Motion Field: Exploring the Dynamics of Computer Vision: Motion Field Unveiled

By Fouad Sabry

What is Motion Field

In computer vision, the motion field is an ideal representation of motion in three-dimensional space (3D) as it is projected onto a camera image. Given a simplified camera model, each point  in the image is the projection of some point in the 3D scene but the position of the projection of a fixed point in space can vary with time. The motion field can formally be defined as the time derivative of the image position of all image points given that they correspond to fixed 3D points. This means that the motion field can be represented as a function which maps image coordinates to a 2-dimensional vector. The motion field is an ideal description of the projected 3D motion in the sense that it can be formally defined but in practice it is normally only possible to determine an approximation of the motion field from the image data.

How you will benefit

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

Chapter 1: Motion Field

Chapter 2: Chain Rule

Chapter 3: Curl (Mathematics)

Chapter 4: Polar Coordinate System

Chapter 5: Green's Theorem

Chapter 6: Line Element

Chapter 7: Camera Matrix

Chapter 8: Pinhole Camera Model

Chapter 9: Derivation of the Navier-Stokes Equations

Chapter 10: Relativistic Lagrangian Mechanics

(II) Answering the public top questions about motion field.

(III) Real world examples for the usage of motion field 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 Motion Field.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: May 12, 2024

Book preview

Chapter 1: Motion field

The motion field is the perfect projection of 3D motion onto a camera's image, and it is used extensively in computer vision.

Assuming a minimally complex camera model,, each point (y_{{1}},y_{{2}}) in the image is the projection of some point in the 3D scene but the position of the projection of a fixed point in space can vary with time.

Given that all image points correspond to fixed 3D coordinates, the motion field can be formally defined as the time derivative of the image position of all image points.

The motion field can then be expressed as a function that transforms image coordinates into a two-dimensional vector.

Formally speaking, the motion field is the perfect description of the projected 3D motion, but in practice, it is usually only possible to estimate the motion field from the image data.

A camera model maps each point (x_{{1}},x_{{2}},x_{{3}}) in 3D space to a 2D image point (y_{{1}},y_{{2}}) according to some mapping functions {\displaystyle m_{1},m_{2}} :

{\displaystyle {\begin{pmatrix}y_{1}\\y_{2}\end{pmatrix}}={\begin{pmatrix}m_{1}(x_{1},x_{2},x_{3})\\m_{2}(x_{1},x_{2},x_{3})\end{pmatrix}}}

For a dynamic scene, where objects are in motion with respect to one another, where objects undergo deformation, and where the camera itself is in motion with respect to the scene, a fixed point in 3D space is mapped to different places in the image. The result of time-differentiating the previous expression is

{\displaystyle {\begin{pmatrix}{\frac {dy_{1}}{dt}}\\[2mm]{\frac {dy_{2}}{dt}}\end{pmatrix}}={\begin{pmatrix}{\frac {dm_{1}(x_{1},x_{2},x_{3})}{dt}}\\[2mm]{\frac {dm_{2}(x_{1},x_{2},x_{3})}{dt}}\end{pmatrix}}={\begin{pmatrix}{\frac {dm_{1}}{dx_{1}}}&{\frac {dm_{1}}{dx_{2}}}&{\frac {dm_{1}}{dx_{3}}}\\[2mm]{\frac {dm_{2}}{dx_{1}}}&{\frac {dm_{2}}{dx_{2}}}&{\frac {dm_{2}}{dx_{3}}}\end{pmatrix}}\,{\begin{pmatrix}{\frac {dx_{1}}{dt}}\\[2mm]{\frac {dx_{2}}{dt}}\\[2mm]{\frac {dx_{3}}{dt}}\end{pmatrix}}}

Here

{\displaystyle \mathbf {u} ={\begin{pmatrix}{\frac {dy_{1}}{dt}}\\[2mm]{\frac {dy_{2}}{dt}}\end{pmatrix}}}

is the motion field and the vector u is dependent both on the image position (y_{{1}},y_{{2}}) as well as on the time t.

Similarly, {\displaystyle \mathbf {x'} ={\begin{pmatrix}{\frac {dx_{1}}{dt}}\\[2mm]{\frac {dx_{2}}{dt}}\\[2mm]{\frac {dx_{3}}{dt}}\end{pmatrix}}}

represents the 3D point's motion in relation to the motion field, and

{\displaystyle \mathbf {u} =\mathbf {M} \,\mathbf {x} '}

where {\mathbf {M}} is the image position dependent 2\times 3 matrix

{\displaystyle \mathbf {M} ={\begin{pmatrix}{\frac {dm_{1}}{dx_{1}}}&{\frac {dm_{1}}{dx_{2}}}&{\frac {dm_{1}}{dx_{3}}}\\[2mm]{\frac {dm_{2}}{dx_{1}}}&{\frac {dm_{2}}{dx_{2}}}&{\frac {dm_{2}}{dx_{3}}}\end{pmatrix}}}

According to this connection, the motion field, at a fixed location in the image, is invariant to 3D motions which lies in the null space of {\mathbf {M}} .

For example, All motion components in the 3D space that are perpendicular to the camera's focal point will be missed by a pinhole camera.

The motion field \mathbf {v} is defined as:

{\displaystyle \mathbf {v} =f{\frac {Z\mathbf {V} -V_{z}\mathbf {P} }{Z^{2}}}}

where

{\displaystyle \mathbf {V} =-\mathbf {T} -\mathbf {\omega } \times \mathbf {P} } .

where

\mathbf {P} is a point in the scene where Z is the distance to that scene point.

\mathbf {V} is the relative motion between the camera and the scene, \mathbf {T} is the translational component of the motion, and

\mathbf {\omega } is the angular velocity of the motion.

As explained above, the motion field is a theoretical creation based on the assumption that the direction of motion for each pixel in an image can be determined. However, in reality, measurements on picture data can only serve as an approximation of the underlying motion field. The issue is that, in most circumstances, the mobility of each image point is unique and must be assessed locally using some sort of neighborhood operation on the image data. Therefore, for some types of neighborhoods, an approximation, generally referred to as the optical flow, must be utilized in place of the true motion field. A neighborhood with a constant intensity, for instance, may correspond to a motion field that is not zero, but the optical flow in such a region would be zero because no local image motion would be measurable. Similarly, the optical flow can only record the normal component of a motion field, even if a neighborhood that is intrinsically 1-dimensional (such as an edge or line) can correspond to any motion field. Image noise, 3D occlusion, and temporal aliasing are all factors that arise naturally in any optical flow measurement technique and lead to discrepancies between the measured and actual motion fields.

To sum up, the motion field is an approximation of the optical flow because it is impossible to measure the motion field precisely for all picture locations. The optical flow can be calculated in a number of different ways, each of which takes into account a unique set of criteria for determining the accuracy of an optical estimate.

{End Chapter 1}

Chapter 2: Chain rule

In calculus, The chain rule is a mathematical expression that may be used to define the derivative of the combination of two differentiable functions, f and g, in terms of the derivatives of f and g.

More precisely, if {\displaystyle h=f\circ g} is the function such that {\displaystyle h(x)=f(g(x))} for every x, if this is the case, the chain rule is, using the notation of Lagrange, {\displaystyle h'(x)=f'(g(x))g'(x).}

or, equivalently,

{\displaystyle h'=(f\circ g)'=(f'\circ g)\cdot g'.}

The chain rule may also be stated using the notation used by Leibniz. If one variable, z, is dependent on another variable, y, and both of those variables are dependent on a third variable, x (that is, y and z are dependent variables), then z is also reliant on x via the use of the variable y as an intermediary. In this particular instance, the chain rule is stated as

{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},}

and

{\displaystyle \left.{\frac {dz}{dx}}\right|_{x}=\left.{\frac {dz}{dy}}\right|_{y(x)}\cdot \left.{\frac {dy}{dx}}\right|_{x},}

for the purpose of signaling at what points the derivatives need to have their values assessed.

When it comes to integration, the replacement rule is the one that corresponds to the chain rule.

Intuitively, the chain rule states that if one knows the instantaneous rate of change of z relative to y and that of y relative to x, then they are able to calculate the instantaneous rate of change of z relative to x as the product of the two rates of change. If one knows the instantaneous rate of change of z relative to x, then they are able to calculate the instantaneous rate of change of z relative.

In the words of George F.

If a vehicle can move twice as quickly as a bicycle, and a bicycle can travel four times as fast as a person walking, then Simmons says this:, then the car travels 2 × 4 = 8 times as fast as the man."

The following describes the connection between this particular illustration and the chain rule:.

Let z, y and x be the (variable) positions of the car, the bicycle, and the guy who was walking, respectively.

The rate of change of relative positions of the car and the bicycle is {\textstyle {\frac {dz}{dy}}=2.} Similarly, {\textstyle {\frac {dy}{dx}}=4.} So, the pace at which the relative locations of the automobile and the walking guy are changing is

{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}=2\cdot 4=8.}

The rate of change in position is equal to the ratio of the speeds,