Inter–modality Registration of NMRi and Histological Section

Images using Neural Networks Regression in Gabor Feature Space

Felix Bollenbeck, Rainer Pielot, Diana Weier, Winfriede Weschke, and Udo Seiffert

Abstract—Image registration is amongst the most prominentproblems in image processing and computer vision. Particularlyin biomedical applications, automated alignment of image datafrom different imaging modalities has received great attention,delivering a high value added for analysis and diagnosis byintegrating spatial information of two or more assays. Inthis context, the use of entropy based mutual informationbetween images has been widely propagated to capture therelation between differential intensity distributions. In thiswork we address the problem of matching two differentintensity distributions in a supervised learning scenario: Weapproximate a function relating both intensity distributionsusing a regression neural network predicting intensity valuesof one modality to the other, thereby allowing direct intensitydifference registration. Predictions are based on a Gabor spacerepresentation of the input image, in order to capture localimage structures. In experiments we show that the approach isi) able to learn a function to predict intensity values and ii) thepredictions can be used to correctly register images by directintensity differences minimization. The latter has the advantageof being computationally appealing and more stable concerningthe optimization framework, which we exploit in registeringhistological section and NMRi data of plant specimen.

I. INTRODUCTION

Registration is a fundamental problem in image pro-

cessing. Matching corresponding structures and features of

images coming from different timepoints, different subjects

and viewpoints, or from different sensors is a problem

encountered in virtually any field of application. Correct

alignment is the crucial in various image processing and com-

puter vision tasks such as image fusion, change detection,

and image restoration, on various scales and applications.

remote sensing, image mosaicing, super-resolution imaging,

and multispectral analysis, often requires registration of

large two-dimensional image data. In medicine, registration

problems are ubiquitous with the large availability and

application of tomographic methods for diagnosis, relating

to three-dimensional registration problems for intervention,

tumor monitoring, anatomical atlases, dose calculation etc.,

as well two-dimensional macro- and microscopic image data.

With the advent of new technology such as laser scanning

microscopy or optical projection tomography, images on a

Felix Bollenbeck and Udo Seiffert are with the Department ofBiosystems Engineering, Fraunhofer IFF, Magdeburg, Germany(felix.bollenbeck,udo.seiffert@iff.fraunhofer.de). Rainer Pielot iswith the Leibniz Institute for Neurobiology, Magdeburg, Germany(pielot@ifn-magdeburg.de). Diana Weier and Winfriede Weschke are withthe Department of Molecular Genetics, Leibniz Institute for Plant Geneticsand Crop Plant Research, Gatersleben, Germany (weier,weschke@ipk-gatersleben.de). This work was supported by German Federal Ministryof Education and Research Grant 0313821A and German ResearchFoundation grant WE-1608/2-1.

cellular level are now available, allowing e.g. co-localization

analysis and tracking of molecular markers, all requiring

automatic image registration algorithms.

Accordingly, a large set of methodology for various image

registration settings has been published. In [1] an initial

overview of early image registration techniques is given.

The authors of [2] review classic as well as more recent

image registration techniques and give a classification of

registration techniques. Reference [3] is a comprehensive

description of the numerical schemes concerning established

image registration algorithms.

The authors of [2] distinguish feature–based and area–

based registration methods. While the former relates to the

matching of manual or automatically selected, non-uniform

control points (e.g. see [4]), we herein address area–based

registration, e.g. matching uniform grids of image intensities.

Following [2], registration in the spatial domain comprises

three components:

• Image Metric. The degree of homology or the quality

of the alignment of two or more images is captured

numerically by an image-to-image metric based on

intensity at corresponding image grid points, which is

to be optimized by the registration.

• Transform. Based on the physical properties of used

sensors and sensing setting, different distortions or

transformation can be assumed between images, e.g.

simple translations, affine mappings, spline-, or regular-

ized free-form transforms. In many applications, only

the transformation parameters, not aligned images are

of interest.

• Optimization Framework. Based on the choice of metric

and transform, different schemes for finding the best

alignmend in the space of transformation parameters

arise, e.g. exhaustive searches, derivation-based meth-

ods, genetic algorithms etc..

Here, [5] provides an overview of freely available implemen-

tations.

Sum of squared image intensity differences (SSD) as an

image metric is intuitive and directly proportional to the

likelihood of correct registration, when images identical

intensity distributions and i.i.d. gaussian noise. Therefore, for

images of the same domain or modality SSD is widely used

for being a robust and computationally appealing criterion.

The extension of direct intensity difference towards cross

correlation-based (CC) formulations has been extensively

suggested in the literature (see [1]), to address changes in

intensity distributions of intra-modality images, e.g. caused

by changing illumination, with significant increase in compu-978-1-4244-2760-4/09/$25.00 ©2009 IEEE

tational costs. While differences in intensity levels can also

be addressed by equalization preprocessing steps, CC based

metrics have the disadvantage of producing relatively flat

maxima, making numerical optimization difficult.

A direct linear correlation can generally not be assumed

when registering images from different modalities, which

renders the use of SSD related CC formulations ineffective.

Metrics invariant to intensity magnitudes, such as edge or

gradient methods have been suggested for inter-modality

registration using CC ([6]). Such preprocessing is crucial, yet

application-specific. A more general approach to registration

using CC is described in [7]: By constructing a mapping

function, ultrasound and magnetic resonance images are

registered by maximizing the correlation ratio. Naturally,

such mapping is critical. In [8] a polynomial fit for esti-

mating intensities is used, which is iteratively updated along

with the optimization, with additional computing costs and

instabilities.

Mutual information (MI) has been suggested to capture more

general relations of intensities in different imaging modali-

ties: Instead of maximizing coherence of image intensities by

correlation, optimal image alignment is described by mini-

mizing the entropy between intensity probabilities observed

at corresponding image grid points. MI based registration,

initially described in [9] and [10], has been widely used in

inter-modality registration in medical imaging as an image

metric which can be applied on direct intensity data without

application specific preprocessing.

The MI criterion, originating from communication theory

based on Shannon’s entropy describes the mutual dependence

of two random variables, measuring the information in bits

transmitted over a communication channel, regardless of the

actual form of dependency. For two images R, T : Ω ⊂R

D 7→ R MI is defined as

MI(R, T ) =

∫

Ω

h(R(x)) + h(T (x)) − h(T (x), R(x))dx.

(1)

Where h denotes (joint) entropy of random variable to binary

(bits in communications theory) or natural base. In image

registration MI is assumed to be maximal, when two images

are aligned, e.g. when relation between image intensities is

least random.

The usage of MI as an image metric introduces additional

challenges in comparison to SSD based registration, which

have been addressed in past works: How to estimate joint and

marginal probability densities to a differentiable form, how

to address the high computational costs, and how to avoid

strong oscillations of MI values.

In applications, the marginal and joint probability density

functions are unknown, and must be estimated from data.

In [9] the authors use Parzen windows with a Gaussian

kernel for iterated density estimation on a subset of grid

point samples. Recently Mattes et al. ([11]) have suggested

keeping a fixed subset of samples and discrete bins of

intensities, in connection with a B-spline kernel for density

estimation to reduce the computational complexity of MI

value calculation.

While the MI metric can be used with rigid ([9]) and also

non-rigid ([11]) transforms, depending on the application, the

choice of the optimization regime is critical, since metric

function produces high frequencies due to the stochastic

nature of the MI criterion, often resulting in an increased

number of iterations during optimization.

In this contribution we address inter-modality in a sense as

motivated by Viola et al. in their initial publication ([9]) on

MI registration, by means of constructing a function F (·)that predicts intensity values of an image R of one modality

to a different modality image T , and registering F (R) and

T using SSD-based intra-modality schemes, exploiting the

advantage in robustness and computational complexity.

Finding a closed parametric form of F , e.g. based on physicaltheory, relating both modalities can by can be considered

impossible from our knowledge. We therefore build an ap-

proximation function by learning the mapping from reference

inter-modality registrations.

II. METHODS

In the following we describe our method for constructing

an approximation function F for predicting image intensity

in order to reduce the inter-modality registration problem to

an intra-modality setting, by proprosing a regression neural

network model for intensity prediction.

Predictions are based on a representation of the input image

in Gabor feature space. The regression network is adapted on

the basis of reference registration pairs of images from both

modalities. Inter-modality registration is addressed by using

the intensity prediction made by F to register images using a

SSD metric. To evaluate the approach, prediction and SSD-

based registration are benchmarked against MI approaches,

as described in the results section (III).

A. Gabor Feature Space

Gabor filters have been widely used in feature extraction

for object recognition ([12]) and texture classification ([13]),

allowing multi-scale decomposition similar to the human

visual system. Multi-channel Gabor filters have similar char-

acteristics to (Gabor) wavelet expansions ([14], [15]), when

a filter bank representing multiple scales and orientations is

used, while being less computational costly than a complete

orthogonal Gabor wavelet expansion.

The Gabor filter is a harmonic function multiplied by 2-D

Gaussian envelope, having the form

ψ(x, y; f, θ) =f2

πγηexp

−

“

f2

γ2x′2

”

+

“

f2

η2y′2

”

x′ = x cos θ + y sin θ

y′ = −x sin θ + y sin θ (2)

in spatial domain, where f, θ, γ, η denote the sinus frequency,rotation of the Gaussian and sinusoid, and spatial widths of

the filter. With the filter response for an image function I(~x) :

R2 7→ R

RI(~x; f, θ) = ψ(~x; f, θ) ∗ I(x)

=

∫ ∫

∞

infty

ψ(x− x1, y − y1; f, θ)I(x1, x2)dx1dx2 (3)

Gabor features of an image for different orientations and

frequencies are calculated to compile an informative space

of filter responses. Orientations θk are equally spaced

θk :=kπ

nk = 1, . . . , n− 1 (4)

while exponential sampling is used for frequency selection

fk := a−kfmax k = 0, . . . ,m− 1 (5)

for the scaling factor a with octave-spacing a = 2 in

scale-space. A Gabor-space representation, similar to Gabor

wavelet expansion, or feature matrix is obtained at each

image grid point ~x

G(~x) =

R(~x; f0, θ0)R(~x; f0, θ1)

...

R(~x; f0, θn−1)...

(~x; fm−1, θ0)...

(~x; fm−1, θn−2)(~x; fm−1, θn−1)

(6)

with the particular filter responses at different scales and

orientations.

Capturing local features, e.g. texture, the Gabor space rep-

resentation of an image provides the basis for estimation

function, where the information within the multi-dimensional

Gabor representation can be learned implicitly by a suitable

regression model.

B. Non-parametric Regression Feed-Forward Network

Artificial neural networks (ANN) are a powerful tool in

non-parametric regression. Using the adaption ability of the

ANN connectionism paradigm, the problem of fitting a model

for Y from a set of explanatory variablesX1, . . . , XM can be

addressed even when the relationship between both variables

is complex, and a closed parametric form is hard to find ([16],

[17], and [18]).

Non-parametric regression models have the general form

Yi = f(Xi) + ǫ (7)

for a class F ∋ f of regression functions and residuals

ǫ. Regardless of the particular f in different regression

functions, the function class F needs to be expressive enough

to approximate a set of regression functions f .ANN implement non-parametric regression models in a sense

that they compose a basis over function space by a set

of logistic functions in the hidden nodes. As stated by

Kolmogorov’s theorem, multi layer perceptrons (MLP) are

capable performing any function approximation with arbi-

trary accuracy, given a suitable network architecture. In [19]

a universal proof for two-layer MLP is given. In practice, a

reduced set of hidden nodes and layers is used to approximate

a regression function.

For a set of paired observations (Xi, Yi) , i = 1, . . . ,M of

the regressor and criterion variables X ∈ Rp and Y ∈ R

m

of the form

X :=[

x′1, x′

2, . . . , x′

a, . . . , x′

p

]T

Y := [y′1, y′

2, . . . , y′

a, . . . , y′

m]T

(8)

the regression model explains Y with X by means of the

MLP network and obtained outputs.

Presented to a k-layer feed-forward network, an input xa

generates input ikaj and output okaj , j = 1, . . . , nk to all

nk, k = 1, . . . , n neurons in hidden and output layers,

where n1 = p and nn = m, by each neuron. The basis

function represented by a neuron is based on the inputs to

non-linearity, or activation function okaj = ψ(ikaj). In here

we use a logistic-sigmoid transfer function for ψ, mapping

to inputs [0, 1] by the form ψ(x) = 1/(

1 + exp(−ikaj))

.

By introducing weights wkij, biases θk

j , elementwise

logistic function θ(·) and relating output k − 1 and input

k linearly by wkij

ikaj =

nk−1∑

i=1

ok−1ai wk

ij + θkj (9)

denoting in vector notation

ik

a:= o

k−1a W

k + Θk

Ik =

(

ik

1, . . . , ik

m

)T(10)

and the layers input and output by

Ok := Ψ(Ik)

Ik := O

k−1W

k + 1Θk

Ok := Γn(On−1)

= Γn(Γn−1(On − 2))

= Γ(O1) (11)

the regression prediction based on the observations is given

by Y ′ = Γ(X).Relating to the non-parametric problem (7), the regression

model has the form Y = Γ(X) + E. Thus, the prediction

error ‖Y − Y′‖ can not be directly minimized as in para-

metric regression, which demands numerical optimization of

networks weights and biases. Although multiple alternatives

have been suggested for fitting regression networks, the error

back-propagation method initially described by Rummelhart

et al. in [20], is de facto standard for network adaption.

Using a set of (X,Y ) pairs, the network is iteratively trained

by presenting input and output data, back-propagating the

observed error through the network, and updating the weights

based on the calculated derivatives of the error function.

C. Inter-modality Intensity Prediction

Given two-dimensional aligned intensity images R, T :Ω ⊂ R

2 7→ R of two different imaging modalities M1,M2,

intensity characterstics and distribution for an identical scan

are differential, i.e. R(~x) and T (~x) are scattered and can not

be described by a closed relation.

We are constructing a mapping function to predict intensity

values for modality M2 given R, that is to find an approxi-

mation function F so that

T = F (R) + E. (12)

Since the relation of intensity values produced by M1 and

M2 is complex, and possibly cannot be extracted by simple

intensity relation, we base the approximation on the Gabor

space features G(R) extracted from every grid point x ∈ R.An MLP is set up and adapted to predict T on input of G(R).The network parameters are adapted by backpropagation

using samples from a reference pair R∗, T ∗ of manually

aligned images, and in turn used to predict intensities of

unseen, unaligned pairs R, T allowing direct SSD based

intra-modality registration schemes for aligning Γ(G(R))and T .

III. RESULTS

A. Data

In this section we describe our initital experiments using

the proposed method for inter-modality registration. For the

sake of simplicity, we performed the experiments with 2-D

images, where the methods is not limited to two-dimensional

data. Datasets used for method evaluation comprised histo-

logical section images and NMRi data1 of developing barley

(Hordeum vulgare) grains.

Histological serial section images were obtained from poly-

mer embedded, contrasted material, microtomed into 3µmthick sections. Scans where obtained at a spatial resolution

of 1.5×1.5µm (1600×1200 pixels) with a color CCD camera

in a conventional light microscope (LM), and converted from

RGB to 8 bit graylevels 2. Images where pre-processed by

embedding the sectioned object in uniform (0) backgroundand intensity equalization of the foreground (see fig. 1a).

NMRi data of hydrogen nuclei density (1H-NMRi) were

obtained from the intact grain object (as described in [21]) in

specific device with an isotropic spatial resolution of 16µm(173 × 257 × 512 voxels), and resampled to 8 bit dynamic

range. For registration experiments x, y-slices were sampled

from the voxel datasets (see fig. 1b), which were manually

aligned by an expert.

Aligning and fusing histological section and NMRi images

is of great interest to scientists in plant biology: It allows the

analysis of molecular properties such as water or assimilate

distribution in relation to histological structures, i.e. different

cell types or tissues.

1Well aware of the fact that MRI is more commonly used, we usethe abbreviation NMRi data for the physically more correct term nuclearmagnetic resonance imaging

2Using NTSC convention I = 0.2989R + 0.5870G + 0.1140B

(a) (b)

Fig. 1: Image data: LM histological section image (a) and

x, y-slice of a 1H-NMRi volume (b) of plant material (barley

grains). Images are preprocessed by background segmen-

tation and embedding and scaled to identical coordinate

spacing.

B. Approximation and Registration Setup

To evaluate intensity prediction for registration, we fol-

lowed a three-step procedure: i) address learning of the

regression model by the MLP, based on extracted features

and reference data, ii) using the regression model to register

unaligned images, and iii) compare the intensity prediction

approach to MI based registration in term of robustness and

computational efficiency.

For a (histology, NMRi) pair of reference alignments R, T ,(bicubic) resampled for identical spacing, we calculated the

Gabor space representation G(R) (eqn. (6)) using a set of

frequencies 1, 1

2, 1

4, 1

8, 1

16, 1

32, 1

64

1

128, 1

256, and orientations

0, 45, 90, 135 according to equations (5) and (4),

comprising a 36 dimensional feature vector of filter response

magnitudes.

We chose a regression network architecture with two hidden

layers with 20 neurons each and log-sigmoid non-linearities,

and a linear transfer function for the output neuron. 5, 000corresponding (G(R(x)), T (x)) feature, intensity pairs were

randomly sampled from foreground pixels and used for

training using backpropagation with Levenberg-Marquardt

optimization for minimal mean square error (MSE). Con-

vergence by minimal gradient was reached in less than 150epochs on average.

In 10 random test-train cycles we observed the prediction of

NMRi intensity values based on extracted features. Figure 2b

shows the scattering of between original and predicted inten-

sities and original inter-modality scattering (2a) by displaying

the joint histograms of image intensities. The prediction

thereby delivers an approximation of NMRi intensities, with

an MSE value of 4.15 in [0, 255] intensities.Testing performance for registration, we compared MI based

registration and our approach for images pairs with known

transformation parameters. For direct MI registration we used

the approach of Mattes et al. ([11]), which is computation-

ally more efficient than the original ([9]) for a competitive

benchmark, provided within the ITK framework ([5]).

Images where transformed by rigid mapping: translation and

rotation around the geometric center. For intensity prediction,

extracted feature data of transformed images was given

into the model, which was adapted using data from one

(a) (b)

Fig. 2: Learning intensity predictions: 2-D joint histograms

of a histological section and NMRi slice. Initially pixel in-

tensities between both images are scattered, with an average

intensity difference of 31.88, showing no linear or higher

order relation (a). The network-predicted intensities show a

direct relation to NMRi intensities with an MSE value of

4.15 (b).

single aligned image as described. Predicted intensity images

were registered using direct SSD based registration, using

gradient descent optimization, which was also used for MI

registration.

Instead of quantitatively registering arbitrary image pairs,

we evaluated the registration performance amongst different

transformations of a single image pair for a qualitative bench-

mark. Therefor, rotations and transformations of different

magnitudes were introduced to compile the test set.

Following MI registration setup as described in the literature:

20 gray level bins were used for MI estimation, reported as

a robust choice. The number of spatial samples was set to

20, 000. The execution times per iteration are approximately

equal to SSD-based schemes, for which the number of

iterations spent serves as a general indicator of complexity.

Table I summarizes the comparison in terms of correct regis-

tration and efficiency averages. Tests were 10-fold repeated.

Both approaches produce acceptable error rates when small

transformations are present, nevertheless the MI produces

high variances, particular for rotations. The number of it-

erations spent in transform search is in the same order of

magnitude as SSD registration, though with high variances.

With larger transformation, the accuracy of MI registration

decreases along with a magnitude higher number of iterations

spent in search space. Here, the charactersitcs of the MI

estimation make optimization more costly and unstable. Fig.

IV shows the trace of search for T6 in search space. Along

with faster convergence, the plot illustrates the instability of

MI runs in terms of variance and local minima solutions.

IV. DISCUSSION

In this work we have introduced a novel method to

address inter-modality registration by means of learning the

relationship between intensity values produced by the spe-

cific modalities. We address the task of intensity predictions

by utilizing informative features and a powerful model for

non-parametric regression, allowing to solve inter-modality

registration as an intra-modality task.

The prediction or mapping of intensities for inter-modality

IMG True MI Predicted

T1

[0, 0] µm [1.56, 1.13]µm [0.02, 0.13]µm±[1.55, 0.84] ±0.0

−5 −4.72 ± 2.39 −5.02 ± 0.00 46 ± 27 29 ± 0.0

T2

[−5, 15]µm [−4.06, 13.34]µm [−4.99, 14.98]µm±[1.08, 1.69] ±0.0

0 2.77 ± 1.25 −0.04 ± 0.00 38 ± 13 29 ± 0.0

T3

[−5,−15]µm [−3.15,−14.18]µm [−3.81,−15.34]µm±[0.49, 2.91] ±0.0

−5 −3.23 ± 1.59 −5.07 ± 0.00 40 ± 21 26 ± 0.0

T4

[0, 0] µm [1.78,−0.27]µm [−0.31, 0.40]µm±[2.37, 1.51] ±0.0

−180 −176.94 ± 3.72 −179.68 ± 0.00 388 ± 108 19 ± 0.0

T5

[−50, 50]µm [−48.51, 49.65]µm [−49.92, 49.99]µm±[1.47, 8.39] ±0.0

0 −1.94 ± 2.04 0.15 ± 0.00 321 ± 137 34 ± 0.0

T6

[−50, 50]µm [−37.21, 37.88]µm [−46.97, 49.59]µm±[19.13, 24.67] ±0.0

−180 −190.82 ± 4.17 −179.53 ± 0.00 400 ± 138 45 ± 0.0

TABLE I: Comparison of our approach with MI-based reg-

istration: Entries display average and variance in translation,

rotation angle and iterations within 10 runs. Both MI and

SSD registration based on predicted intensities give suffi-

cient accuracy in finding the correct transformation, while

the latter slightly outperforms MI, particularly for large

transformations. Concerning both efficiency and robustness,

intensity prediction clearly outperforms MI: Due to the

stochastic nature of MI concerning sample selection and PDF

estimation, registration is unstable in terms of high variances

in the best transformation result. This also holds for the

number of iterations spent during optimization. (SSD based

registration is deterministic. For MI estimation 20 bins, and

20, 000 grid points were used.)

Fig. 3: Registration results: Correct alignment of a NMRi and

histological section pair. Fusion and visualization of different

data sources is often relevant to analysis. In the biological

context here, it allows to relate functional measurements

(proton distribution) to histology (tissue types).

registration has been addressed before by others, for which

the approach can be considered feasible for general applica-

tion. We briefly demonstrate this, in terms of robustness and

computational efficiency, by using a sufficiently accurate pre-

diction and intra-modality registration scheme. The particular

contribution here is the use of machine learning paradigms

and pattern recognition methods in the image registration

problem context.

While we setup initial experiments only herein, more com-

prehensive studies and applications are desireable: Investi-

gating other modalities, as well as different image represen-

tations and non-parametric regression algorithms and regis-

tration applications exploiting the advantage in computation

time, such as 2-D/3-D or 3-D/3-D applications. Here a model

can be adapted on small subsets of the data, e.g. individual

slices, and used for registering the full data set. Additionally,

learning relations between modalities can be interesting from

a broader perspective, while the neural model here has the

disadvantage of having a black-box characteristic.

V. ACKNOWLEDGEMENT

The authors would like to thank U. Siebert and B. Zeike

(IPK Gatersleben, Germany) for providing manual reference

alignments of images and material preparation, and Frank

Volke and Bertram Manz (IBMT St. Ingbert, Germany) for

NMRi measurements.

REFERENCES

[1] L. Brown, “A survey of image registration techniques,” ACM Com-puting Surveys (CSUR), vol. 24, no. 4, pp. 325–376, 1992.

[2] B. Zitova and J. Flusser, “Image registration methods: a survey,” Imageand Vision Computing, vol. 21, no. 11, pp. 977–1000, 2003.

[3] J. Modersitzki, Numerical Methods for Image Registration. OxfordUniversity Press, 2004.

[4] K. Rohr, Landmark-Based Image Analysis: Using Geometric andIntensity Models. Kluwer Academic Publishers, 2001.

[5] L. Ibanez, W. Schroeder, L. Ng, and J. Cates, The ITKSoftware Guide, 2nd ed., Kitware, Inc. ISBN 1-930934-15-7,http://www.itk.org/ItkSoftwareGuide.pdf, 2005.

[6] J. Hipwell, G. Penney, R. McLaughlin, K. Rhode, P. Summers,T. Cox, J. Byrne, J. Noble, and D. Hawkes, “Intensity-based 2-D-3-D registration of cerebral angiograms,” Medical Imaging, IEEETransactions on, vol. 22, no. 11, pp. 1417–1426, 2003.

[7] J. Maintz, P. Van den Elsen, and M. Viergever, “Comparison offeature-based matching of CT and MR brain images,” Lecture Notesin Computer Science, pp. 219–219, 1995.

[8] A. Roche, X. Pennec, M. Rudolph, D. Auer, G. Malandain, S. Ourselin,L. Auer, and N. Ayache, “Generalized correlation ratio for rigidregistration of 3-D ultrasound with MR images,” in Proceedings ofthe Third International Conference on Medical Image Computing andComputer-Assisted Intervention. Springer-Verlag London, UK, 2000,pp. 567–577.

[9] P. Viola and W. Wells III, “Alignment by maximization of mutualinformation,” International Journal of Computer Vision, vol. 24, no. 2,pp. 137–154, 1997.

[10] F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, and P. Suetens,“Multimodality image registration by maximization of mutual infor-mation,” Trans. on Med. Imaging, vol. 16(2), pp. 187–198, 1997.

[11] D. Mattes, D. Haynor, H. Vesselle, T. Lewellen, and W. Eubank, “PET-CT image registration in the chest using free-form deformations,”Medical Imaging, IEEE Transactions on, vol. 22, no. 1, pp. 120–128,2003.

[12] V. Kyrki, J. Kamarainen, and H. Kalviainen, “Simple Gabor featurespace for invariant object recognition,” Pattern Recognition Letters,vol. 25, no. 3, pp. 311–318, 2004.

[13] D. Clausi and M. Ed Jernigan, “Designing Gabor filters for optimaltexture separability,” Pattern Recognition, vol. 33, no. 11, pp. 1835–1849, 2000.

(a)

(b)

Fig. 4: Observing the registration process of both methods in

transform space for T6 ([−50µm, 50µm, 180]). The com-

puted transformations are marked (see Table I). 4a illustrates

the instability of MI in terms of oscillations of metric values

along with ten-fold more iterations spent, and instability of

the optimization resulting in individual endpoints far away

from the true solution compared to SSD based registration

(4b).

[14] I. Daubechies, “The wavelet transform, time-frequency localiza-tion and signalanalysis,” Information Theory, IEEE Transactions on,vol. 36, no. 5, pp. 961–1005, 1990.

[15] T. Lee, “Image representation using 2-D gabor wavelets,” PatternAnalysis and Machine Intelligence, IEEE Transactions on, vol. 18,no. 10, pp. 959–971, 1996.

[16] H. Stern, “Neural networks in applied statistics,” Technometrics,vol. 38, pp. 205–214, 1996.

[17] C. Bishop, Neural Networks for Pattern Recognition. Oxford Uni-versity Press, USA, 1995.

[18] H. White, “Connectionist nonparametric regression: multilayer feed-forward networks can learn arbitrary mappings,” Neural Networks,vol. 3, no. 5, pp. 535–549, 1990.

[19] V. Kurkova, “Kolmogorov’s theorem and multilayer neural networks,”Neural Networks, vol. 5, no. 3, pp. 501–506, 1992.

[20] G. De Rumelhart and R. Williams, “Learning internal representationsby error propagation,” Parallel Distributed Processing: Explorationsin the Microstructure of Cognition, vol. 1, p. 319, 1896.

[21] R. Pielot, U. Seiffert, B. Manz, D. Weier, F. Volke, and W. Weschke,“4D warping for analysing morphological changes in seed develop-ment of barley grains,” in VISAPP (1), A. Ranchordas and H. Araujo,Eds. INSTICC - Institute for Systems and Technologies of Informa-tion, Control and Communication, 2008, pp. 335–340.