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# Conversion to world coordinates

#### 3D IMAGE (VOLUME) ORIENTATION AND LOCATION IN SPACE

There are 3 different methods by which continuous coordinates can attached to voxels. The discussion below emphasizes 3D volumes, and

the continuous coordinates are referred to as (x,y,z). The voxel index coordinates (i.e., the array indexes) are referred to as (i,j,k),

with valid ranges:

i = 0 .. dim[1]-1

j = 0 .. dim[2]-1 (if dim[0] >= 2)

k = 0 .. dim[3]-1 (if dim[0] >= 3)

The (x,y,z) coordinates refer to the CENTER of a voxel. In methods 2 and 3, the (x,y,z) axes refer to a subject-based coordinate system,

with +x = Right +y = Anterior +z = Superior.

This is a right-handed coordinate system. However, the exact direction these axes point with respect to the subject depends on qform_code

(Method 2) and sform_code (Method 3).

N.B.: The i index varies most rapidly, j index next, k index slowest. Thus, voxel (i,j,k) is stored starting at location (i + j*dim[1] + k*dim[1]*dim[2]) * (bitpix/8) into the dataset array.

N.B.: The ANALYZE 7.5 coordinate system is +x = Left +y = Anterior +z = Superior which is a left-handed coordinate system. This backwardness is too difficult to tolerate, so this NIFTI-1 standard specifies the coordinate order which is most common in functional neuroimaging.

N.B.: The 3 methods below all give the locations of the voxel centers in the (x,y,z) coordinate system. In many cases, programs will wish to display image data on some other grid. In such a case, the program will need to convert its desired (x,y,z) values into (i,j,k) values in order to extract (or interpolate) the image data. This operation would be done with the inverse transformation to those described below.

N.B.: Method 2 uses a factor 'qfac' which is either -1 or 1; qfac is stored in the otherwise unused pixdim[0]. If pixdim[0]=0.0 (which should not occur), we take qfac=1. Of course, pixdim[0] is only used when reading a NIFTI-1 header, not when reading an ANALYZE 7.5 header.

N.B.: The units of (x,y,z) can be specified using the xyzt_units field.

#### METHOD 1 (the "old" way, used only when qform_code = 0):

The coordinate mapping from (i,j,k) to (x,y,z) is the ANALYZE 7.5 way. This is a simple scaling relationship:

x = pixdim[1] * i

y = pixdim[2] * j

z = pixdim[3] * k

No particular spatial orientation is attached to these (x,y,z) coordinates. (NIFTI-1 does not have the ANALYZE 7.5 orient field, which is not general and is often not set properly.) This method is not recommended, and is present mainly for compatibility with ANALYZE 7.5 files.

#### METHOD 2 (used when qform_code > 0, which should be the "normal case):

The (x,y,z) coordinates are given by the pixdim[] scales, a rotation matrix, and a shift. This method is intended to represent "scanner-anatomical" coordinates, which are often embedded in the image header (e.g., DICOM fields (0020,0032), (0020,0037), (0028,0030), and (0018,0050)), and represent the nominal orientation and location of the data. This method can also be used to represent "aligned" coordinates, which would typically result from some post-acquisition alignment of the volume to a standard orientation (e.g., the same subject on another day, or a rigid rotation to true anatomical orientation from the tilted position of the subject in the scanner). The formula for (x,y,z) in terms of header parameters and (i,j,k) is:

[ x ] [ R11 R12 R13 ] [ pixdim[1] * i ] [ qoffset_x ]

[ y ] = [ R21 R22 R23 ] [ pixdim[2] * j ] + [ qoffset_y ]

[ z ] [ R31 R32 R33 ] [ qfac * pixdim[3] * k ] [ qoffset_z ]

The qoffset_* shifts are in the NIFTI-1 header. Note that the center of the (i,j,k)=(0,0,0) voxel (first value in the dataset array) is just (x,y,z) = (qoffset_x,qoffset_y,qoffset_z).

The rotation matrix R is calculated from the quatern_* parameters. This calculation is described below.

The scaling factor qfac is either 1 or -1. The rotation matrix R defined by the quaternion parameters is "proper" (has determinant 1). This may not fit the needs of the data; for example, if the image grid is

i increases from Left-to-Right

j increases from Anterior-to-Posterior

k increases from Inferior-to-Superior

Then (i,j,k) is a left-handed triple. In this example, if qfac=1, the R matrix would have to be

[ 1 0 0 ]

[ 0 -1 0 ] which is "improper" (determinant = -1).

[ 0 0 1 ]

If we set qfac=-1, then the R matrix would be

[ 1 0 0 ]

[ 0 -1 0 ] which is proper.

[ 0 0 -1 ]

This R matrix is represented by quaternion [a,b,c,d] = [0,1,0,0] (which encodes a 180 degree rotation about the x-axis).

#### METHOD 3 (used when sform_code > 0):

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

x = srow_x[0] * i + srow_x[1] * j + srow_x[2] * k + srow_x[3]

y = srow_y[0] * i + srow_y[1] * j + srow_y[2] * k + srow_y[3]

z = srow_z[0] * i + srow_z[1] * j + srow_z[2] * k + srow_z[3]

The srow_* vectors are in the NIFTI_1 header. Note that no use is made of pixdim[] in this method.

#### 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. Image display software can use either transform,

depending on its purposes and needs.

In Method 2, the origin of coordinates would generally be whatever the scanner origin is; for example, in MRI, (0,0,0) is the center of the gradient coil.

In Method 3, the origin of coordinates would depend on the value of sform_code; for example, for the Talairach coordinate system, (0,0,0) corresponds to the Anterior Commissure.

Information from nifti1.h by RW Cox, NIH