Difference between revisions of "Degrees of freedom"
m (Created page with 'The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom (DOF). In this guide, DOF are given for 3D images. == B...') |
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In general, the degrees of freedom of an estimate is equal to the number of independent scores that go into the estimate minus the number of parameters estimated as intermediate steps in the estimation of the parameter itself. In image registration, a transformation matrix establishes geometrical correspondence between coordinate systems of different images. It is used to transform one image into the space of the other. | In general, the degrees of freedom of an estimate is equal to the number of independent scores that go into the estimate minus the number of parameters estimated as intermediate steps in the estimation of the parameter itself. In image registration, a transformation matrix establishes geometrical correspondence between coordinate systems of different images. It is used to transform one image into the space of the other. | ||
| − | === Transformations | + | === Transformations generally used in biomedical imaging === |
==== Rigid-body transformations ==== | ==== Rigid-body transformations ==== | ||
| − | Rigid-body transformations include translations and rotations. They preserve all lengths and angles. These are 6 DOF transformations, and the transformation matrix is as follows: | + | Rigid-body transformations include translations and rotations. They preserve all lengths and angles. These are '''6 DOF''' transformations, and the transformation matrix is as follows: |
<math> | <math> | ||
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R_x & R_y & R_z \\ | R_x & R_y & R_z \\ | ||
T_x & T_y & T_z | T_x & T_y & T_z | ||
| + | \end{matrix} | ||
| + | </math> | ||
| + | |||
| + | ==== Global rescale transformations ==== | ||
| + | |||
| + | Include translations, rotations, and a single scale parameter S=Sx=Sy=Sz. They preserve all angles and relative lengths. These are '''7DOF''' transformations and the transformation matrix is as follows: | ||
| + | |||
| + | <math> | ||
| + | \begin{matrix} | ||
| + | R_x & R_y & R_z \\ | ||
| + | T_x & T_y & T_z \\ | ||
| + | S | ||
\end{matrix} | \end{matrix} | ||
</math> | </math> | ||
Revision as of 18:48, 2 May 2012
The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom (DOF). In this guide, DOF are given for 3D images.
Contents
Basics
In general, the degrees of freedom of an estimate is equal to the number of independent scores that go into the estimate minus the number of parameters estimated as intermediate steps in the estimation of the parameter itself. In image registration, a transformation matrix establishes geometrical correspondence between coordinate systems of different images. It is used to transform one image into the space of the other.
Transformations generally used in biomedical imaging
Rigid-body transformations
Rigid-body transformations include translations and rotations. They preserve all lengths and angles. These are 6 DOF transformations, and the transformation matrix is as follows:
<math> \begin{matrix}
R_x & R_y & R_z \\
T_x & T_y & T_z
\end{matrix}
</math>
Global rescale transformations
Include translations, rotations, and a single scale parameter S=Sx=Sy=Sz. They preserve all angles and relative lengths. These are 7DOF transformations and the transformation matrix is as follows:
<math> \begin{matrix}
R_x & R_y & R_z \\
T_x & T_y & T_z \\
S
\end{matrix}
</math>
