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Lecture12

# Lecture12 - CS221 Lecture notes Geometric vision In the...

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CS221 Lecture notes Geometric vision In the last lecture, we discussed how to compute various visual features, such as edges, corners, and lines, which are useful for tasks such as face recognition or object recognition. Now, we are going to take a step back and look at the geometry of image formation . How does a 3-D scene in the world translate into the 2-D image that is picked up by the human retina or a camera? An understanding of the geometry of image formation will lead to some very useful algorithms. 1 Perspective camera model Our model for image formation will be the perspective camera , where all of the points in the world are projected onto an imaginary 2-D plane. As shown in Figure 1, we have an image plane π , and a camera center , labeled O . The optical axis is the line which passes through the camera center O and is perpendicular to the image plane π . We call the distance from O to π the focal length and denote it as f . The point o where the optical axis intersects π is called the principal point , or image center . Typically, we choose a Cartesian ( x, y, z ) coordinate system such that the camera center O is at the origin (0 , 0 , 0), and the z -axis is the optical axis. The image is generated by projecting points in 3-space onto the image plane π . More specifically, consider the point P shown in Figure 1. To project P onto π , we take the line connecting O to P , and let the projection p of P be the point where that line intersects π . (In general, we will denote points in 3-space with capital letters, and projections onto the image plane with 1

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2 Figure 1: The perspective camera model. lowercase letters.) We can express projection algebraically as follows. Let P = X Y X . Since we took the camera center O to be the origin (0 , 0 , 0), all the points on the line connecting O and P will be scalar multiples of P . In other words, we have p = x y z = α X Y Z . To determine the value of α , note that αZ = z , and that z = f , because the image plane π is at Z coordinate f . This implies that α = f/Z , and so, x = f X Z , y = f Y Z . (1) We will use the two equations (1) as our definition of projection for the rest of this lecture.
3 Note that projection is a non-linear function of P because of the factor 1 z . (Recall that a function g is linear if for any vectors u and v and scalars α and β , g ( αu + βv ) = αg ( u ) + βg ( v ).) Also, projection does not preserve distances between points or angles between lines. (The projections of two trees far off in the distance might be very close together in the image plane, and the projection of a chessboard might not have 90 angles in the image plane.) Also, as you show in Problem Set 3, circles appear as ellipses in the image plane. 1.1 Projection preserves straight lines While we can list many properties not preserved under projection, it turns out that projection does preserve straight lines. Consider three points P 1 , P 2 , and P 3 , with projections p 1 , p 2 , and p 3 , as shown in Figure 2. For simplicity, assume that P 2 is the midpoint of P 1 and P 3 , i.e. P 2 = ( P 1 + P 3 ) / 2. We will

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Lecture12 - CS221 Lecture notes Geometric vision In the...

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