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    Pinhole Camera Model: Understanding Perspective through Computational Optics
    Pinhole Camera Model: Understanding Perspective through Computational Optics
    Pinhole Camera Model: Understanding Perspective through Computational Optics
    Ebook180 pages1 hour

    Pinhole Camera Model: Understanding Perspective through Computational Optics

    By Fouad Sabry

    ()

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    About this ebook

    What is Pinhole Camera Model


    The pinhole camera model is a mathematical representation of the relationship between the coordinates of a point in three-dimensional space and its projection onto the picture plane of an ideal pinhole camera. In this model, the camera aperture is portrayed as a point, and there are no lenses employed to concentrate light. By way of illustration, the model does not take into account geometric distortions or the blurring of unfocused objects that can be brought about by lenses and apertures of a finite size. The fact that the majority of practical cameras only have discrete picture coordinates is another thing that is not taken into consideration. Because of this, the pinhole camera model can only be utilized as a first-order approximation of the mapping from a three-dimensional scene to a two-dimensional graphical representation. Its validity is contingent on the quality of the camera, and in general, it diminishes from the center of the image to the edges as the effects of lens distortion rise.


    How you will benefit


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


    Chapter 1: Pinhole camera model


    Chapter 2: Cartesian coordinate system


    Chapter 3: Spherical coordinate system


    Chapter 4: Isometric projection


    Chapter 5: Matrix representation of conic sections


    Chapter 6: Fourier optics


    Chapter 7: 3D projection


    Chapter 8: Transformation matrix


    Chapter 9: Graphics pipeline


    Chapter 10: Three-dimensional space


    (II) Answering the public top questions about pinhole camera model.


    (III) Real world examples for the usage of pinhole camera model 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 Pinhole Camera Model.

    LanguageEnglish
    PublisherOne Billion Knowledgeable
    Release dateApr 30, 2024
    Pinhole Camera Model: Understanding Perspective through Computational Optics

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      Book preview

      Pinhole Camera Model - Fouad Sabry

      Chapter 1: Pinhole camera model

      When there are no lenses involved in focusing light, the pinhole camera model describes the mathematical relationship between the location of a point in three-dimensional space and its projection onto the image plane. Geometric distortions and out-of-focus blur due to lenses and fixed aperture sizes are not accounted for in the model. Furthermore, it ignores the fact that most real-world cameras only use discrete picture coordinates. This means that the mapping from a 3D scene to a 2D image produced by the pinhole camera model is simply an estimate at best. Its reliability reduces from the image's center to its edges due to lens distortion effects and varies depending on the camera's quality.

      If a high-quality camera is employed, some of the impacts that the pinhole camera model ignores can be accounted for, such as by executing suitable coordinate transformations on the image coordinates. Therefore, in fields like computer vision and computer graphics, where accurate descriptions of how a camera represents a 3D scene are necessary, the pinhole camera model is often enough.

      The figure shows how a pinhole camera's mapping geometry works. The following are the building blocks of the illustration::

      An O-centric, three-dimensional, orthogonal coordinate system. The camera's aperture is found here as well. X1, X2, and X3 are the names given to the three coordinate axes. The optical axis, principal axis, or principle ray points in the direction of the camera's field of view. The primary plane, or front side of the camera, is the space defined by axes X1 and X2.

      Picture plane, where the world, in three dimensions, is projected via the lens of a camera.

      The image plane is parallel to axes X1 and X2 and is located at distance f from the origin O in the negative direction of the X3 axis, where f is the pinhole camera's focal length.

      For a pinhole camera to work in practice, the picture plane must be positioned so that it crosses the X3 axis at -f, where f is greater than zero.

      The picture plane and the optical axis meet at a position denoted by R. This is the focal point, or the heart of the picture.

      A point P somewhere in the world at coordinate (x_1, x_2, x_3) relative to the axes X1, X2, and X3.

      The line that point P uses to project itself onto the film plane. Connecting points P and O, this green line represents this connection.

      This is the image plane onto which the point P is projected, denoted Q.

      The image plane and the green projection line intersect at this position.

      In any practical situation we can assume that x_{3} > 0 which means that the intersection point is well defined.

      In addition to the 3D world, the image plane has its own set of coordinates, with the center at R and the axes perpendicular to each other (X1 and X2), respectively.

      The coordinates of point Q relative to this coordinate system is (y_1, y_2) .

      All projection lines are supposed to pass via an infinitesimally small point at the camera's pinhole aperture. The term optical center is used to describe this location in three dimensions.

      Next we want to understand how the coordinates (y_1, y_2) of point Q depend on the coordinates (x_1, x_2, x_3) of point P.

      The next figure will aid with this process by displaying the identical scene as the previous one, only this time seen from above, Having one's eyes pointed downward, along the X-axis negative direction.

      The figure depicts a pair of congruent triangles, both of whose hypotenuses are segments of the green projection line.

      The catheti of the left triangle are -y_1 and f and the catheti of the right triangle are x_{1} and x_3 .

      The similarities between the two triangles suggest that

      \frac{-y_1}{f} = \frac{x_1}{x_3} or y_1 = -\frac{f \, x_1}{x_3}

      The results of a similar inquiry when viewed counterclockwise around the X1 axis are

      \frac{-y_2}{f} = \frac{x_2}{x_3} or y_2 = -\frac{f \, x_2}{x_3}

      In a nutshell, this means

      \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = -\frac{f}{x_3} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}

      which is an expression that describes the relation between the 3D coordinates (x_1,x_2,x_3) of point P and its image coordinates (y_1,y_2) given by point Q in the image plane.

      The mapping from 3D to 2D coordinates described by a pinhole camera is a perspective projection followed by a 180° rotation in the image plane.

      This is in keeping with the workings of a conventional pinhole camera; the resulting image is rotated 180° and the relative size of projected objects depends on their distance to the focal point and the overall size of the image depends on the distance f between the image plane and the focal point.

      To obtain a picture that hasn't been rotated,, which is to be expected from a photographic device, It could go either way:

      Rotate the coordinate system in the image plane 180° (in either direction).

      This is the solution that any functional pinhole camera would use; When viewing a photo taken using a camera, the image is rotated before being viewed, In the case of a digital camera, the image is rotated as a result of the order in which the pixels are read out.

      Image plane must be moved such that it meets the X3 axis at f, rather than -f, and all prior computations must be redone. Since this cannot be done in fact, a theoretical camera is created that may be easier to analyze than the actual camera.

      Without the negation, the previous expression gives the mapping from 3D to 2D picture coordinates in both circumstances.

      \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} = \frac{f}{x_3} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}

      Homogeneous coordinates are another way to describe the mapping from 3D point locations in space to 2D picture locations.

      Let \mathbf {x} be a representation of a 3D point in homogeneous coordinates (a 4-dimensional vector), and let \mathbf{y} be a representation of the image of this point in the pinhole camera (a 3-dimensional vector).

      Then the subsequent relationship is true.

      \mathbf{y} \sim \mathbf{C} \, \mathbf{x}

      where \mathbf{C} is the 3\times 4 camera matrix and the \, \sim means equality between elements of projective spaces.

      This means that any non-zero scalar multiplication between the left and right sides is also equal.

      A consequence of this relation is that also \mathbf{C} can be seen as an element of a projective space; If a scalar multiplication of two camera matrices yields the same result, then the matrices are comparable.

      Pinhole camera mapping as described here, as a linear transformation \mathbf{C} instead of as a fraction of two linear expressions, allows for several relations between 3D and 2D coordinates to be derived with fewer steps of calculation.

      {End Chapter 1}

      Chapter 2: Cartesian coordinate system

      In geometry, a Cartesian coordinate system (UK: /kɑːˈtiːzjən/, US: /kɑːrˈtiʒən/) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, where two perpendicular lines meet at a fixed point, what are the signed distances between them?, lines of coordinates, for short, system's coordinate axes, or axes (plural of axis).

      The intersection point, also known as the origin, is marked with a zero, 0) as coordinates.

      Cartesian coordinates, the signed distances from a point to three perpendicular planes, can be used to describe the location of a point in three-dimensional space. In general, for every dimension n, a point in an n-dimensional Euclidean space can be described using n Cartesian coordinates. These are the coordinates of a point, expressed as the signed distances to n fixed, perpendicular hyperplanes.

      Cartesian coordinates are named for René Descartes, to the mathematician whose development of them in the 17th century established the first systematic connection between geometry and algebra and thereby sparked a mathematical revolution.

      Applying the Cartesian system of coordinates, Equations containing the coordinates of points on a geometric shape (such a curve) can be used to describe the shape in detail.

      For example, radius 2 circle, located focus on the plane's genesis, may be described as the set of all points whose coordinates x and y satisfy the equation x² + y² = 4.

      Because of its central role in analytic geometry, Cartesian coordinates also provide light on other areas of study in mathematics, including linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and many more. The function graph is a well-known example of such a concept. Most practical fields that deal with geometry rely heavily on Cartesian coordinates, including astronomy, physics, engineering, and many more. They are the de facto standard for data processing in fields like computer graphics, CAD, and other forms of geometric modeling.

      The Cartesian refers to the French mathematician and philosopher René Descartes, who, while living in the Netherlands, published this theory in 1637.

      Pierre de Fermat found it on his own, who were also involved in multidimensional work, Despite the fact that Fermat did not share his finding,.

      Polar coordinates for the plane, and spherical and cylindrical coordinates for three-dimensional space, are only a few of the many that have been established since Descartes.

      Choosing a point O on the line (the origin), a length unit, and an orientation for the line are the three components of a Cartesian coordinate system for a straight line in a two-dimensional space. When a line is orientated (or points) from the negative half to the positive half, it means that the orientation chose which of the two halves of the line given by O should be considered positive. Then, the distance from O to any given point P on the line can be stated using a plus (+) or minus (-) symbol, depending on which half-line contains P.

      A number line is a line that uses a specific Cartesian coordinate system. There is a specific place on the line for every real number. On the other hand, each point on the line can be thought of as a discrete element of a continuous number system like the real numbers.

      An ordered pair of perpendicular lines (axes), a common unit of length for both axes, and an orientation

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