INTERPOLATION ARTIFACTS IN BIOMEDICAL IMAGE REGISTRATION G K Rohde 1 D M Healy Jr 2 C A Berenstein 2 A Aldroubi 3 1 Department of Biomedical Engineering Carnegie Mellon University Pittsburgh PA 15213 2 Department of Mathematics University of Maryland College Park MD USA 20742 3 Mathematics Department Vanderbilt University Nashville TN USA 37235 ABSTRACT Using the generic framework of B spline image interpolation we then show that the energy of interpolated translated and sampled data sequences oscillates with respect to the translation parameter We show that Monte Carlo integration methods can be used to compute objective functions without artifactual oscillations Finally results and conclusions are offered We identify a cause for interpolation artifacts in objective functions by observing that the energy of interpolated and translated 2 sequences oscillates with respect to the translation parameter Using the B spline interpolation framework we show that such oscillations in the energy of the signals affect the sum of squared differences cross correlation and mutual information objective functions We describe several approaches that can be used to avoid interpolation artifacts such as higher degree interpolation as well as stochastic sampling and explain in detail why these are effective in eliminating the artifacts 2 THEORY Index Terms Image registration interpolation artifacts local optima 1 INTRODUCTION Image interpolation methods have been known to introduce artifacts local optima in objective functions used for automatic intensity based image registration To date most consider such artifacts to be a feature speci c to the mutual information similarity measure 1 2 3 A common strategy for avoiding such artifacts has been through the introduction of a randomizing sampling operation see 2 4 3 for examples though a precise cause for the oscillatory artifacts when regular sampling is used has not been mathematically demonstrated In 5 6 we showed that the covariance properties of an image undergoing spatial transformations can cause artifactual oscillations and in some cases local optima in the sum of squared differences SSD cross correlation CC and mutual information MI objective functions Our arguments however were entirely stochastic in that only the effects of additive noise were considered Here we provide a more general and concise explanation for oscillation artifacts in the objective functions used above and show that even interpolation on signals or images with no noise can cause oscillation artifacts We explain three approaches for avoiding them based on low pass ltering the image higher degree interpolation and Monte Carlo integration We begin by showing that the energy of interpolated data sequences is present in all objective functions named above 1 4244 0672 2 07 20 00 2007 IEEE Let 2 denote the space of square summable in nite dimensional real valued sequences The inner product between two sequences a b in 2 is de ned by a b 2 k Z a k b k while b 2 2 b b 2 A convolution between two 2 sequences is denoted a b and thus b can be though of as a discrete operator characterized by its transfer function B z k If B has no zeros on the unit circle then the k Z b k z inverse operator b 1 exists and is uniquely de ned by b 1 1 B A sequence s 2 can be translated by an arbitrary amount R by rst tting an interpolation or approximation model to s and then sampling the model at a translated set of coordinates normally organized on a regular grid Let 0 refer to the centered normalized rectangle Then the B spline function of degree n at value x is given by n x n 1 0 x Let s x s k n x k 1 k Z n 1 where x k n x k and bn k k Z b n x x k The translated data sequence can then be represented by s k n s k 2 n where n k n k These operations can be extended to dimensions 2 and over via tensor products of 2 1 Objective functions The goal in image registration is to compute a function f Rd Rd where are parameters that determine f such 648 ISBI 2007 that two signals images are aligned This procedure can be automated as an optimization problem f arg max Q s t f f where Q is the objective function chosen for a speci c problem An intuitive gure of merit for quantifying how well two sequences s and t align is the squared 2 norm of its difference SSD Let s s f k k Z The SSD objective function to be minimized in this case is Q s t 2 2 t 2 2 s t 2 s 2 2 Another objective function often used is the cross correlation between the signals Q pS T log S2 pS T pS pT d d where 1 IG T S log 1 2 T S 2 stands for the covariance between random variables S and T and T2 and S2 represent their respective variances R is a residual term Assuming stationarity and ergodicity these quantities can be estimated from a set of samples s f k and t k k 1 s f k t k N k T2 1 t k 2 and N k S2 1 s f k 2 N k 1 1 s f k 2 s 2 2 N N k Using the Gram Charlier series expansion for the pdfs above the MI between two random variables can also be written as 9 M I IG R assuming the signals are zero mean where represents a set of prede ned coordinates and N Clearly S2 is related to s 2 2 via s t 2 t 2 s 2 Finally the registration between two images can also be computed by maximizing the MI between them Originally introduced as a similarity measure for registration problems in 7 8 the MI between two signals s and t quanti es the amount of information due to s is present in t Let S and T represent random variables associated with the discrete signals s f k k Z and t k k Z respectively with probability density functions pdf pS pT and joint pdf pS T The mutual information is de ned as Q Fig 1 Frequency response for cardinal B spline translation lters for different translation values a and degrees b 2 2 Oscillation artifacts Note that the term s 2 is present in all three objective functions discussed above Using the B spline interpolation framework de ned earlier we now show that this quantity can oscillate with respect to translations f k k Using 2 we have 1 s 2 s n 2 d 3 2 with n n k e i k k In Figure 1 we plot n for different values of and n It is clear that for a xed degree n n is not constant with respect to the translation parameter When Z n 1 and thus the frequency content of s is not modi ed and the signal s energy remains intact As approaches 0 5 n attenuates any high frequency components in s more than …