MONT 104S Perception and Virtual Reality

    College of the Holy Cross, Fall 2008

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    Assignment 1

    Due: Wednesday, September 10, in class

    Problem 1: Perspective Projection in Art

    Long before there were computers, there was graphic design. Painters have been studying how to make realistic pictures for thousands of years. Sometime during the Renaissance artists discovered that for objects in the distance to look realistic, they need to be smaller in size in the painting than objects nearby. Based on your knowledge of the projection system in a pinhole camera (Perspective projection), explain why this is so. Sketch a diagram that illustrates your point.

    Problem 2: Height in the Visual Field as an Indicator of distance

    In class we learned that the brain uses height in the image to give an indication of distance. The assumption is that objects that are higher in the image are more distant in the scene. In this problem we will see why this is a reasonable assumption to make.

    Imagine a person (the observer) standing on a flat surface (the ground) and looking straight ahead at 2 other people who are standing at different distances from the observer on the same flat surface. We will use a world coordinate system, in which distance in front of the observer (depth) is given by the Z value, distance to the right or left (horizontal position) is given by the X value and distance up and down (vertical position) is given by the Y value. Assume the observer's eye is located at (X, Y, Z) position (0, 0, 0). The observer's eye is 170 cm (about 5 ft 7 inches) above the ground, so the vertical position of the ground is Y = -170 cm for all the people. The first person is standing at a distance of Z = 100 cm in front of the observer. The second person is standing at a distance of Z = 200 cm in front and X = 100 cm to the right of the observer. Both people are the same physical size.

    The following give the (X, Y, Z) coordinates for different parts of person 1 and person 2:

      Person 1:
      Left Eye: (-5, 0, 100)
      Right Eye: (5, 0, 100)
      Left Shoulder: (-20, -30, 100)
      Right Shoulder: (20, -30, 100)
      Left Foot: (-20, -170, 100)
      Right Foot: (20, -170, 100)

      Person 2:
      Left Eye: (95, 0, 200)
      Right Eye: (105, 0, 200)
      Left Shoulder: (80, -30, 200)
      Right Shoulder: (120, -30, 200)
      Left Foot: (80, -170, 200)
      Right Foot: (120, -170, 200)

    Part a: Remember that we can approximate the formation of the image on the retina as a pinhole camera. Using the perspective projection in the pinhole camera, the coordinates of projection onto an image plane are given by:

      y = Yd/Z
      x = Xd/Z

    The 3D coordinates of the point are given by uppercase letters (X,Y,Z) and the 2D coordinates of the projected image are given by lowercase letters (x,y). The distance of the image plane from the center of projection is given by d.

    • Compute the x and y coordinates for each point projected onto and x-y image plane located at a distance of 1 cm from the center of projection (d = 1).

    • On a piece of graph paper, plot the projected image of each point onto the x-y projection plane, and label each of the projected points with their (x,y) coordinates. For each person, connect the 4 points for the shoulders and feet to form a rectangle.

    Part b: Answer the following questions:

    • How does the size of the rectangle formed for person 1 compare to the size of rectangle for person 2?
    • Which person's feet are higher in the image? Is that person closer or farther from the observer than the other person? Are their feet acutally at different heights in the world?
    • What can you conclude about how image size and height in the image relate to distance in the real world?

    Problem 3: Mt. Monadnock and Cues to Distance

    In class we learned that the images of objects in the distance are smaller than those that are nearby. We also learned that the images of objects that are farther away tend to be higher than those of objects that are nearby (see also problems 1 and 2 of this assignment). These two facts are used by our brain as cues to the distance to an object. While we are on our climb, observe the landscape around you. Write a description of some examples you notice on our hike where these cues would be useful and some examples where the cues would not be useful for determining distance. For each example, explain why you believe it would or would not be useful. You should write at least one page, typed and double-spaced.

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    Constance Royden--croyden@mathcs.holycross.edu
    MONT 104S, Perception and VR
    Date Created: January 26, 2008
    Last Modified: September 4, 2008
    Page Expires: September 4, 2009