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    Velocity Moments: Capturing the Dynamics: Insights into Computer Vision
    Velocity Moments: Capturing the Dynamics: Insights into Computer Vision
    Velocity Moments: Capturing the Dynamics: Insights into Computer Vision
    Ebook134 pages1 hourComputer Vision

    Velocity Moments: Capturing the Dynamics: Insights into Computer Vision

    By Fouad Sabry

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    About this ebook

    What is Velocity Moments


    In the field of computer vision, velocity moments are weighted averages of the intensities of pixels in a sequence of images, similar to image moments but in addition to describing an object's shape also describe its motion through the sequence of images. Velocity moments can be used to aid automated identification of a shape in an image when information about the motion is significant in its description. There are currently two established versions of velocity moments: Cartesian and Zernike.


    How you will benefit


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


    Chapter 1: Velocity Moments


    Chapter 2: Navier-Stokes equations


    Chapter 3: Mean squared error


    Chapter 4: Rigid rotor


    Chapter 5: Directional statistics


    Chapter 6: Circular distribution


    Chapter 7: Von Mises distribution


    Chapter 8: Rice distribution


    Chapter 9: Wrapped normal distribution


    Chapter 10: Variance gamma process


    (II) Answering the public top questions about velocity moments.


    (III) Real world examples for the usage of velocity moments 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 Velocity Moments.

    LanguageEnglish
    PublisherOne Billion Knowledgeable
    Release dateMay 4, 2024
    Velocity Moments: Capturing the Dynamics: Insights into Computer Vision

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      Book preview

      Velocity Moments - Fouad Sabry

      Chapter 1: Velocity Moments

      Similar to image moments, velocity moments are weighted averages of the intensities of pixels in a sequence of photographs. However, in addition to defining an object's shape, velocity moments also characterize its mobility across the sequence of images. Moments of velocity can be utilized to aid in the automated identification of a shape in an image when the description of the motion is significant. Currently, there are two accepted versions of velocity moments: Cartesian

      Calculating the Cartesian moment of a single picture

      {\displaystyle m_{pq}=\sum _{x=1}^{M}\sum _{y=1}^{N}x^{p}y^{q}P_{xy}}

      where M and N are the dimensions of the image, {\displaystyle P_{xy}} is the intensity of the pixel at the point (x,y) in the image, and {\displaystyle x^{p}y^{q}} is the basis function.

      These Cartesian moments are the basis for Cartesian velocity moments.

      A Cartesian velocity moment {\displaystyle vm_{pq\mu \gamma }} is defined by

      {\displaystyle vm_{pq\mu \gamma }=\sum _{i=2}^{images}\sum _{x=1}^{M}\sum _{y=1}^{N}U(i,\mu ,\gamma )C(i,p,g)P_{i_{xy}}}

      where M and N are again the dimensions of the image, {\displaystyle images} is the number of images in the sequence, and {\displaystyle P_{i_{xy}}} is the intensity of the pixel at the point (x,y) in image i .

      {\displaystyle C(i,p,q)} is taken from Central moments, added in order to make the equation translation invariant, defined as

      {\displaystyle C(i,p,q)=(x-{\overline {x_{i}}})^{p}(y-{\overline {y_{i}}})^{q}}

      where \overline {x_{i}} is the x coordinate of the centre of mass for image i , and similarly for y .

      {\displaystyle U(i,\mu ,\gamma )} introduces velocity into the equation as

      {\displaystyle U(i,\mu ,\gamma )=({\overline {x_{i}}}-{\overline {x_{i-1}}})^{\mu }({\overline {y_{i}}}-{\overline {y_{i-1}}})^{\gamma }}

      where {\displaystyle {\overline {x_{i-1}}}} is the x coordinate of the centre of mass for the previous image, i-1 , and again similarly for y .

      After calculating the Cartesian velocity moment, it can be normalized by

      {\displaystyle {\overline {vm_{pq\mu \gamma }}}={\frac {vm_{pq\mu \gamma }}{A*I}}}

      where A is the average area of the object, in pixels, and I is the number of images.

      Now the value is independent of the number of photographs in a sequence and the size of the item.

      As both Cartesian moments and Cartesian velocity moments are non-orthogonal, distinct moments can be closely connected. However, these velocity moments offer translation and scale invariance (unless the scale changes within the sequence of images).

      A single image's Zernike moment is computed by

      {\displaystyle A_{mn}={\frac {m+1}{\pi }}\sum _{x}\sum _{y}[V_{mn}(r,\theta )]^{*}P_{xy}}

      where ^{*} denotes the complex conjugate, m is an integer between {\displaystyle 0} and \infty , and n is an integer such that {\displaystyle m-|n|} is even and {\displaystyle |n|

      To determine Zernike moments, the image, A relevant portion of the image is mapped to the unit disc, then {\displaystyle P_{xy}} is the intensity of the pixel at the point (x,y) on the disc and {\displaystyle x^{2}+y^{2}\leq 1} is a restriction on values of x and y .

      The coordinates are then converted to polar form, and r and \theta are the polar coordinates of the point (x,y) on the unit disc map.

      {\displaystyle V_{mn}(r,\theta )} is derived from Zernike polynomials and is defined by

      {\displaystyle V_{mn}(r,\theta )=R_{mn}(r)e^{jn\theta }}{\displaystyle R_{mn}(r)=\sum _{s=0}^{\frac {m-|n|}{2}}(-1)^{s}F(m,n,s,r)}{\displaystyle F(m,n,s,r)={\frac {(m-s)!}{s!({\frac {m+|n|}{2}}-s)!({\frac {m-|n|}{2}}-s)!}}r^{m-2s}}

      Moments of Zernike velocity are based on these Zernike moments.

      A Zernike velocity moment {\displaystyle A_{mn\mu \gamma }} is defined by

      {\displaystyle A_{mn\mu \gamma }={\frac {m+1}{\pi }}\sum _{i=2}^{images}\sum _{x=1}\sum _{y=1}U(i,\mu ,\gamma )[V_{mn}(r,\theta )]^{*}P_{i_{xy}}}

      where {\displaystyle images} is again the number of images in the sequence, and {\displaystyle P_{i_{xy}}} is the intensity of the pixel at the point (x,y) on the unit disc mapped from image i .

      {\displaystyle U(i,\mu ,\gamma )} introduces velocity into the equation in the same way as in the Cartesian velocity moments and {\displaystyle [V_{mn}(r,\theta )]^{*}} is from the Zernike moments equation above.

      Similar to the Cartesian velocity moments, the Zernike velocity moments can be normalized using the same formula.

      {\displaystyle {\overline {A_{mn\mu \gamma }}}={\frac {A_{mn\mu \gamma }}{A*I}}}

      where A is the average area of the object, in pixels, and I is the number of images.

      Due to the fact that Zernike velocity moments are derived from orthogonal Zernike moments, they offer less correlated and more compact descriptions than Cartesian velocity moments. Additionally, Zernike velocity moments offer translation and scale invariance (even when the scale changes within the sequence).

      {End Chapter 1}

      Chapter 2: Navier–Stokes equations

      The Navier–Stokes equations (/nævˈjeɪ stoʊks/ nav-YAY STOHKS) are partial differential equations which describe the motion of viscous fluid substances, Claude-Louis Navier and George Gabriel Stokes were the namesakes. They were both French engineers and physicists.

      They are the culmination of decades of research and incremental theory development, spanning from 1822 (Navier) to 1842-1850 (Stokes).

      Momentum conservation and balancing are quantitatively expressed by the Navier-Stokes equations for Newtonian fluids. An equation of state connecting pressure, temperature, and density is sometimes provided alongside them. Assuming that the stress in the fluid is equal to the product of a diffusing viscous component (proportional to the gradient of velocity) and a pressure term, they result from applying Isaac Newton's second law to fluid motion. The Navier-Stokes equations are similar to the Euler equations, but the Euler equations only model inviscid flow, whereas the Navier-Stokes equations also account for viscosity. With this trade-off in mathematical structure comes the improved analytic features of the Navier-Stokes, which are a parabolic equation (e.g. they are never completely integrable).

      The physics of many phenomena of scientific and engineering interest may be described by the Navier-Stokes equations, making them a useful tool. You may use them to simulate everything from weather to ocean currents to water flow in a conduit to air flow over a wing. Aircraft and automobile design, blood flow research, power plant construction, environmental impact assessment, and many more fields all benefit from the full and simplified Navier-Stokes equations. They can be used to model and analyze magnetohydrodynamics when coupled with Maxwell's equations.

      In a strictly mathematical sense, the Navier-Stokes equations are also of enormous interest. The existence of smooth solutions in three dimensions, that is, solutions that are endlessly differentiable (or even just limited) at all locations in

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