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ORIENTATION IN MEDICAL IMAGING¶

![Coordinate Systems][Coordinate Systems]

World Coordinate System:¶

Coordinate system in which the patient is positionned in the PET, CT or MR

Anatomical Coordinate System:¶

Axial plane is parallel to the groud and separates the head (Superior) from the feet (Inferior). It corresponds with the $z$ axis in the world coordinate system.
Coronal plane is perpendicular to the ground and separates the front (Anterior) from the back (Posterior). It corresponds with the $y$ axis world coordinate system.
Sagittal plane separates the left from the right. It corresponds with the $x$ axis world coordinate system.

Most common definitions are:

LPS (Left, Posterior, Superior): used in DICOM format and by (Simple)ITK $$ LPS = \begin{Bmatrix} \text{from right towards left} \\ \text{from anterior towards posterior} \\ \text{from inferior towards superior} \end{Bmatrix} $$
RAS (Right, Anterior, Superior) or Neurological convention: used by NIfTI format and by NiBabel and SPM $$ RAS = \begin{Bmatrix} \text{from left towards right} \\ \text{from posterior towards anterior} \\ \text{from inferior towards superior} \end{Bmatrix} $$
LAS (Left, Anterior, Superoir) or Radiological convetion: used by ANALYZE format $$ LAS = \begin{Bmatrix} \text{from right towards left} \\ \text{from posterior towards anterior} \\ \text{from inferior towards superior} \end{Bmatrix} $$

Image Coordinate System¶

The image coordinate system describes how an image was acquired with respect to the anatomy. Medical scanners create regular, rectangular arrays of points and cells which start at the upper left corner. The i axis increases to the right, the j axis to the bottom and the k axis backwards.

The file standard assumes that the voxel coordinates refer to the center of each voxel, rather than at any of its corners.

In addition to the intensity value of each voxel $(i,j,k)$ the origin and spacing of the anatomical coordinates are stored too.

The origin of coordinates represents the position of the first voxel (0,0,0) in the anatomical coordinate system, e.g. (100mm, 50mm, -25mm)

The spacing specifies the distance between voxels along each axis, e.g. (1.5mm, 0.5mm, 0.5mm)

Image Transformation¶

The transformation from an image space vector to an anatomical space vector is an affine transformation, consists of a linear transformation followed by a translation.

Quarternions:¶

The orientation of the $(x,y,z)$ axes relative to the $(i,j,k)$ axes in the image space is specified using a unit quaternion $[a,b,c,d]$, where $a*a+b*b+c*c+d*d=1$. The $(b,c,d)$ values are all that is needed, since it is required that $a = \sqrt{1.0-(b*b+c*c+d*d)}$ be non negative. The $(b,c,d)$ values are stored in the $(quatern_b,quatern_c,quatern_d)$ fields.

The Purpose of the qform and sform in NIfTI files¶

The qform and sform stored in a NIfTI file header are intended to fulfil the following functions:

specify the handedness of the coordinate system (important for getting left and right correct)
specify the original scanner coordinates (qform only)
specify standard space coordinates (sform only)
specify a relationship with another image's coordinates (sform only)

The qform and sform should never specify coordinates with different handedness (i.e. have determinants of different signs). Otherwise it is not possible to tell left from right.

According to NIfTI-1 Standard [1], the qform code should be set to either unknown (0) or scanner_anat (1). While, The sform code should be set to either unknown (0), aligned_anat (2), tailarach (3) or mni_152 (4).

Orientation information¶

Name	Code	Description
unknown	0	Arbitrary coordinates.
scanner_anat	1	Scanner-based anatomical coordinates.
aligned_anat	2	Coordinates aligned to another file, or to the “truth” (with an arbitrary coordinate center).
talairach	3	Coordinates aligned to the Talairach space.
mni_152	4	Coordinates aligned to the mni space.

Method 1 (The "old" way)¶

Form	Code
qform_code	0
sform_code	Not used

The coordinate mapping from $(i,j,k)$ to $(x,y,z)$ is the ANALYZE 7.5 way. This is a simple scaling relationship: $$ \left[ \begin{array}{c} x\\ y\\ z \end{array} \right]= \left[ \begin{array}{c} i\\ j\\ k \end{array} \right]\odot \left[ \begin{array}{c} \mathtt{pixdim[1]}\\ \mathtt{pixdim[2]}\\ \mathtt{pixdim[3]}\\ \end{array} \right] $$

No particular spatial orientation is attached to these $(x,y,z)$ coordinates. This method is not recommended, and is present mainly for compatibility with ANALYZE 7.5 files.

Method 2: qform_code > 0 (Recommended)¶

It is intended to be used to indicate the scanner coordinates, in a way that resembles the coordinates specified in the DICOM header. It can also be used to represent the alignment of an image to a previous session of the same subject (such as for coregistration).

$$ \mathbf{R} = \left[ \begin{array}{ccc} a^2+b^2-c^2-d^2 & 2(bc-ad) & 2(bd+ac) \\ 2(bc+ad) & a^2+c^2-b^2-d^2 & 2(cd-ab) \\ 2(bd-ac) & 2(cd+ab) & a^2+d^2-b^2-c^2 \end{array} \right] $$ $$ \left[ \begin{array}{c} x\\ y\\ z \end{array} \right]=\mathbf{R} \left[ \begin{array}{c} i\\ j\\ q\cdot k\\ \end{array} \right]\odot \left[ \begin{array}{c} \mathtt{pixdim[1]}\\ \mathtt{pixdim[2]}\\ \mathtt{pixdim[3]}\\ \end{array} \right]+ \left[ \begin{array}{c} \mathtt{qoffset\_x}\\ \mathtt{qoffset\_y}\\ \mathtt{qoffset\_z}\\ \end{array} \right] $$

Method 3: sform_code > 0¶

The $(x,y,z)$ coordinates are given by a general affine transformation of the $(i,j,k)$ indexes:

$$ \left[ \begin{array}{c} x\\ y\\ z\\ 1 \end{array} \right]=\left[ \begin{array}{cccc} \mathtt{srow\_x[0]} & \mathtt{srow\_x[1]} & \mathtt{srow\_x[2]} & \mathtt{srow\_x[3]}\\ \mathtt{srow\_y[0]} & \mathtt{srow\_y[1]} & \mathtt{srow\_y[2]} & \mathtt{srow\_y[3]}\\ \mathtt{srow\_z[0]} & \mathtt{srow\_z[1]} & \mathtt{srow\_z[2]} & \mathtt{srow\_z[3]} \\ 0 & 0 & 0 & 1\end{array} \right]\cdot\left[ \begin{array}{c} i\\ j\\ k\\ 1 \end{array} \right] $$

Why 3 Methods?¶

Method 1 is provided only for backwards compatibility. The intention is that Method 2 (qform_code > 0) represents the nominal voxel locations as reported by the scanner, or as rotated to some fiducial orientation and location. Method 3, if present (sform_code > 0), is to be used to give the location of the voxels in some standard space. The sform_code indicates which standard space is present. Both methods 2 and 3 can be present, and be useful in different contexts (method 2 for displaying the data on its original grid; method 3 for displaying it on a standard grid).

In this scheme, a dataset would originally be set up so that the Method 2 coordinates represent what the scanner reported. Later, a registration to some standard space can be computed and inserted in the header.

Main Software packages¶

Row- and column-major order¶

Please, be aware that different programming languages or libraries store the data in different order. This might be of importance, for example, when reading raw data (e.g. Volumes-of_interest created in ACCURATE) or when multiple libreries are used (SimpleITK and NiBabel)

Programming languages and libraries	Row-major	Column-major
Python: SimpleITK	x
Python: Numpy	x
C/C++	x
Matlab		x
ACCURATE

Transposition of the data solved the differences.

ITK & SimpleITK¶

ITK and SimpleITK use the LPS convention. After reading a file, the data matrix is stored as $(z,y,x)$. However, when writing a NIfTI file they convert the data matrix to RAS.

NiBabel¶

NiBabel uses the RAS convention. After reading a file, the data matrix is stored as $(x,y,z)$.

LPS (DICOM) to RAS (NIfTI)¶

DICOM's coordinate system is 180 degrees rotated about the z-axis from the neuroscience/NIFTI coordinate system. To transform between DICOM and NIFTI, you just have to negate the x- and y-coordinates.

(Simple)ITK converts the data matrix to RAS when writing the NIfTI

Further details in:

[//]: # (Links & Images) [Coordinate Systems]: images/Coordinate_systems.png [1]: https://nifti.nimh.nih.gov/

Python: NiBabel¶

For this example we will use a NIfTI file and the NiBabel library

In [2]:

!cls
# Libraries
import os
import copy
import numpy as np
import nibabel as nib
import matplotlib.pyplot as plt

# Input data
cwd = os.getcwd()
os.chdir(cwd)

Neurological Convention RAS (RL)¶

In [3]:

filename = os.path.join(cwd,'data','nifti1','avg152T1_RL_nifti.nii.gz') # Neurological Convention RAS
# Load file with NiBabel
img      = nib.load(filename)

# Load the data matrix
#img_data = img.get_data()  # uint8
img_data = img.get_fdata() # float32

# Display data matrix
display('Raw data matrix:')
plt.imshow(img_data[:,:,50].T, cmap="gray", origin="lower")  # Data needs to be transposed for visualization
plt.show()

# Orientation information
print('qform_code:', img.header['qform_code']) # 0: unknown
print('qform matrix: \n', img.get_qform(), '\n')

print('sform_code:', img.header['sform_code']) # 4: mni_152
print('sform matrix: \n', img.get_sform(), '\n')

'Raw data matrix:'

No description has been provided for this image

qform_code: 0
qform matrix: 
 [[2. 0. 0. 0.]
 [0. 2. 0. 0.]
 [0. 0. 2. 0.]
 [0. 0. 0. 1.]] 

sform_code: 4
sform matrix: 
 [[   2.    0.    0.  -90.]
 [   0.    2.    0. -126.]
 [   0.    0.    2.  -72.]
 [   0.    0.    0.    1.]]

Radiological Convention LAS (LR)¶

In [4]:

filename = os.path.join(cwd,'data','nifti1','avg152T1_LR_nifti.nii.gz') # Radiological Convention LAS
# Load file with NiBabel
img      = nib.load(filename)

# Load the data matrix
#img_data = img.get_data()  # uint8
img_data = img.get_fdata() # float32

# Display data matrix
display('Raw data matrix:')
plt.imshow(img_data[:,:,50].T, cmap="gray", origin="lower")  # Data needs to be transposed for visualization
plt.show()

# Orientation information
print('qform_code:', img.header['qform_code']) # 0: unknown
print('qform matrix: \n', img.get_qform(), '\n')

print('sform_code:', img.header['sform_code']) # 4: mni_152
print('sform matrix: \n', img.get_sform(), '\n')

'Raw data matrix:'

qform_code: 0
qform matrix: 
 [[2. 0. 0. 0.]
 [0. 2. 0. 0.]
 [0. 0. 2. 0.]
 [0. 0. 0. 1.]] 

sform_code: 4
sform matrix: 
 [[  -2.    0.    0.   90.]
 [   0.    2.    0. -126.]
 [   0.    0.    2.  -72.]
 [   0.    0.    0.    1.]]

NiBabel always uses RAS output space

Conversion of DICOM to NIfTI¶