MONT 104S Perception and VR
College of the Holy Cross, Fall 2008
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Assignment 2Due: Friday, September 19, in class
Problem 1: Role of neurons in primary visual cortex Describe the receptive field of a typical complex cell in primary visual cortex. (You might want to review the reading, "The Primary Visual Cortex," by David Hubel. What are properties of the visual stimuli that determine the firing rate of the cell? What do these properties tell us about how the visual system analyzes the images on the retina?
Problem 2: Brightness perception In class we learned that perceived brightness of a surface does not correspond exactly to its luminance (i.e. the physical intensity of light reflected from the surface). Describe at least three situations or experiments that show this is the case. Explain why it might sometimes be important for the brain to assess brightness differently from the physical luminance (i.e. in what situations might it give us a more accurate assessment of the scene?) Problem 3: Finding edges with Center-Surround receptive fields We can analyze how a retinal ganglion cell will respond to a given visual stimulus by dividing its receptive field into regions that give a specific response to light within that region. For the purposes of this problem, we are going to approximate the receptive field as a square region rather than a circular one (because it makes the math simpler). Imagine a cell that has a receptive field as diagrammed below.
Each positive number indicates the cell will increase its firing rate by the given amount when light covers that entire region. That is, if the region is labeled +3, the cell will increase its firing rate by 3 spikes per second when the region is completely covered by the light. If a fraction of the region is covered by light, it will have that fraction of the effect. For example, if the light covers only 1/2 the region labeled +3, the cell will increase its firing rate by 3/2 or 1.5 spikes per second. Each negative region indicates that the cell will decrease its firing rate by that amount when that region is covered by light. If the region is labeled -1, the cell will decrease its firing rate by 1 spike per second when the region is completely covered by light. As with the positive regions, if a fraction of the region is covered by light, it will have that fraction of the effect. (Note that this is a vast simplification of the behavior of these cells, but the simplified model will help us understand their behavior). Imagine that there is a vertical edge between light and dark in the middle of the scene (We will call the horizontal position of the edge, x = 0). The purpose of this problem is to calculate the different responses of neurons whose receptive fields have different positions with respect to this edge. The figure below shows a receptive field with its right edge positioned at x = 0 (and left edge at x = -4). Notice how the entire receptive field is in the light. The change in the cell's firing rate (the rate at which it fires action potentials) will be the sum of all the values in each region of the receptive field.
The images below show receptive fields positioned with right edge at x = 0.5 and 1.0 (and with the left edge at -3.5 and -3, respectively).
For the above images, the change in the firing rate will be the sum of the values of only those regions (and partial regions) that are in the light. The total firing rate of the cell will be the background firing rate plus the change in firing rate caused by the light covering portions of the receptive field. a) Calculate the change in firing rate of cells whose receptive fields are positioned with their right edges at x = 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5 and 4.0. b) Draw a graph of the responses, with the Y axis showing the change in firing rate and the X axis indicating the position of the right edge. c) David Marr suggested that the response of these retinal ganglion cells is related to the process of smoothing and differentiation that he proposed for finding edges in an image. He pointed out that one could locate edges by finding the "zero-crossings" in an image that had been smoothed and differentiated twice. How do the responses of the cells that you calculated in part a show such a zero-crossing? Why is it that the cell whose receptive field is centered on the edge cannot signal the presence of the edge by itself?
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Constance Royden--croyden@mathcs.holycross.edu
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