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    Homography: Homography: Transformations in Computer Vision
    Homography: Homography: Transformations in Computer Vision
    Homography: Homography: Transformations in Computer Vision
    Ebook95 pages47 minutesComputer Vision

    Homography: Homography: Transformations in Computer Vision

    By Fouad Sabry

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    About this ebook

    What is Homography


    In the field of computer vision, any two images of the same planar surface in space are related by a homography. This has many practical applications, such as image rectification, image registration, or camera motion-rotation and translation-between two images. Once camera resectioning has been done from an estimated homography matrix, this information may be used for navigation, or to insert models of 3D objects into an image or video, so that they are rendered with the correct perspective and appear to have been part of the original scene.


    How you will benefit


    (I) Insights, and validations about the following topics:


    Chapter 1: Homography (computer vision)


    Chapter 2: Affine transformation


    Chapter 3: Transformation matrix


    Chapter 4: Image stitching


    Chapter 5: Line-plane intersection


    Chapter 6: Fundamental matrix (computer vision)


    Chapter 7: Camera resectioning


    Chapter 8: Image rectification


    Chapter 9: Camera matrix


    Chapter 10: Camera auto-calibration


    (II) Answering the public top questions about homography.


    (III) Real world examples for the usage of homography in many fields.


    Who this book is for


    Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Homography.

    LanguageEnglish
    PublisherOne Billion Knowledgeable
    Release dateApr 28, 2024
    Homography: Homography: Transformations in Computer Vision

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      Book preview

      Homography - Fouad Sabry

      Chapter 1: Homography (computer vision)

      As used in computer vision, a homography is a relationship between any two images of the same planar surface in space (assuming a pinhole camera model). This can be used in a variety of contexts, including image rectification, image registration, and detecting and correcting rotational and translational camera motion between two images. After an estimated homography matrix has been used for camera resectioning, the resulting information can be put to use for navigation or for inserting 3D models of objects into an image or video so that they are rendered in the correct perspective and appear to have always been a part of the original scene (see Augmented reality).

      A and B are our two cameras, looking at points P_{i} in a plane.

      Passing from the projection {\displaystyle {}^{b}p_{i}=\left({}^{b}u_{i};{}^{b}v_{i};1\right)} of P_{i} in b to the projection {\displaystyle {}^{a}p_{i}=\left({}^{a}u_{i};{}^{a}v_{i};1\right)} of P_{i} in a:

      {\displaystyle {}^{a}p_{i}={\frac {{}^{b}z_{i}}{{}^{a}z_{i}}}K_{a}\cdot H_{ab}\cdot K_{b}^{-1}\cdot {}^{b}p_{i}}

      where {\displaystyle {}^{a}z_{i}} and {\displaystyle {}^{b}z_{i}} are the z coordinates of P in each camera frame and where the homography matrix {\displaystyle H_{ab}} is given by

      {\displaystyle H_{ab}=R-{\frac {tn^{T}}{d}}} .

      R is the rotation matrix by which b is rotated in relation to a; From point a to point b, t represents the direction of translation; n is the normal vector of the plane, and d is the distance in radians from the plane's center to the origin.

      Ka and Kb are the cameras' intrinsic parameter matrices.

      In the diagram, camera b is positioned at a distance of d from the plane.

      Taken from the diagram above:, assuming n^{T}P_{i}+d=0 as plane model, n^{T}P_{i} is the projection of vector P_{i} along n , and equal to -d .

      So {\displaystyle t=t\cdot 1=t\left(-{\frac {n^{T}P_{i}}{d}}\right)} .

      And we have {\displaystyle H_{ab}P_{i}=RP_{i}+t} where {\displaystyle H_{ab}=R-{\frac {tn^{T}}{d}}} .

      This formula holds only in the case where camera b does not rotate or translate.

      In the general case where R_{a},R_{b} and t_{a},t_{b} are the respective rotations and translations of camera a and b, R=R_{a}R_{b}^{T} and the homography matrix {\displaystyle H_{ab}} becomes

      {\displaystyle H_{ab}=R_{a}R_{b}^{T}-{\frac {(-R_{a}*R_{b}^{T}*t_{b}+t_{a})n^{T}}{d}}}

      where d is the horizontal separation between camera b and the plane.

      An affine homography is a better model of image displacements when the picture region where the homography is computed is tiny or the image was recorded with a large focal length. In contrast to general homographies, affine homographies have a fixed last row.

      h_{{31}}=h_{{32}}=0,\;h_{{33}}=1.

      {End Chapter 1}

      Chapter 2: Affine transformation

      An affine transformation (from the Latin affinis, connected with) is a geometric transformation in Euclidean geometry that maintains straight lines and parallelism but changes the lengths and directions of the angles and distances involved.

      A more general definition of an affine transformation is an automorphism of an affine space (Euclidean spaces are special cases of affine spaces), that is, a function which maps an affine space onto itself while maintaining the ratio of the lengths of parallel line segments. Therefore, after an affine transformation, sets of parallel affine subspaces retain their parallelism. Distances and angles between lines are not always preserved by an affine transformation, but distance ratios along a straight line are preserved.

      Assuming X is the point set of some affine space, we can write every affine transformation on X as the combination of a linear transformation on X and a translation of X. The starting point of the affine space is not required to be kept the same during an affine transformation, unlike a linear one. Accordingly, every affine transformation is linear, but not every linear transformation is affine.

      Affine transformations include translation, enlargement, reduction, homology, similarity, reflection, rotation, shear mapping, and any combination or sequence of these.

      Affine transformations are those projective transformations of a projective space that preserve the invariance of the hyperplane at infinity, defining the affine space as the complement of the hyperplane at infinity.

      An affine map is a more general form of an affine transformation.

      Imagine a field k and an affine space X, Let V denote the vector space to which it belongs.

      A bijection f from X onto itself is called an affine transformation; this means that a linear map g from V to V is well defined by the equation {\displaystyle g(y-x)=f(y)-f(x);} here, as usual,

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