Geometric Transformations,8DOF Transform

Lecture #5Friday January 26, 2018

Classes of Image Transformations• Rigid transformations– Combine rotation and translation– Preserve relative distances and angles– 3 Degrees of freedom

• Similarity transformations– Add scaling to rotation and translation– Preserves relative angles– 4 Degrees of freedom

Affine Transformations• Any transform of the form:!"1=

% & '( ) *0 0 1

,-1

• 6 DOF• Arbitrary combination of– translations– rotations– scales (uniform or non-uniform)– shears

Similarity vs. Affine Matrices• Similarity : 4 DOF

• Affine : 6 DOF

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Specifying Affine Transformations

• There are six unknowns in the matrix (a through f)• If you specify one point in the source image and a

corresponding point in the target image, that yields two equations:

• So providing three point-to-point correspondences specifies an affine matrix

feydxvcbyaxu

iii

iii

++=

++=

Perspective Transformations

• We can go beyond affine transformations.• We can do any perspective transformation of a

one 3D view of a plane to another view.• Therefore, we can model an image as a plane

in space, and project it onto any other image.– How does this differ from the perspective

projection pipeline in CS410?

Perspective Matrix

• Why does element [3,3] = 1?• How many points are needed to specify this matrix?

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Solving for Perspective

• Four corresponding points produce eight equations, eight unknowns --- but we can�t observe w

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Solving (cont.)

• Multiply to get rid of the fraction…

• Now, remember that the u’s, v’s, x’s & y’s are known; group the unknown terms

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Solving (III)

• And express the result as a system of linear equations

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Solving (IV)

• Finally, invert the constant matrix and solve

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Solving (V) : Questions

• Is there always a solution?• Under what conditions is the matrix invertible?• Is the solution always unique?

Perspective Image Transforms (Intuition)• What does the following matrix do?

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Matrix Decomposition

• Note that such decompositions are:– not unique (why?)– difficult to intuit

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More Intuitions

• What will the following matrix do?

• More specifically, what will it do to the giraffe image?

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Check Your Intuitions

• What�s going on here?

More Intuition Checking

• Part of what you are seeing is a scale effect– positive terms in the bottom row create larger w values, and

therefore smaller u, v values

• Something much weirder is also going on:– What happens when y = x+1?– How do you interpret this geometrically?– Isn’t the perspective transform linear?

• So how do you select transformation matrices?

Perspective Transform of 2D Planes Continued.

• Recall the basic equation for the perspective transform

• The only practical way to specify an image transform is by providing four point correspondences

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Computing Transformations

• Remember how to build a transformation from four point correspondences….

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Computing...• So if we want the following mapping:

(0,0)®(0,0), (0,144)®(0,144), (152,0)®(152,50), (152,144)®(152,94)

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What doesThis say about x?

How doesThis alter it?

Remember the earlierWLOG? c = u1,…

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Back to Attention…

• In PA1, you map attention windows to a target output video size– Using an affine transformation• Why affine? Why not rigid? Why not perspective?