RESAMPLING OF 2D/3D IMAGES

Grégoire Malandain

Code

A small library (written in C) allowing 2D or 3D image resampling using a 4x4 matrix. The value of a point in the result image is estimated

either by [bi|tri]linear interpolation,
or by using the value of the nearest point.

is available for non-commercial uses only.

Examples obtained with the code

Original grey level image

A 4 by 4 transformation matrix M (with homogenous coordinates) is of the form

M =	R₀₀	R₀₁	R₀₂	T₀
	R₁₀	R₁₁	R₁₂	T₁
	R₂₀	R₂₁	R₂₂	T₂
	0	0	0	1

The R_ij define an affine matrix (rotation in case of rigid transformation), while the T_i define a translation vector. The transformation of a point M = (x,y,z) into M' = (x',y',z') is done using

x'	= M	x
y'		y
z'		z
1		1

which is equivalent to

x'	= R	x	+ T
y'		y
z'		z

The principle of resampling is then to compute, for each point (of integer coordinates) in the result image its correspondant (of real coordinates) in the original image, and to estimate the grey level value of the correspondant. Thus, the transformation from the original image into the result image is the inverse of the given matrix. Consider the matrix

M =	0.8660254	-0.5	0.0	80.5147742
	0.5	0.8660254	0.0	-46.4852258
	0.0	0.0	1.0	1.0
	0	0	0	1

It correspond to a 2D rigid transformation with a rotation of 30 degrees (X axis is from left to right, Y axis is from top to bottom) of center (127,127) (invariant point for the transformation). Using directly this matrix with the resampling program yields a rotation of -30 degrees as shown below.

%test-reech-pnm greg-rmn.pgm greg-rmn-reech1.ppm -mat matrix.example

By inverting the matrix, one may obtained the desired 30 degrees rotation

%test-reech-pnm greg-rmn.pgm greg-rmn-reech2.ppm -mat matrix.example -inv

Interpolation using the nearest value
Command lines as the same as above except that the option -nearest has been added

This option is quite useful when one want to resample labels images (see example in the connected components page).

Initial image

Linear resampling
New grey levels appear at borders

Nearest neighbor resampling
Only the original grey levels are used

Gregoire Malandain

Last modified: Mon Jun 26 10:43:00 MET DST 2000