Multiecho enhancement for 2D/3D (N-D) images

Enhances images using a weighted sum or MEFIC.

This multiecho_enhance function enhances images along the nth dimension using a weighted sum. The purpose of this code is also to demonstrate the equivalence of image space summation and fourier transform maximization (and then inverse transform). A helpful hint is that if you want to weight your images by SNR, the noise measurement is not needed if the noise distribution is similar at each element in the nth dimension. This is true in the case of a multiecho (CPMG or GRME) in MRI. The function help is included below:


multiecho_enhance is a simple function that performs a weighted sum or the Multi Echo Fourier Image Contrast (MEFIC) technique for 2D, 3D or N-D images. The enhancement is taken along the last dimension of the image (e.g., along the echo time for for a multiecho MRI image). For instance, in a 100x100x100x8(echoes) image, the enhancement would be computed along the 8 echoes. The second purpose of this function is to REVEAL that the summation in image space is EQUIVALENT to finding the max of the image in the fourier domain (k-space) and then inverse transformed. This equivalence is a property of the fourier transform. Below are two references where MEFIC was used to enhance a multiecho image:
Enhanced T2 MRH. MRM, 2006 Oct;56(4):717-25.
MEFIC: improved T2 (T2*) weighting. MRM, 1999 Feb;41(2):423-8.

Usage:
img2=multiecho_enhance(img,str,weights)

Arguments:
img - image for enhancement
str - enhancement type 'sum' or 'mefic' (case insensitive)
weights - weights for summing (vector input, e.g., [1 1.2 1.1 2 5]. Must match number of elements in last dimension of the image)

Returns:
img2 - the enhanced image (dimension reduced by 1)

Examples:
img2=multiecho_enhance(img)
img2=multiecho_enhance(img,'sum')
img2=multiecho_enhance(img,'mefic')
img2=multiecho_enhance(img,'sum',[0.1 0.4 0.3 0.8 0.1])

2014/01/23: written by Luke Xie and Russell Dibb

To see an example of multiecho enhancement in the kidney (via MRI gradient multiecho dataset), please refer to the supplemental material available on our CIVMspace: http://www.civm.duhs.duke.edu/lx201210/. A demonstration of multiecho enhancement for MRI phase data can be found in Magnetic susceptibility with multi-echo, MRM, 2012 Jan 2;59(1):297-305 (http://www.sciencedirect.com/science/article/pii/S1053811911007804).

keywords: MRI, post processing, multiecho, multiple echoes, MEFIC, CPMG, GRME, weighted mean, weighted sum, phase/susceptibility enhancement