# Degrees of freedom

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{bmatrix}

R_x & R_y & R_z \\ T_x & T_y & T_z \end{bmatrix}

</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{bmatrix}

R_x & R_y & R_z \\ T_x & T_y & T_z \\ S \end{bmatrix}

</math>

#### Affine transformations

Include translations, rotations, scales, and/or skewing parameters. They preserve straight lines but necessarily not angles or lengths. Transformation matrixes for affine transformations are as follows:

**9 DOF** transformation matrix which includes scale parameters Sx, Sy and Sz looks as follows

<math> \begin{bmatrix}

R_x & R_y & R_z \\ T_x & T_y & T_z \\ S_x & S_y & S_z \end{bmatrix}

</math>

**12 DOF** transformation matrix which includes both scale and skew parameters. In the matrix, skewing parameters are presented as Skx, Sky, and Skz:

<math> \begin{bmatrix}

R_x & R_y & R_z \\ T_x & T_y & T_z \\ S_x & S_y & S_z \\ S_kx & S_ky & S_kz \end{bmatrix}

</math>

## References

- Lisa Gottesfeld Brown, A survey of image registration techniques (abstract), ACM Computing Surveys (CSUR) archive, Volume 24 , Issue 4, December 1992), Pages: 325 - 376
- Simonson, K., Drescher, S., Tanner, F., A Statistics Based Approach to Binary Image Registration with Uncertainty Analysis. IEEE Pattern Analysis and Machine Intelligence, Vol. 29, No. 1, January 2007
- Domokos, C., Kato, Z., Francos, J., Parametric estimation of affine deformations of binary images. Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, 2008