2.image geometry

8/11/2019 2.Image Geometry

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World everything is in 3D [X,Y,Z]

Lose Z when projected on to 2D detector Methods to recover Z is called as SHAPE

from X where X stands for Stereo, Shaping,

Focus, Defocus etc ..


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Some Basic Transformation

TranslationScalingRotation

Perspective Transformation


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Translation

Let (X1,Y1,Z1) displaced to new location givenby


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Scaling


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Rotation

Rotation around Z theta/ gamma Rotation around X Phi / alpha Rotation around Y beta


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Rotation of a point around Z axis


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Angles measured counter clockwise


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Clock wise direction


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Can you try the ROTATION MATRIX for X and Ydirections [ consider clock wise]


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Perspective Transformation

Also known as Imaging Transformationprojects a 3D point onto a 2D plane.

Here X,Y,Z represent the world coordinatesand x,y,z represent the camera coordinates.


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X,Y,Z maps to x, y, by considering the two

equivalent triangles


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If origin moved to the image plane


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To represent the camera coordinates in a

Matrix form as in case of Translation, Rotationetc.. we need define perspective matrix ,which is accomplished by converting theworld coordinates (Cartesian) to homogenous

world coordinates


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We define Perspective Transformation


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Cartesian or Euclidean space parallel linesDONT intersect.

in projective space the tracks in the picturebecomes narrower while it moves far awayfrom eyes. Finally, the two parallel rails meetat the horizon, which is a point at infinity.


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The Cartesian coordinates of a 2D point can be

expressed as ( x, y). What if this point goes far away to infinity? The point at infinity would be (,), and it

becomes meaningless in Euclidean space. The parallel lines should meet at infinity in

projective space, but cannot do in Euclidean

space. Mathematicians have discovered a way to solve

this issue.


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Homogeneous coordinates,

introduced by August Ferdinand Mbius,make calculations of graphics and geometrypossible in projective space.

Homogeneous coordinates are a way ofrepresenting N-dimensional coordinates withN+1 numbers.


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To make 2D Homogeneous coordinates, simplyadd an additional variable, w, into existingcoordinates.

Therefore, a point in Cartesian coordinates, (X, Y)becomes (x, y, w) in Homogeneous coordinates. And X and Y in Cartesian are re-expressed with x,

y and w in Homogeneous as; X = x/w Y = y/w


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2.image geometry

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