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Mutual Informa-on Algorithm applied to Rigid Registra-on
Vibha Chaswal, Ph.D.
Registra-on
• Image registra-on – Define geometric transforma-ons T that will map co-‐ordinats between one image onto another image such that some image quality criterion is maximized.
– also referred to as image fusion, superimposi-on, matching or merge
Registra-on algorithms
• Used to find the transforma-on • Rigid & affine – Landmark based – Edge based – Voxel intensity based – Informa,on theory based
• Non-‐rigid – Registra-on using basis func-ons – Registra-on using splines – Physics based
• Elas-c, Fluid, Op-cal flow, etc.
Rigid Body Registra-on of medical images
• The anatomical and pathological structures do not deform during image acquisi-ons
• Tissue deforma-ons ignored and register images using rigid body transforms
• only rota-ons and transla-ons • 6 degrees of freedom: 3 transla-ons and 3 rota-ons
• Key Characteris-c: All distances are preserved
3D Rigid-‐body Transforma-ons
• A 3D rigid body transform is defined by: – 3 transla-ons -‐ in X, Y & Z direc-ons – 3 rota-ons -‐ about X, Y & Z axes
• The order of the opera-ons maZers
Transla-ons Pitch about x axis
Roll about y axis
Yaw about z axis
Informa-on theory based Rigid body Registra-on
• Image registra-on is considered as to maximize the amount of shared informa-on in two images – reducing the amount of informa-on in the combined
image
• Algorithms used – Joint entropy
• Joint entropy measures the amount of informa-on in the two images combined
– Mutual informa,on • A measure of how well one image explains the other, and is maximized at the op,mal alignment
– Normalized Mutual Informa-on
Measures of Informa-on • Hartley defined the first informa-on measure:
– H = n log s – n is the length of the message and s is the number of
possible values for each symbol in the message – Assumes all symbols equally likely to occur
• Shannon proposed variant (Shannon’s Entropy)
• weighs the informa-on based on the probability that an outcome will occur
• second term shows the amount of informa-on an event provides is inversely propor-onal to its probability of occurring
Three Interpreta-ons of Entropy
• The amount of informa-on an event provides – An infrequently occurring event provides more informa-on than a frequently occurring event
• The uncertainty in the outcome of an event – Systems with one very common event have less entropy than systems with many equally probable events
• The dispersion in the probability distribu-on – An image of a single amplitude has a less disperse histogram than an image of many greyscales • the lower dispersion implies lower entropy
Joint Entropy for Image Registra-on
• Define a joint probability distribu-on: – Generate a 2-‐D histogram where each axis is the number of possible greyscale values in each image
– each histogram cell is incremented each -me a pair (I_1(x,y), I_2(x,y)) occurs in the pair of images • If the images are perfectly aligned then the histogram is highly focused. As the images mis-‐align the dispersion grows
• recall Entropy is a measure of histogram dispersion
Entropy for Image Registra-on
• Using joint entropy for registra-on – Define joint entropy to be:
– Images are registered when one is transformed rela-ve to the other to minimize the joint entropy
– The dispersion in the joint histogram is thus minimized
Joint entropy: overlap problem
• Joint entropy very sensi-ve to mapping of posi-on and intensity
• ‘blur’ with increasing misregistra-on
• May lead to incorrect solu-on
aligned 2mm 5mm
MR/MR
MR/CT
MR/PET Figure from Hill et.al., Voxel Similarity measures for automated image registra3on, 1994, Proc. SPIE, 2359
Solu-on: Mutual Informa-on
A solu-on to the overlap problem from which joint entropy suffers is to consider the informa-on contributed to the overlapping volume by each image being registered together with the joint informa-on. The informa-on contributed by the individual images is simply the entropy of the por-on of the image that overlaps with the other image volume
Defini-ons of Mutual Informa-on • Three commonly used defini-ons: – 1) MI(A,B) = H(B) -‐ H(B|A) = H(A) -‐ H(A|B)
• Mutual informa-on is the amount that the uncertainty in B (or A) is reduced when A (or B) is known.
– 2) MI(A,B) = H(A) + H(B) -‐ H(A,B) • Maximizing the mutual info is equivalent to minimizing the joint entropy (last term)
• Advantage in using mutual info over joint entropy is it includes the individual input’s entropy
• Works beZer than simply joint entropy in regions of image background (low contrast) where there will be low joint entropy but this is offset by low individual entropies as well so the overall mutual informa-on will be low
Defini-ons of Mutual Informa-on II
– 3)
• This defini-on is related to the Kullback-‐Leibler distance between two distribu-ons
• Measures the dependence of the two distribu-ons
• In image registra-on I(A,B) will be maximized when the images are aligned
• In feature selec-on choose the features that minimize I(A,B) to ensure they are not related.
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I(A,B) = p(a,b) ⋅ log p(a,b)p(a)p(b)
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⎝ ⎜
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a,b∑
MI algorithm
Registra-on using MI Maximiza-on
Rigid Registra,on Affine Registra-on Non-‐rigid Registra-on