC curves differ? Then assume that the "dark light" corresponds to a different mean rate (10 photoisomerizations). In what way will the internal response and ROC curves differ? Based on this, outline how you would estimate the mean rates of photoisomerizations for the "dark light" and test flashes, from the experimental data. Don't forget that you need to do this in such a way that you can either estimate the criterion or make sure that the criterion doesn't affect your estimates of the mean rates of photoisomerizations for the "dark light" and test flashes.

(e) The above analysis is described effectively as if you did the detection with a single rod receptor that produce all the light-induced and spontaneous photoisomerizations. Suppose instead that the light is distributed equally across 5 receptors (each getting, on average, 1/5 of the photons) and that you know the average dark light for each receptor, but these dark-light values (average number of spontaneous photoisomerizations) are not identical across the 5 receptors. What decision rule would be optimal under these circumstances?

(f) This is for extra credit (i.e., not required). If you can, try to use Matlab to simulate the internal response and ROC curves for various different choices of "dark light" and test flash intensity. Better yet, use Matlab to generate a simulated data set ("yes" and "no" responses for a bunch of trials) and then fit the simulated data to estimate the mean rates of photoisomerizations for the "dark light" and test flashes. Although this part of the assignment is for "extra credit", implementing the simulations will guarantee that your answers are correct, and you might find that simulating it will help get an intuition for it.

Assignment 2: Spatial Vision and Color (due 11/29 *** extended deadline ***)

As we discussed in class, standard theories of spatial vision suggests that spatial patterns are analyzed in the human visual system with a small number of "channels" varying in the spatial frequency to which they are maximally sensitive and the peak temporal frequency as well. Each "channel" consists of a set of elements with linear receptive fields identical in shape, differing only in spatial position. The models then typically include nonlinearities, noise (to simulate variability in human responses) and a decision mechanism that combines the noisy responses from each location to compute a decision in each simulated experimental trial.

Similarly, in color vision it has been suggested that colors are also encoded by separate "channels": the opponent mechanisms that encode blue-vs.-yellow, red-vs.-green, and black-vs.-white. Again, each of these "channels" begins with a receptive field that linearly combines the responses of cones, with different linear weights for each cone class (S, M, and L).

In two papers, Poirson & Wandell modeled and measured how spatial and chromatic tuning are combined in terms of the effect on color appearance (Poirson & Wandell, 1993) and detection threshold (Poirson & Wandell, 1996).

Please read the 2nd (detection) paper (you might scan the appearance paper for context if you like, but that's not necessary), and summarize what they did and what they found.

(1) Stimuli: Describe clearly and carefully their "chromatic" Gabor stimuli in terms of the responses such a stimulus engenders in each of the three cone classes. Then, describe as best you can how you would determine the RGB values at each pixel required to produce that stimulus.

(2) Model: Their most-constrained model is "pattern-color separable". That is, the corresponding receptive fields are f(x,y,lambda)=g(x,y)h(lambda), where x,y are the spatial position and lambda is wavelength of a light. Give some details of how you would build such separable receptive fields as a sequence of linear receptive fields building on the initial cone representation of the input, each stage linearly combining units at the preceding stage.

(3) Results: Describe the three channels found by Poirson & Wandell in terms of the color and pattern selectivity of each, as well as a description of the spatial receptive field corresponding to each.

(4) Suppose you wanted to verify the results of Poirson & Wandell using a standard experiment for identifying channel tuning properties (i.e., a masking, adaptation or summation experiment, as discussed in class). Describe the experiment you would perform (stimuli, conditions, task) and data analysis (estimation of threshold, etc.) required to establish estimated spatiochromatic tuning.