Efficient cosine-windowed cross-correlation for intermediate deformable image registration

Abstract

Deformable image registration is commonly included in population and longitudinal medical imaging analysis pipelines. Initializing deformable registration with the results of affine registration, where global misalignments have been reduced, can improve overall registration accuracy. Affine registration, however, uses a limited linear transformation model that does not align nonlinear anatomical variations, such as those between pre- and post-operative images or across different individual anatomies. In this work, we introduce a new intermediate deformable image registration (IDIR) method that corrects large deformations via cosine-windowed cross-correlation, and provide an efficient implementation via the fast Fourier transform. We evaluate our general-purpose approach qualitatively and quantitatively on 2D bone X-ray images, 3D brain magnetic resonance images, and 3D abdominal computed tomography images, demonstrating its ability to align large nonlinear anatomical variations within a few iterations and its suitability for initializing standard deformable registration.

Introduction

Spatial correspondences among medical images are often needed in population and longitudinal imaging studies, and are typically obtained via the image registration step in the processing pipeline. Affine (linear) transformation is often insufficient to account for temporal changes within a subject or anatomical variations across subjects, thereby necessitating the two-step affine + deformable image registration for many analyses.

Conventional deformable registration techniques¹ commonly optimize a cost function, whereby the displacement field is updated iteratively according to the very local information provided by the image gradient. In the presence of large deformations, however, the locality of this information may cause the registration to converge to the wrong local optimum, and/or require many iterations leading to increased computational burden. To remedy these issues, the displacement field is usually regularized or smoothed^2,3, hence corrected at each location per information from a neighborhood as large as the smoothing kernel. Furthermore, some approaches pyramidally exploit lower-resolution versions of the images, which have larger pixel sizes, by down-sampling in a coarse-to-fine registration scheme^4,5 or implicitly through contracting layers of a convolutional neural network^6–10. Despite these efforts, deformable registration is still commonly driven by local image information (making it flexible but prone to entrapment in local optima), in sharp contrast to affine registration that strives for big-picture alignment (but is far from able to align fine details). In the affine + deformable registration pipeline, thus, an intermediate registration step that is inherently neither as global as the affine model nor as local as most current deformable models remains desirable. Such an intermediate step would provide a smooth transition from affine to deformable registration, which, by providing the final deformable registration with an initial solution closer to the global optimum, might improve the overall process.

Patch-based, such as block-matching, registration methods have been introduced that deform the image through affine transformations in many smaller regions^11–15. Such methods compute the displacement for a pixel by considering its surrounding region (i.e., the patch), which can be chosen as large as desired to reduce the locality of deformation computation. Among the various similarity metrics that a block-matching algorithm can use to locate similar blocks between two images, peak cross-correlation (CC) of the blocks has gained popularity—particularly for speckle tracking in ultrasound imaging¹⁶— due to its efficient implementation via the fast Fourier transform (FFT). However, given that each pixel for which the displacement is computed requires a dedicated block, the spatial resolution of the deformation field depends on the number of (possibly overlapping) blocks. As a result, the computational complexity of the block-matching algorithm, which grows with both the number of blocks and the size of each block, becomes a limiting factor, creating a tradeoff between the resolution of the deformation field and the size of the region considered for displacement computation. Block-matching registration with large blocks can indeed be computationally prohibitive at the full resolution.

In this work, inspired by the CC-based block-matching approach, we present a new deformable registration algorithm based on CC of images windowed with the cosine function. We compute local translations (i.e., the displacement field) by masking the two images with a smooth sliding window function that is effectively half the size of the image (in each dimension). The large size of the neighborhood that is focused on to compute the deformation field at each pixel (in contrast to the local region that standard deformable registration uses) makes our method a suitable intermediate deformable image registration (IDIR) approach, particularly in the presence of large deformations, to be used prior to standard (local) deformable registration. In contrast to recent deep-learning-based methods, which are often tailored to a specific anatomy or imaging modality depending on the data they are trained on, our method does not require training and is therefore general-purpose and agnostic to image type. We validate our IDIR approach qualitatively and quantitatively by applying it to 2D bone X-ray, 3D brain magnetic resonance imaging (MRI), and 3D abdominal computed tomography (CT) images.

Our Matlab implementation of the proposed IDIR method is publicly available (see the “Code availability” section). This article extends a preliminary conference version¹⁷. In particular, we present a new set of quantitative empirical evaluations and comparisons, as well as more clarifications. In the following, we describe the proposed method in detail in the “Methods” section, present and discuss experimental results in the “Results and discussion” section, and finish with some concluding remarks in the “Conclusions”.

Methods

For simplicity, we first describe our method for one-dimensional (1D) input signals. Let be two 1D discrete-domain signals of length to be registered, where for . We would like to compute the transformation , or equivalently, the displacement field , which results in the image that is similar to .

Global cross-correlation

To avoid wrap-around effects during circular CC (see below), we first zero-pad and on their right sides, resulting in and of length , which are now considered periodic outside their domain (i.e., and for ). This zero-padding makes the circular CC practically equivalent to a regular CC. We then use FFT to divide the frequency components of each signal by their magnitudes, thus benefiting from the fact that the degree of alignment is reflected in CC more effectively and with greater contrast when the image contains only the phase information^18,19.

To find the global translation that best aligns two images, the following global CC, has been traditionally used:^18–20

1

where denotes the CC operation. The sum is equivalent to a circular convolution, , which can be efficiently computed in the frequency domain as via the FFT operator in . Computing the CC via FFT substantially reduces the number of necessary calculations, which makes the operation not only faster but also more precise thanks to less accumulated round-off error²¹. The shift maximizing is then regarded as the translation best aligning the two signals. This global translation-only registration, therefore, simply produces the constant displacement field . Next, we will extend the CC so that it produces a spatially varying 1D deformation field.

Cosine-windowed cross-correlation

As described above, global CC considers the entire domain of both images to produce a single optimal global translation. A deformation field, on the other hand, is a set of local translations that are optimized while focusing on smaller regions in the image. CC can still be used for this purpose if the images are masked with a window function centered on the desired region, so the displacement (i.e., local translation) is computed with a focus on the image contents of that region (along the same lines as how local correlation coefficient has been utilized²²). To compute the displacement for a region centered at , we propose the following smooth cosine window function, :

2

This is practically a positive window function, given that it is chosen in such a way that when multiplied by the zero-padded image, for any , the image contents (at ) only coincide with the central (positive) lobe of the cosine window, which is thanks to the inequality . The deformation field is then computed as:

3

This deformation field can be used in an iterative process until convergence. To that end, we keep an overall transformation (initialized as identity) and update it as (recall that ), and repeat the next iteration with the updated image .

One can yet notice a severe practical drawback for the deformation update presented in Eq. (3). Even when employing efficient FFT to compute CC, computing the deformation field for all values of would require operations, which is prohibitively large for images. Accordingly, in the next section, we will show how to reduce the computational complexity, thereby making the proposed IDIR feasible.

Computational complexity reduction

To reduce the computational cost of the deformation update, we exploit the expansion of the cosine function and rewrite Eq. (2) as:

4

The term now expands to four CCs, which can be simplified into three terms,

5

where and are derived in Table 1.

Table 1.

Components of .

1
2
3

Next, we need to apply the operator in Eq. (3). For a function that is symmetric around its peak, would be the same as the center of mass (CoM) of the function. (As an example for clarification, if the function were a probability density function, and CoM would be the mode and mean of the distribution, respectively, which would coincide for a function symmetric around its peak.) Nevertheless, when symmetry is not guaranteed, such as in our case of in Eq. (3), can still be approximated with CoM. The error of such an approximation can be reduced by sharpening the peak of the function²³, e.g., via raising it to a power, which would concentrate the data points closer to the peak. This will make the magnitudes of the two tails – and their difference – smaller compared to the peak, thus reducing the effect of asymmetry on CoM and pushing CoM closer to the peak. Accordingly, we compute Eq. (3) by first raising to the power of (with ) to amplify and sharpen its peak,

6

and then approximating its peak location as its CoM:

7

Since CC of phase images can occasionally contain small negative values, needs to be an odd natural number, so such negative values do not change sign (we used in our experiments presented in the “Results and discussion” section).

Multinomial expansion of Eq. (6) results in terms (e.g., ),

8

where (resp. ) is computed following the multinomial theorem as the product of s (resp. s) for selections of with repetitions allowed. Combining Eqs. (7) and (8), the field becomes:

9

Thanks to the separation of functions of from those of in Eq. (9), the deformation field can now be computed for the entire image simultaneously. We compute the inner sums independently of , and then the outer sums for all values of . A fixed number of FFTs is sufficient to produce and store all s. Consequently, the entire is computed in one pass in , i.e. significantly less than mentioned in the previous subsection, thus making our approach practical. (The factor 7 in the complexity term arises from the 4 FFTs and 3 inverse FFTs necessary to compute the in Table 1).

Next, we will extend the proposed method to images.

Extension to higher dimensions

For a -dimensional image of size , with pixels, we generalize the window function in Eq. (2) as:

10

This generates terms in Eq. (5) and terms in Eq. (8) (e.g., and ). The vector field is then computed similarly to Eq. (9), in the more general complexity of .

Note that IDIR is “intermediate” between affine and deformable registration, and, as such, operates with the assumption that global affine misalignments (e.g., large rotations) have already been corrected.

Implementation

In our implementation, which is publicly available (see the “Code availability” section), the following are considered.

We apply a compositive scheme similar to the diffeomorphic demons framework²⁴ to update the deformation field. Briefly, instead of directly applying the update transformation , we create a diffeomorphic approximation of it by initializing a transformation as , composing it with itself subsequently 10 times, and using the result as to update the overall transformation (see the “Cosine-windowed cross-correlation” section).

When dividing the Fourier transform by its magnitude to create phase-only images, for more robustness, we divide it instead by the magnitude plus 0.001 times its norm, which reduces the weighting of low-magnitude frequency components (that often have a lower signal-to-noise ratio).

To further increase stability, we reduce the effects of the peaks in the CC far from the origin by weighting the numerator and denominator of Eq. (9) by an isotropic Gaussian centered at the origin of the CC (i.e., at ) with a width equal to times the mean image dimension, which can be efficiently done by equivalently multiplying each by a Gaussian with times the variance. We found this CC weighting to be beneficial for images where the object was cropped by the borders (such as in our “Experiments on 2D bone X-ray images” and our “Experiments on 3D abdominal computed tomography images”), but did not notice an improvement when the object did not touch the borders (such as in our “Experiments on 3D brain magnetic resonance images”). This is possibly because the CC weighting reduces the update to the deformation field, potentially diminishing the effects of the artificially created edges at the border (which may especially affect phase-only registration approaches like ours). Further regularization (e.g., smoothing) of the update field³ is not necessary, given the inherent smoothness of the basis functions used to represent the field (Table 1, right column).

Instead of keeping the native space of as the reference space, one could alternatively define forward and backward transformations, and , and perform symmetric updates of and .²⁵ However, we empirically found the asymmetric update, which does not deal with the mid-space drift issue²⁶, to be more stable.

Computing the terms during the summation in Eq. (9) (despite no need to store them all in memory) is the most computationally expensive step of the proposed IDIR implementation, as each or requires image multiplications. Nevertheless, by properly sorting the multinomial expansion, the results of the first multiplications in a term can be kept and reused for many subsequent terms, effectively reducing the number of necessary multiplications to for most terms. Alternatively, these terms can be computed in parallel threads and summed (e.g., using a reduction variable in a Matlab parfor-loop).

Results and discussion

We evaluated our IDIR method on various medical images acquired with three different imaging modalities, the results of which are reported as follows. Since the data used in this work are anonymized and publicly available, obtaining consent was not necessary.

The smallest odd natural power for peak sharpening (refer to the “Computational complexity reduction” section) is , which is the value we chose for all experiments; using larger (which leads to more computation, see the “Extension to higher dimensions” section) did not yield significant improvement in our initial benchmarking experiments.

Experiments on 2D bone X-ray images

We applied our method first on public 2D pre- and post-operative X-ray images (see the “Data availability” section for data), with a heuristically chosen CC weighting of (refer to the “Implementation” section). In one experiment, we used a pair of jaw images of the size 170 × 194 acquired before and after an orthognathic surgery,²⁷ and in another, we used a pair of foot images of the size 424 × 248 acquired before and after a metatarsus adductus surgery²⁸. The original images, along with the results at iterations 1, 2, and 20, are shown in Fig. 1. The large moving window in CC computation brings about the benefit of correcting large deformations within only a few iterations, but also the caveat of not fully aligning finer details that require sharp transitions in the deformation field. The displacement fields, depicted as blue arrows, appear smooth despite the absence of explicit regularization. We observed, however, that running the algorithm for many more iterations could sometimes make it unstable and even cause it to diverge. For comparison with conventional deformable registration, we previously applied the demons method to the same jaw images, achieving accurate alignment but after about a thousand iterations.²⁹ These results suggest the potential advantage of initializing standard (local) deformable registration with the proposed IDIR method.

Fig. 1.

Registration of pre- (green) to post-surgery (red) X-ray images using the proposed IDIR method, shown at different iterations. (Jaw images courtesy of Dr. Yang Zonbang. Foot images courtesy of the Chelsea and Westminster Hospital.)

Regarding the algorithm runtime, each iteration took 0.1 and 0.2 s in the first and second experiments, respectively, on a computer with a 12-core 3.6 GHz (4.5 GHz Turbo) Intel Xeon Gold CPU. We did not explicitly parallelize the code; however, Matlab often inherently multi-threads its internal operations.

Experiments on 3D brain magnetic resonance images

Next, we tested our algorithm on a 3D longitudinal fetal brain MRI atlas with T2-weighted images of the size 145 × 125 × 121, provided to the public by the Computational Radiology Laboratory of the Boston Children’s Hospital³⁰ (see the “Data availability” section for data). We registered the first time-point of the atlas (week 21) to its last (week 38), without applying the CC weighting (i.e., ; refer to the “Implementation” section). The original images, along with iterations 1, 2, 3, and 50, are shown in Fig. 2 from three viewpoints. As expected, our windowed-CC-based IDIR method captures the bulk of the deformation that maps the smaller to the larger brain within a few iterations, eventually achieving reasonable alignment that can be further refined with standard deformable registration initialized with the IDIR result. Each iteration took 7.5 s on a graphics processing unit (Nvidia RTX A6000 GPU) or 40.6 s on the CPU (same hardware as in our “Experiments on 2D bone X-ray images”).

Fig. 2.

Central sagittal (top), coronal (middle), and axial (bottom) slices of the 3D fetal brain atlas are shown at different registration iterations (from left to right), while the first time-point (week 21, green) is registered to the last time-point (week 38, red) using the proposed IDIR method. Throughout the registration, parts of the 3D green brain enter or leave the visualized 2D slices (from or to adjacent slices).

The dataset also contained 116 labels for different regions and structures of the brain. After propagating the labels using nearest-neighbor interpolation, we computed the portion of the voxels where the labels from the two images correctly overlapped, which is plotted for each iteration in Fig. 3. The increasing nature of the plot indicates consistent improvement at each iteration, with most of the alignment occurring during the first few iterations. The average of the 116 regions’ Dice scores increased from the initial 0.02 to 0.32.

Fig. 3.

Label overlap ratio between the moving and reference fetal brain images at each IDIR iteration.

For comparison, we ran the asymmetric diffeomorphic demons registration algorithm²⁴ (see the “Code availability” section for toolbox) on this image pair with empirically optimized (smoothing and step size) parameters, once for 100 iterations at a single resolution, and again with a coarse-to-fine (multi-resolution) scheme for 100, 50, and 25 iterations at three resolution levels, resulting in the label overlap ratios of 0.63 and 0.76 (average Dice of 0.05 and 0.30), respectively, i.e. lower than the 0.77 seen in Fig. 3 (average Dice of 0.32).

When we initialized the single-level diffeomorphic demons with the IDIR result, however, the label overlap was increased to 0.80 (average Dice of 0.38).

Experiments on 3D abdominal computed tomography images

Finally, we evaluated our method through a set of experiments on abdominal CT images from the public Multi-Atlas Labeling Beyond the Cranial Vault³¹ database (see the “Data availability” section for data). All scans were acquired for routine clinical care and randomly selected from a combination of a colorectal cancer chemotherapy trial and a ventral hernia study at the Vanderbilt University Medical Center. We used the training set, which included images from 30 subjects, each accompanied by manual labels for 13 organs (Table 2, left column). The number of slices in the images ranged from 90 to 198, with slice thicknesses of 2.5 to 5.0 mm. Each slice was 512 × 512 pixels, with pixel sizes ranging from 0.59 to 0.98 mm.

Table 2.

Dice scores (Mean ± SEM) from the abdominal CT image registration experiments.

Organ	Pre-registration	IDIR		MSI		IDIR + MSI		NiftyReg
		Optimal	CV	Optimal	CV	Optimal	CV
Spleen	0.07 ± 0.04	0.28 ± 0.05	0.20 ± 0.05	0.23 ± 0.05	0.21 ± 0.04	0.44 ± 0.06	0.34 ± 0.07	0.37 ± 0.08
Right kidney	0.05 ± 0.03	0.25 ± 0.05	0.19 ± 0.06	0.16 ± 0.05	0.15 ± 0.05	0.38 ± 0.07	0.27 ± 0.08	0.37 ± 0.07
Left kidney	0.09 ± 0.04	0.27 ± 0.05	0.13 ± 0.03	0.22 ± 0.05	0.16 ± 0.05	0.38 ± 0.07	0.22 ± 0.06	0.28 ± 0.08
Gallbladder	0.04 ± 0.02	0.05 ± 0.03	0.03 ± 0.02	0.06 ± 0.03	0.01 ± 0.00	0.09 ± 0.04	0.09 ± 0.04	0.07 ± 0.04
Esophagus	0.03 ± 0.02	0.19 ± 0.05	0.12 ± 0.04	0.03 ± 0.02	0.01 ± 0.00	0.22 ± 0.03	0.09 ± 0.03	0.26 ± 0.05
Liver	0.23 ± 0.07	0.55 ± 0.05	0.44 ± 0.07	0.52 ± 0.04	0.45 ± 0.04	0.68 ± 0.05	0.60 ± 0.06	0.71 ± 0.05
Stomach	0.11 ± 0.04	0.31 ± 0.04	0.22 ± 0.04	0.24 ± 0.05	0.13 ± 0.04	0.40 ± 0.05	0.27 ± 0.05	0.31 ± 0.06
Aorta	0.04 ± 0.02	0.26 ± 0.04	0.23 ± 0.05	0.17 ± 0.05	0.09 ± 0.03	0.39 ± 0.06	0.31 ± 0.07	0.43 ± 0.07
Inferior vena cava	0.04 ± 0.01	0.29 ± 0.04	0.23 ± 0.05	0.13 ± 0.04	0.11 ± 0.03	0.34 ± 0.05	0.27 ± 0.06	0.34 ± 0.06
Portal+splenic veins	0.03 ± 0.02	0.09 ± 0.03	0.05 ± 0.02	0.08 ± 0.03	0.03 ± 0.02	0.14 ± 0.03	0.10 ± 0.03	0.08 ± 0.03
Pancreas	0.05 ± 0.03	0.12 ± 0.03	0.10 ± 0.03	0.11 ± 0.03	0.03 ± 0.02	0.16 ± 0.03	0.12 ± 0.03	0.12 ± 0.03
Right adrenal gland	0.01 ± 0.01	0.04 ± 0.02	0.03 ± 0.01	0.02 ± 0.01	0.01 ± 0.01	0.12 ± 0.04	0.10 ± 0.04	0.14 ± 0.04
Left adrenal gland	0.01 ± 0.01	0.10 ± 0.03	0.02 ± 0.02	0.07 ± 0.03	0.02 ± 0.01	0.19 ± 0.05	0.05 ± 0.02	0.07 ± 0.03
Mean	0.06 ± 0.02	0.22 ± 0.03	0.15 ± 0.03	0.16 ± 0.02	0.11 ± 0.02	0.30 ± 0.03	0.22 ± 0.03	0.28 ± 0.04

In each column, all entries are derived from the same set of experiments.

We grouped the 30 images into 15 pairs and registered the second image to the first image in each pair in separate experiments using the proposed IDIR method with 50 iterations. For comparison, we also registered the images using the multi-resolution mid-space-independent (MSI) deformable registration method²⁶ (at three resolution levels with 100, 50, and 25 iterations; see the “Code availability” section for toolbox), as well as the MSI method (100 iterations at full resolution) initialized with the IDIR results (“IDIR + MSI”). We zero-padded the input image with fewer slices, so both images ended up the same size. For IDIR, we repeated the experiments with a range of values from 0.03 to 0.1 for the CC weighting parameter (refer to the “Implementation” section). Furthermore, since the contribution of regions with very high image intensity (such as bones) can overshadow the important information provided by the soft tissue, we dampened the effects of those regions by smoothly thresholding the image as , where the original is measured in Hounsfield units, and repeated the experiments with a range of values from 0.05 to 1 for the threshold . For the MSI method, we repeated the registration experiments over a range of 0.0001 to 0.5 for the regularization parameter and 0.1 to 100 for the step size. The parameter values were chosen within each range roughly uniformly on the logarithmic scale. In addition, the dataset included images already registered using the NiftyReg nonlinear registration method^5,32 (see the “Code availability” section for the toolbox), which we used for further comparison.

To evaluate registration accuracy, we calculated Dice similarity coefficients between the manual labels of the reference image and those of the transformed image propagated using the computed deformation field. Cross-subject mean Dice scores along with the standard error of the mean (SEM) are reported in Table 2. For the methods run repeatedly with the aforementioned ranges of different parameter values, we report two sets of results: the “optimal” results and the leave-one-out cross-validation (“CV”) results. The optimal results are those obtained by the single experiment that produces the highest mean Dice score (i.e., Dice averaged across all organs) among all tested parameter values. The CV results, on the other hand, were calculated as follows: for each of the 15 subject pairs, we selected the experiment run with the set of parameter values that most frequently produced the highest mean Dice across the other 14 pairs. Note that in each column of Table 2, all entries are derived from the same set of experiments. As a baseline, pre-registration Dice scores are also reported. Two subjects did not have a gallbladder, for whom the gallbladder was omitted from the mean Dice calculation.

As expected, for each method, the CV Dice results were lower than the optimal results, with larger organs typically showing higher Dice scores. IDIR produced higher Dice scores than MSI did for most organs. Notably, IDIR + MSI produced a higher Dice score than each of IDIR and MSI alone did for every organ. Dice scores by NiftyReg were generally higher than those by IDIR and MSI and the CV Dice scores by IDIR + MSI, but lower than the optimal Dice scores by IDIR + MSI. For the CV results by IDIR, the parameter value pair of and was consistently chosen through the CV, reflecting the robustness of the optimal parameter values.

One factor contributing to diminished organ Dice values reported in Table 2 is basing the optimization of the registration parameters (i.e., choice of the optimal/CV experiment) on the cross-organ mean Dice, rather than optimizing them for each organ separately. For instance, the value at the “Optimal” column of a method and the “spleen” row is the cross-subject average of the spleen Dice from the experiments resulting in the best cross-organ mean Dice, which is not necessarily as high as the average of the subjects’ best spleen Dice. Higher Dice scores would therefore have been achieved for the organs had we optimized the registration focusing on each organ independently (rather than considering all organs combined), which is the suitable approach in cases where a specific organ of interest is being studied.

Conclusions

We have introduced a new intermediate step for deformable image registration that leverages efficient FFT-based cosine-windowed CC computation for alignment. The large window used by the proposed IDIR method facilitates the retrieval of large deformations in only a few iterations – as is apparent from our results on various medical images – but may also prevent eventual capturing of fine-grained deformations. Our method is therefore not meant to be directly compared to standard deformable registration; instead, as we have shown, its potential chiefly lies in initializing standard deformable registration, giving it a head start toward a more accurate optimal solution in fewer iterations. Unlike most deep-learning-based registration techniques, the IDIR approach is free of training and agnostic to anatomy and image modality. Nonetheless, combining it with deep-learning-based registration may be beneficial, which is part of our future research.

Author contributions

I.A. designed the methods, ran the experiments, analyzed the results, and wrote the article.

Funding

Support for this research was provided by the National Institutes of Health (NIH), specifically the National Institute on Aging (RF1AG068261, R01AG068261). Additional support was provided by the Michael J. Fox Foundation for Parkinson’s Research (MJFF-021226). Computational resources were provided through a Microsoft Azure Credit Grant by the Harvard Data Science Initiative, the Enterprise Research Infrastructure and Services at Mass General Brigham, the Massachusetts Life Sciences Center, and NIH Shared Instrumentation Grants (S10RR023401, S10RR019307, S10RR023043).

Data availability

All the data used in our experiments are publicly available, enabling the reproducibility assessment of our results. We applied our method to the following publicly available data: The X-ray images used in our “Experiments on 2D bone X-ray images” are available at: jaw images (courtesy of Dr. Yang Zonbang):²⁷ http://lytornado.pixnet.net/blog/post/266239880 and foot images (courtesy of the Chelsea and Westminster Hospital):²⁸ www.chelwest.nhs.uk/your-visit/patient-leaflets/surgery-services/midfoot-problems. The fetal brain MRI atlas of the Boston Children’s Hospital³⁰ used in our “Experiments on 3D brain magnetic resonance images” is publicly available at: http://crl.med.harvard.edu/research/fetal_brain_atlas. The Multi-Atlas Labeling Beyond the Cranial Vault³¹ abdominal CT image database used in our “Experiments on 3D abdominal computed tomography images” is publicly available at: www.synapse.org/Synapse:syn3193805/wiki/217789.

Code availability

The Matlab function, registerIDIR, implementing the proposed IDIR method, is included in our deformable image registration toolbox that is publicly available at: www.nitrc.org/projects/msi-register. This toolbox also implements our mid-space-independent (MSI) deformable registration method²⁶ used in our “Experiments on 3D abdominal computed tomography images” as well as the asymmetric diffeomorphic demons method²⁴ used in our “Experiments on 3D brain magnetic resonance images”. The NiftyReg nonlinear image registration method,^5,32 the results of which were included in our “Experiments on 3D abdominal computed tomography images”, is publicly available at: https://sourceforge.net/p/niftyreg/git/ci/master/tree.

Declarations

Competing interests

The authors declare no competing interests.