#### Summary of Velocity Kinematics in 2D

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We summarise the important points from this masterclass.

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We summarise the important points from this masterclass.

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We learn how to describe the orientation of an object by a 2×2 rotation matrix which has some special properties. Try your hand at some online MATLAB problems. You’ll need to watch all the 2D “Spatial Maths” lessons to complete the problem set.

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Since an image in MATLAB is just a matrix of numbers, we could write code to fill in the elements of the matrix. Let’s look at some simple examples such as squares, circles and lines and more complex images formed by pasting these shapes together.

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The orientation of a body in 3D can be described by three angles, examples of which are Euler angles and roll-pitch-yaw angles. Note that in the MATLAB example at 8:24 note that recent versions of the Robotics Toolbox (9.11, 10.x) give a different result: >> rpy2r(0.1,0.2,0.3)ans = 0.9363 -0.2751 0.2184 0.2896 0.9564 -0.0370 -0.1987 0.0978 […]

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We learn how to describe the 2D pose of an object by a 3×3 homogeneous transformation matrix which has a special structure. Try your hand at some online MATLAB problems. You’ll need to watch all the 2D “Spatial Maths” lessons to complete the problem set.

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The image Jacobian depends not only on the image plane coordinates but also the distance from the camera to the points of interest. If this distance is not known, what can we do? Let’s look at how we can determine this distance, and how the optical flow equation can be rearranged to convert from observed […]

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We can describe the relationship between a 3D world point and a 2D image plane point, both expressed in homogeneous coordinates, using a linear transformation – a 3×4 matrix. Then we can extend this to account for an image plane which is a regular grid of discrete pixels.

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Images contain many pixels and the normal way to process them is with nested for loops that index each pixel in turn. This is slow and somewhat cumbersome to write. MATLAB has a facility called vectorization that allows us to perform complex matrix operations without any loops.

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Let’s recap some of the most important topics we’ve covered about treating an image as a matrix within MATLAB which we can display or index into.

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We can also derive a Jacobian which relates the velocity of a point, defined relative to one coordinate frame, to the velocity relative to a different coordinate frame.