CS5350: Possible Errata and Clarifications
Here we list possible errata in the readings. In general these may not be confirmed with the author, hence we call them “possible” errata.
We also list some clarifications for possibly tricky sections.
- Top of page 53: “collinear” should be “concurrent”.
- Page 75: eq. 6.1 should be .
- Page 75: eq. 6.2 should be .
- Middle of page 77: “” should be .
- Top of page 3: “points in the ordinary plane” should be “infinite points”.
- Bottom of page 6: “” should be .
- Tottom of page 12: “” should be .
- Top of page 13: “” should be .
- Page 246, first paragraph: “position” should be “orientation”, i.e. a tumbling brick both translates and rotates in general; Euler’s rotation theorem only deals with rotation, not translation. Chasle’s theorem deals with both.
- Page 248, top right: the expression “” may be confusing. Here, and are quaternions, say and similarly for . However, for the purposes of taking the dot product , they are temporarily considered simply 4-dimensional vectors: .
- Page 248, second from last paragraph. An excerpt from the referenced book by Misner et al should shed more light on the idea of “entanglements” as used here.
- Page 249, figure “constructing a point for tangent”: the lower-left symbol in the figure may be hard to read; it should be “”.
- Page 249, lower right: in the formula for and are quaternions but temporarily considered as vectors to evaluate the dot product , just as was done for on p. 248.
- Page 6: “” should be , and similar for all the other expansions of on the page.
- Page 60, bottom: every term in the equations for should be negative
- Page 13: the first figure in section 4 is actually figure 5 in the paper, not figure 1 as it is labeled.
- Page 16: the third equation in equation group (14) should be .
Official confirmed and unconfirmed errata.
- Page 131, two code blocks: change
- Page 131, last line: change to .
- Page 164, last paragraph before start of section “Affine Transform”: change “a perspective transform can turn a rectangle into a trapezoid” to “a perspective transform can turn a rectangle into any quadrilateral”.
- Page 164, last line of second footnote: delete the word “orthogonal”.
- Page 165, figure 6–13: change “trapezoids” to “arbitrary quadrilaterals”.
- Page 167, footnote: change “trapezoid” to “arbitrary quadrilateral”.
- Page 317—318: the description of Harris corners, starting with the last paragraph on p. 317 and continuing for the first three paragraphs of p. 318, is unnecessarily confusing, and possibly erroneous. The wikipedia entry for Harris corners is much better (and start by reading the section on the Moravec corner dectector, which is the origin of the idea, and quite easy to understand). The actual Harris algorithm is (1) calculate the matrix for each pixel according to the equation at the top of p. 318 (note that in this equation and are the first derivatives of the image in the horizontal resp. vertical directions) and then (2) classify pixel as a corner iff both eigenvalues (there will be two because is ) of are “relatively large”. Harris suggested an approximation to this condition based on the determinant and trace of which saves a little computation vs actually computing the eigenvalues, but later Shi and Tomasi pointed out that it gives better results to actually compute the eigenvalues and verify that the smaller of the two is greater than some threshold.
- Page 329, top paragraph: replace both instances of “eigenvectors” with “eigenvalues”.
- Page 352, paragraph before second displayed equation: change “first measure ” to “first measurement ”.
- Page 352, second displayed equation: replace with and with .
- Page 356: Note some other texts use the opposite naming convention for matrices Q and R (i.e. what Bradski and Kaehler call “Q” some other texts call “R” and vice versa).
- Page 357: the second equation actually displays the transpose of .
- Page 358, first displayed equation: change to .
- Page 371—373: Note that in the equations on these pages, lower-case variables such as are in units of pixels, and upper-case variables such as are in physical units, e.g. millimeters (any choice of physical units works as long as it is used consistently).
- Page 373: in the second paragraph, is the actual focal length in physical units (e.g. millimeters), and and are effective horizontal and vertical focal lenghths in pixels. These differ only when the pixels are not actually square, i.e. when . In practice, despite what the book says, nowadays most cameras have square pixels, even cheap ones. Also, it is usually possible to get accurate specifications from the camera manufacturer of the designed values of and . The designed focal length is also often specified, but the actual as-built value may differ somewhat.
- Page 376: the lower two equations on the page should be and . Also, note that all equations on this page are given without derivation or detailed explanation; we’ll just take them at face value. The quantity appearing in the equations is defined as the radial distance of the pixel from the optical center : .
- Page 380: replace the equation with , and note that is calculated in object frame coordinates. This is fairly nonstandard, in particular note that the vector here is not the same as the vector used later in the chapter (starting on p. 386). The matrix is the same though. See further discussion below on a similar issue with the vector on p. 422—423.
- Page 381: “intrinsic corrections” and “intrinsics matrix”, terms used in the first and second paragraph, seem to not have been previously defined. Here, “intrinsic corrections” refers to both the pinhole model of the camera (equation at the top of p. 374), which is defined by four parameters , combined with the radial and tangential lens distortion models (equations on p. 376), which are defined by five parameters . Perhaps confusingly, “intrinsics matrix” refers only to the matrix defined at the top of p. 374, i.e. the pinhole camera model. The discussion of the required number of equations on p. 381 mostly ignores the distortion parameters. Fortunately, the discussion is repeated in more detail, including the distortion parameters, on p. 388.
- Page 385: note in the equation , there is not a single which works for every . The equation would be better stated as , where means “is proportional to”.
- Page 385: note that the point is measured in physical units (e.g. millimeters) in a coordinate frame fixed to the moving object (i.e. the chessboard). Thus, if the chessboard has square cells of side length millimeters, its corners would have coordinates of the form for integers , assuming the chessboard is aligned so that one corner is at the origin of , that the axis of is normal to the plane of the chessboard, and that the and axes are aligned with the rows and columns of the chessboard. The point is in units of pixels in the image plane.
- Page 386: in the sentence before the final equation on the page, replace with .
- Page 386: note that in all equations involving that there is not a single which works for every value of the other variables. These equations would be better stated using instead of , where means “is proportional to”.
- Page 387: in the equations and , change to where means “is proportional to”.
- Page 390: delete the final transpose at the upper right of the third displayed equation on the page (i.e. the one beginning ).
Page 391: while technically correct, the block of equations at the top of the page is unnecessarily complicated (this appears to be an artifact of translating the original equations from Zhang’s paper, which handles a slightly more general case). Note that here . Using that fact, and also substituting in the expressions in variables for the derived quantities and in the third and fifth equations, respectively, gives the following simpler set of equations:
.Also, the introduction of this particular is not explained. It comes from the fact that if is a solution to the equation at the bottom of p. 390, then so is any scalar multiple of . Thus, the returned solution is actually multiplied by some unknown scalar factor . Fortunately, here can be recovered by the above equation.
- Page 391: in the second block of equations on the page, note that is not necessarily the same as in the first block of equations on the page.
- Page 391, bottom: the final expression for has a typo—rather than depending on , depends on . Also, it is never explicitly stated, but the coordinates are produced by multiplying a point in object frame by the now-reconstructed extrinsic transformation matrix :
- Page 402, first displayed equation: divide the second term by and the third by (note ); also replace the lower-left entry in the matrix with .
- Page 402, second displayed equation: divide the LHS by and replace the lower-left entry in the matrix with . Also, there are more direct ways to calculate from .
- Page 403, top paragraph: replace the phrase “your use of the jacobian function” with “your use of the cvRodrigues2 function”.
- Page 422, third paragraph: replace “We begin by considering the relationship between and ” with "We begin by considering the relationship between and .
- Pages 422—423: Note that the definition of the translation vector used in the section “Essential matrix math” is different from that used later in the chapter. However, this is ok, because each usage is self-consistent. The usage on 422—423 is relatively non-standard, and is similar to that on p. 380 (see above): is an orthonormal basis for the left camera frame in the right camera frame (as would be typical for defining a rigid transform, as we are doing, from left camera coordinates to right camera coordinates), but (this is the nonstandard part) here is the location of the right camera in the left camera frame; normally (i.e. as when forming the right column of a homogenous transformation matrix taking left camera coordinates to right camera coordinates) the translation vector would simply be the location of the left camera in the right camera frame.
- Page 422, second from last paragraph: replace “all possible points through…” with “all possible points on a plane through…”.
- Page 422, last paragraph: replace and with and .
- Page 422, first footnote: the second mention of and in the sentence should be replaced by and .
- Page 424, top paragraph: replace all instances of with , and replace “(the pixel coordinate)” with “(the camera frame coordinate)”.
- Page 425, footnote: the description of the RANSAC algorithm is really LMedS and vice versa.
- Page 427, second paragraph: The first in should be replaced by .
Page 428, first footnote: replace the existing text with “Let’s be careful about what these terms mean: and denote the locations of the 3D point in the coordinate system of the left and right cameras, respectively. itself is in an object-relative coordinate frame s.t. and are rigid transforms that take coordinates in to the left resp. right camera frame. is a rigid transform that takes coordinates in the left camera frame to the right camera frame; thus encodes the pose of the left camera with respect to the right.”
- Page 314, line 3: here where is the commanded forward velocity and is the commanded rotational velocity. This is a common way to command robots that move in a planar workspace, and the update equation here is well known in robotics.
- Page 314, line 5: here is a covariance matrix representing the noise distribution of the robot state.
- Page 314, line 6: here is a covariance matrix representing the noise distribution of the sensor data. is the standard deviation of the range (distance to sensed object), is the standard deviation of the bearing (relative direction of sensed object), and is the standard devation of the signature of the sensed object.
- Page 322, line 15: the two diagonal entries in the displayed matrix with values –1 and 1 should be replaced with and , respectively.