An image is a 2D or 3D array of pixels, each containing a grey value (typically in 0..255).
Give two images, image registration finds the transformation of one image so that it "best" aligns with the other image.
Image registration can be intra-modal, where both images are acquired with the same modality (e.g. x-ray, CT, MR, PET, US), or can be inter-modal, where the images are acquired with different modalities.
Image registration can be
Any image registration method needs:
We will consider only 2D/2D image registration for the moment.
Given two images, $I_1$ and $I_2$, where one of the images has already been transformed to overlap the other image, a similarity measure,
$S( I_1, I_2 )$
determines how "well" the images align.
Some similarity measures are
RMS Error just measures the squared difference between corresponding pixels:
$S(I_1,I_2) = \sqrt{ {1 \over N} \sum_{ij} (I_1(i,j) - I_2(i,j) )^2 }$
This is a poor measure, as it requires that the two images be of the same modality and captured with the same imaging parameters.
The cross-correlation is the product of corresponding pixels:
$S(I_1,I_2) = \sum_{ij} I_1(i,j)\ I_2(i,j) $
This is really a dot product, where the pixel values of each image are laid out in a one-dimensional "image vector", and the dot product of the two image vectors is computed.
The dot product is maximized when the vectors are parallel. Similary, the correlation is maximized when each pixel of one image is a scalar multiple of the corresponding pixel of the other image ... that is, when the images are perfectly aligned.
This measure is also known as the Pearson correlation coefficient.
NCC determines the correlation of the two images, with each image normalized to so that its mean pixel value is zero and its standard deviation is one:
$S(I_1,I_2) = \large \sum_{ij} ({ I_1(i,j) - \mu_1 \over \sigma_1 }) ({ I_2(i,j) - \mu_2 \over \sigma_2 } )$
where the $\mu$ and $\sigma$ are the means and standard deviations of their respective images.
This is a better measure, as it doesn't matter if the images have different brightnesses (which is corrected by the subtraction of the mean) or different contrasts (which is corrected by the division by the standard deviation). However, there's an expectation that the pixels of the two images have the same normalized values at optimal alignment. This will not be true if the images come from different modalities.
Mutual Information (MI) is a very commonly-used similarity measure. But we need to discuss joint histograms and entropy before considering MI.
Those topics will be discussed in the next two lectures.