Fuzzy-Entropy based Image Congealing - IEEE Xplore

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Aberystwyth University. Aberystwyth, Ceredigion, Wales. UK. Email: [email protected]. Harry Strange. Dept. of Computer Science. Aberystwyth University.
Fuzzy-Entropy based Image Congealing Neil Mac Parthal´ain

Harry Strange

Dept. of Computer Science Aberystwyth University Aberystwyth, Ceredigion, Wales. UK Email: [email protected]

Dept. of Computer Science Aberystwyth University Aberystyth, Ceredigion, Wales. UK Email: [email protected]

Abstract—Group-wise image alignment or image congealing is an image processing technique which allows the joint alignment of a collection of images. Typically, information theoretic metrics have been employed as the objective function for the assessment of the process of alignment of the images for such methods. However, these objective functions rely on their probabilistic foundations and cannot model the underlying vagueness or uncertainty that is captured by approaches such as those based on fuzzy sets. In this paper a novel fuzzy-entropy based approach is presented for the task of image congealing. This approach allows for much flexibility in terms of employing different definitions for both similarity and fuzzy-entropy. Indeed, the existing approach for image congealing is subsumed as a special case of the approach proposed in this paper. The novel fuzzyentropy congealing technique is applied to different benchmark problems and also to a medical imaging dataset with good results. Index Terms—fuzzy sets, fuzzy-entropy, image processing, image alignment

I. I NTRODUCTION Image congealing [1] which may also be known as groupwise image alignment is the process of joint alignment of a set of images of a particular class. It essentially considers a set of images (or more generally, a set of arrays) and transforms them with respect to a continuous set of predefined transformations in order to make them more ‘similar’, according to some measure of assessment. The alignment, which can be considered in terms of a pixel-wise granularity, can be carried-out through an iterative process of spatial and rotational transformations of the images under consideration. Image congealing has been employed for a variety of different tasks and has had numerous different modifications [2], [3], [1]. However, most of these modifications relate to the tranformations employed [2], as well as the parameterisation of the technique itself [3]. There has been little work which considers the modification of the information theoretic objective function or metric. The origins of information theory [4] lie formally with communication systems. However, it has also proven popular in more recent times in other areas. In particular, information theory has been applied with success to a wide variety of machine learning tasks [5] such as: clustering, classification and feature selection. Information theory is concerned with the measurement of information and is defined as the amount of information transmitted in an event, with respect to the probability of that event. The mean information over all events

is known as the information entropy (IE) [6] and essentially deals with the probabilistic uncertainty. Fuzzy set theory (FST) [7] is a way of modelling the uncertainty associated with imprecise information. FST generalises classical set theory and can be used in a wide range of domains in which information is imprecise or noisy. Indeed, such situations often arise when dealing with image data, and more particularly medical imaging data, where the ability to handle noise in a structured manner is an important consideration. Information theory is unable to capture the essential property of partial membership to a given concept or concepts [7], which is the goal of FST. Since IE deals with crisp probabilistic uncertainty, and fuzzy sets with vagueness or imprecision, the motivation for extending the objective function used in image congealing to the fuzzy case is clear. This paper presents a novel fuzzy-entropy [8] based algorithm for the task of image congealing or group-wise image alignment. The algorithm is based on the well-known approach in [1], but uses a fuzzy-entropy framework which offers flexibility in terms of the definitions of fuzzy-entropy [9], [10], [11], and the fuzzy similarity relations which can be employed [12]. The definition of fuzzy-entropy adopted for the work described in this paper collapses to the traditional definition of IE [6] when all of the similarities are crisp. This means that the proposed approach can subsume the original alignment objective function as a special case whilst simultaneously being able to model the fuzziness or uncertainty related to partial membership. Indeed, the proposed framework has the potential to be extended to employ other fuzzy-based metrics or interpretations of fuzzy sets [12]. The remainder of this paper is structured as follows. Section 2 summarises the theoretical basis and concepts of image congealing, information entropy and fuzzy-entropy. Section 3 describes the fuzzy-entropy based image congealing approach and corresponding algorithm. Section 4 shows the results of applying the fuzzy-entropy based approach to a number of datasets, as well as a real-world medical imaging task. Section 5 concludes the paper with some discussion, and suggestions for future work. II. I MAGE C ONGEALING Much work has been carried-out in the area of group-wise image alignment, particularly in the area of medical image analysis [13]. One of the more popular methods for group-wise

The transformation for the j-th image, as denoted Uj is composed of component transforms: x-translation (tx ), ytranslation (ty ), rotation (θ), x-scale (sx ), y-scale (sy ), x-shear (hx ), and y-shear (hy ). The affine transformation matrix Uj can then be built such that U =F (tx , ty , θ, sx , sy , hx , hy )   1 0 tx cos θ − sin θ 0 cos θ 0 =  0 1 ty   sin θ 0 0 1 0 0 1  s   ex 0 0 1 hx 0  0 esy 0   0 1 0   0 0 1 0 0 1

  1 hy 0

 0 0 1 0  0 1

Fig. 1. Image Congealing - a set of pixels drawn from the same location in the the set of n images

image alignment is image congealing [1]. The primary aim of image congealing is to align or reduce the variability within a set of images. In order to achieve this, a metric or objective function is employed. By optimising the objective function, the images are aligned with respect to a set of predefined image transforms. Given a ‘stack’, I, of n images of size m pixels, a single pixel value in this stack is denoted xji where i ∈ [1, m] and j ∈ [1, n] as illustrated in Fig.1. The ‘stack’ of pixels {x1i , x2i , . . . , xni } is denoted xi . Each image, I ∈ I, is independently transformed by an affine transformation U. The transformation for the j-th image is denoted Uj and the transformed pixel stack is denoted xj 0 . The image congealing algorithm seeks to iteratively minimise the entropy across the stack of images by transforming each image by a small amount with respect to a set of possible affine transforms. The transform that gives rise to the greatest decrease in entropy is discovered via a hill-climbing approach. At each iteration of the algorithm, the goal is to minimise the cost function which corresponds to the sum of entropy across all images in the image ‘stack’ (as illustrated in Fig.1):

f (I) =

n X i=1

 −

 X j0 X j0 1 1  x (k)log2 x (k) (1) n j i n j i 

X m

where xji (k) is the probability of the k-th element of the multinomial distribution in xji . Since this minimisation corresponds to minimising the total joint entropy of pixels across the stack of images and can be reformulated simply as f (I) =

n X

H(Di )

(2)

i=1

where H(Di ) is the entropy across the distribution field of probabilities for the i-th image [1]. The minimisation can be performed by finding the transformation matrix U, for each image, that maximises the log-likelihood of the image with respect to the distribution of pixels across the stack.

After each iteration the scale of all the transforms, U1 , . . . , Un , are readjusted by the same amount so that the mean log-determinant of the transforms is 0, that is, the set of all transforms is zero-meaned. The rational for this step is to prevent the algorithm from succumbing to a situation where all of the images are shrunk to a point where they are no longer representative, as a result of an ever decreasing value of entropy. Once the congealing algorithm has converged, or has reached a predefined number of iterations, the congealed versions of the input images can be revealed by multiplying the original images by their associated transformation matrix. A. Entropy and Fuzzy-entropy Classical Entropy may be defined as a measure of the degradation or dispersal of energy and also as the energy form of a system that relates to its internal state of disorder or randomness. Entropy may also be described as a measure of progress of a process of equalisation. It is often used in relation to thermodynamic or metabolic biological processes. High entropy values are indicative of disordered states, and low entropy values are characteristic of ordered states. Information entropy or Shannon entropy [6] is also a measure of the amount of disorder of a system and can be defined as: H(X) = −

N X

pi log2 pi

(3)

i=0

The entropy of the event X is the sum, over all possible outcomes i of X , of the product of the probability of outcome i times the log of the probability of i, and is measured in ‘bits’ of information. This can also be applied to a general probability distribution, rather than a discrete-valued event.The IE value tends to zero with increasing order in any system. There have been many attempts to fuzzify IE, thus resulting in various definitions of fuzzy-entropy [9]. It should be emphasised that these measures are quite different from classical IE, as no probabilistic component is required for such definitions. This is due (as mentioned previously) to the fact that fuzziness

models the uncertainty associated with vagueness. Fuzzyentropy is therefore defined using the concept of membership. The first attempts at defining fuzzy-entropy (FE) were made in [10], and it is expressed based on the concept of membership function, where there are n membership functions (µi ): HA = −K

n X

{µi log(µi ) + (1 − µi )log(1 − µi )}

equivalence relation induces a fuzzy partition. This property is important as it ensures that when all relations are crisp that the resulting partition is also crisp. For a finite set U, A is a fuzzy or real-valued attribute set, which generates a fuzzy equivalence relation R on U. The fuzzy relation matrix is previously defined as M (R), and the fuzzy equivalence class generated by xi and R can be defined thus:

(4) [xi ]R =

i=1

where µi is the membership function and K is a constant (1/n). Note that H is expressed in terms of log10 and not the log2 function ln used for IE. This definition can therefore be interpreted as the average amount of information arising from fuzziness. In addition to this first definition, the work of [10] also proposed four properties that must hold for any fuzzy-entropy definition: • HA = 0, iff A is a crisp set (µi = 0 or 1, ∀xi ∈ A). • HA is maximum iff µi = 0.5, ∀xi ∈ A. ∗ ∗ ∗ • H ≥ H where H is the entropy of A , a sharpened version of A. • H = H, where H is the complement of A (A). Later work on fuzzy-entropy included the consideration of the fuzzy-entropy of a fuzzy set [8]. It is this later work that is used as the basis for the definitions utilised in this paper. III. F UZZY-E NTROPY BASED I MAGE C ONGEALING In this section the new FE-based congealing approach is described. The definition of FE described here is derived from the work in [14] and [8]. Many different definitions for FE can be used, however, the work here utilises a particular interpretation which has one specific property as mentioned previously. The definition of FE which is employed here is based on fuzzy-similarity relations, and as such has the desirable property that it collapses to classical IE [6], when the similarities between objects are crisp. The means that the approach subsumes the original algorithm as well as offering the ability to model fuzzy uncertainty. Also, by adopting such an approach, no additional subjective thresholding information or ‘fuzziness’ parameter is required; only the information in the data is utilised. As mentioned previously, the work of [8] is used as a foundation for the proposed fuzzy extension to image congealing. For a non-empty finite set X, R is a binary relation defined on X, denoted by a relation matrix M (R):   r11 r12 · · · r1n  r21 r22 · · · r2n    M (R) =  . .. ..  ..  .. . . .  rm1 rm2 · · · rmn where rij ∈ [0, 1] is the (similarity) relation value of xi and xj . It is important to note that a crisp equivalence relation will generate a crisp partition of the universe, whereas a fuzzy

ri2 rin ri1 + + ··· + x1 x2 xn

(5)

A number of different similarity relations can be used to induce such a matrix [12]. The cardinality of [xi ]R is then represented by |[xi ]R |. The information quantity or fuzzyentropy of the fuzzy equivalence relation is then defined as: n

1X log λi . H(R) = − n i=0

(6)

where: |[xi ]R | (7) n Using this definition of fuzzy-entropy, a new fuzzy information metric can be defined for the image congealing problem. This metric is employed to measure the uncertainty of the pixel-wise fuzzy-entropy for all of the images under consideration. Recall the concept of ‘image stack’ introduced in section II; by taking all of the pixel grey-level values across the stack, a metric based upon the entropy is used as an assessment of alignment. By modifying this metric, the newly described fuzzy-entropy metric can be employed in its place. The first step in formulating the FE metric is to induce a fuzzysimilarity relation matrix. This is also achieved by using the pixel grey-level values in this particular case, although there are other image characteristics that can be used [1]. The similarity relation matrix is essentially a symmetric matrix which allows the comparison of all of the objects to each other. The matrix can be constructed by employing a fuzzysimilarity relation in order to determine the pairwise similarity of objects. RP is the fuzzy-similarity relation induced by a pixel P : λi =

µRP (x, y) = Ta∈P {µRa (x, y)}

(8)

µRa (x, y) is the degree to which objects (image pixels in the ‘stack’) x and y are similar for that indexed pixel a, and may be defined in many ways [12], thus allowing much flexibility in how the matrix is constructed. For the work in this paper the fuzzy-similarity relations defined in (9), (10) and (11) are employed. µRa (x, y) = 1 −

|a(x) − a(y)| |amax − amin |

(a(x) − a(y))2 µRa (x, y) = exp − 2σa2 

(9)  (10)

  (a(y) − (a(x) − σa )) µRa (x, y) = max min , (a(x) − (a(x) − σa ))   ((a(x) + σa ) − (a(y)) ,0 ((a(x) + σa ) − (a(x))

(11)

These relations are illustrated in Fig.2, and demonstrate the different characteristics of each similarity relation. The relations are referred to as sim1, sim2 and sim3 respectively for the remainder of this paper. The work proposed here is an extension of the congealing algorithm in [1], and modifies the alignment assessment criterion or objective function. The proposed work therefore uses the pixel values in order to construct a fuzzy-similarity relation matrix as described previously. Such matices are generated using the pixel grey-level values and their related-ness is calculated using a fuzzy-similarity relation. This allows all of the pixel values of an image ‘stack’ and their overall relatedness to be considered in the context of all of the images. Each fuzzy-similarity relation matrix generated is then used in order to calculate the fuzzy-entropy as defined in (6). Although beyond the scope of this work, other fuzzy/hybrid similarity metrics could also be employed for the task of congealing at this point - see future work. For the task of assessing the alignment, the fuzzy-entropy measure of each pixel stack is then combined to form the joint fuzzy-entropy for all images. This metric is then used as a guide to aligning the images for the process of congealing at each iteration of the algorithm. IV. E XPERIMENTAL E VALUATION In this section the novel fuzzy-entropy based congealing method is compared with the original approach described in [1]. Two different types of data have been considered for this evaluation: i) benchmark data, which consists of various samples of hand-written digits between zero and eight, from [15] and ii) real-world medical imaging data for prostate cancer. A series of experiments are carried-out on both sets of data and the results are presented which demonstrate the advantages of the proposed method. 1) Experimental Setup: For the MNIST data in [15], two sets of experiments are carried-out. It is important to note that although the MNIST dataset is very large, it would be impractical to attempt to congeal all of the images or even subsets of those images for the work presented here. This is also the experimentation strategy adopted in the work in [1] (which this paper is based upon) where only a small number of randomly chosen images are used, such that the results can be easily illustrated. Also, given that one of the main comparisons in this section relates to IE congealing, it is appropriate that it is compared in a similar manner. The first experiment for the MNIST data compares the results of congealing 36 different (randomly chosen) handwritten instances of the the digit ‘0’ (zero).For the approach proposed in this paper, three different fuzzy-similarity relations sim1, sim2 and sim3 are utilised. The second set of experiments considers a collection of each of the handwritten digits:

’0’ to ’8’, with four randomly chosen instances per digit. For both sets of experiments based on this data, the maximum number of total iterations for both versions of the congealing algorithm was set to a value of 100. It should be noted that the FE-based methods often terminated in ≤15 iterations. For the prostate cancer medical imaging data [16], the maximum number of iterations of the congealing algorithm was set to a value of 100, however, the fuzzy congealing algorithm converged after much fewer iterations. Only seven out of a total 10 patients in the dataset have been included here, as the segmentation information was not included for 3 of the patients in the original dataset. For this particular task, only the results for the FE-based congealing which employs fuzzy-similarity relation sim1 are included here due to space constraints. Also, this produced the most interesting end result. For both types of data, the number of transformations per iteration for congealing was set to a value of 1, whilst the ordering of the transformations was the the default; xtranslation (tx ), y-translation (ty ), rotation (θ), x-scale (sx ), y-scale (sy ), x-shear (hx ), and y-shear (hy ). Since the approach presented in this paper is nondeterministic, any form of numerical assessment is fraught with problems. There are a number of ways in which this could be done, but these are outside the scope of this paper see future work. A. Benchmark Data This section describes the results obtained for the MNIST benchmark data [15]. As can be seen from Fig.3(c), for the first set of experiments, the fuzzy-entropy based congealing using sim1 results in a more faithful representation of the hand written digit ‘0’. This is particularly evident for those highlighted cases, with the instance in the second row showing significant difference with respect to the two metrics. In this case, the FE congealing succeeds in maintaining the correct position with respect to the others. The relations sim2 and sim3 although managing to maintain the position of the same instance in the vertical axis, it does seem to have shifted to the left somewhat. Other examples show that FE with sim1 in particular, managed also to preserve some of the intricacies of the original images, that are otherwise destroyed or at least ignored by the probability-based method. The results shown for sim2 and sim3 in Fig.3 (d) and (e), are not as marked as those for sim1 (c) when compared with IE-based entropy, although there is improved or comparable performance with respect to IE-based congealing. it is interesting to note that sim1 and sim2 are more strict in their treatment of the data and consequently the resulting transformations are limited to some degree. Indeed, looking at the corresponding mean images in Fig.4 it is clear that these particular similarity relations result in different initial and final images. It should noted that the dataset presented here is a relatively simple problem and that similarity relations sim2 and sim3 may be better suited to problems which are a little more complex or where the ‘boundaries’ are more crisp.

(a)

(b)

(c) Fig. 2.

Fuzzy Similarity Relation Functions; (a) represents the function in eqn.(9) or sim1, (b) that of eqn.(10) or sim2, and (c) eqn.(11) or sim3

(a)

(b)

(d)

(c)

(e)

Fig. 3. The results for traditional IE-based congealing (b) and FE-based congealing (using sim1 (c), sim2 (d), and sim3 (e)) on the a random subset of the MNIST zeros dataset (a). Highlighted are the congealed images that differ most for the two approaches. In particular, IE-based congealing inadvertently tends to ‘hide’ the smaller instances within the boundary of others, where as FE-based congealing does not.

Examining the mean images generated for this dataset in Fig.4, it is easy to appreciate how these metrics consider different aspects of uncertainty. In Fig. 4 (a), (b), (c), and (d) the ‘mean’ images for the MNIST ‘0’ dataset are presented. It can be seen that the IE attempts to make the mean image as ‘crisp’ as possible in (e), whilst the FE-based methods in (b),(c) and (d) are concerned with maximising the certainty in the boundaries of the digit as much of the data in the ’centre’ is very similar (f-h).

Another interesting aspect which was observed during this set of experiments is related to the stability of the employed metrics. There is a tendency for IE-based congealing to ’overcongeal’ images if the number of iterations for the algorithm is too-high. That is, IE-based congealing continues to attempt to drive the images to a more transformed state, as the IE value is indicating that each new set of transformations results in a decrease in entropy. However, this decrease in entropy does not necessarily translate into images that are more closely

(a)

(b)

(d)

(c)

(e)

Fig. 5. When attempting to congeal data with a large amount of variation, IE-based congealing tends to ‘over-congeal’, whereas FE-based congealing will produce a stable result (after 100 iterations). Note that FE congealed images produce an alignment within each digit class (that is, the ‘one’s are aligned with themselves, as are the ‘three’s, etc.).

aligned. This situation is demonstrated clearly in Fig.5, where (b) shows the results of congealing the data with IE after 100 iterations. Although the result shown has better score for IE, it clearly does not offer improved alignment. When compared with FE-based congealing in (c), it is clear that once the FE score stabilises, regardless of the number of predefined iterations, the FE-based method does not attempt to drive the congealing process to a state where the images under consideration become over-congealed. Although, for FE-based congealing using sim2 and sim3 there are some cases where the image is over-transformed. When these results are compared with IE-based congealing, the general trend

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

is to preserve the instances under consideration, with sim2 performing better in this respect than sim3. B. Medical Imaging Data To further test the validity of the FE-based congealing approach for the task of image registration, a set of prostate MRI images from different patients were registered using the proposed approach1 . There is a high variability in the shape of the prostate between patients often caused by different patient 1 The XNAT prostate data was used for these experiments. Data available from http://bit.ly/SRobLk (Date Checked: 16/11/2012)

(a) Fig. 4. The mean image for the MNIST ‘zero’ stack prior to performing congealing (a) and after IE-based congealing (e). The fuzzy-similarity image using sim1, sim2 and sim3 prior to congealing (b-d) and the fuzzysimilarity image using the same relations after fuzzy based congealing (f-h).

(b)

Fig. 6. The middle prostate image from a single patient’s volume before (a) and after congealing (b). Size and rotation, etc. has changed but no major artefacts have been introduced. Figures are cropped but not scaled (thus parts of the spine are not visible in (a) that are visible in (b) after congealing has scaled the image).

positions before and during surgery. As such, the registration of prostate MRI images is an important step in the computer aided diagnosis toolchain. The data consists of seven T2-weighted MRI images, each from a different patient, alongside expert segmentations for each MRI image. The middle image from each MRI stack was chosen and the fuzzy congealing algorithm was initially applied on the segmentation images and the learnt transforms were then applied to the original MRI images. The results in Fig.7 show the segmentations before (top set of images) and after (bottom set of images) congealing along with the fuzzy-similarity images of the two sets. The congealed segmentations have a much more compact overlap, with the change in fuzzy-similarity image clearly depicting this. The larger prostates have been reduced in size while the smaller prostates have been enlarged. Due to the fact that the transformations are affine there are no changes to the overall shape of the prostates, that is, the lobes on the lower half of the prostate remain intact. This is further demonstrated by the transformed image in Fig.6. Here, the original image has been transformed using the affine transformation matrix learned as a result of congealing. The resulting image shows that, although undergoing scale, rotation, and translation changes, no undesired artefacts are introduced. V. C ONCLUSION This paper has presented a novel algorithm for the alignment of images using congealing based on a fuzzy-entropy metric. The use of a fuzzy-based metric has demonstrated improved performance over the original information theoretic metric, and the results also demonstrate that different definitions of fuzzy-similarity can be used in order to exploit the flexibility of the approach. Further work in this area will include a more in-depth experimental investigation into the performance of fuzzy-congealing for larger medical imaging datasets such as mammographic data. Also, it would be interesting to see how the choice of similarity relations affects the results obtained from such data. Whilst a particular definition of fuzzy-entropy is employed for the work described in this paper, (due mainly to the fact that it collapses to IE when the similarities are crisp) there are many others which are equally applicable [9]. A comparison of these different definitions in order to explore their different properties, would form a series of topics for further investigation. The work in this paper has focussed on the use of fuzzyentropy as an assessment for alignment. However, given the encouraging results obtained, it would also be interesting to further extend the approach to consider the case of fuzzysimilarity or fuzzy hybrid metrics for alignment. ACKNOWLEDGMENT Neil Mac Parthal´ain would like to acknowledge the financial support for this research through NISCHR (National Institute for Social Care and Health Research) Wales, grant Number: RFS-12-37.

Harry Strange would like to acknowledge the financial support for this research through the NISCHR BRU Advanced Medical Imaging and Visualisation Unit. R EFERENCES [1] E. Learned-Miller, “Data driven image models through continuous joint alignment”, Pattern Analysis and Machine Intelligence, IEEE Transactions on, vol. 28, no. 2, pp. 236-250, 2006. [2] M. Cox, S. Lucey, S. Sridharan, and J. Cohn, “Least-Squares Congealing for Large Numbers of Images”, Proceedings of the 12th IEEE International Conference on Computer Vision (ICCV), pp.1949–1956 2009. [3] E. Learned-Miller and V. Jain, “Many heads are better than one: Jointly removing bias from multiple MRs using nonparametric maximum likelihood”, Proceedings of Information Processing in Medical Imaging, pp. 615-626, 2005. [4] C.E. Shannon, “A mathematical theory of communication”. Bell System Technical Journal, vol. 27, pp. 379423, and pp. 623-656, 1948. [5] R. Kohavi, and M. Sahami, “Error-based and entropy-based discretization of continuous features”. In Proceedings of the second international conference on knowledge discovery and data mining, pp.114–119 1996. [6] C.E. Shannon, “Prediction and entropy of printed English”, Bell System Technical Journal, vol.30, pp.50-64, 1951. [7] L.A. Zadeh, “Fuzzy sets”, Information and Control, vol. 8, no. 3, pp. 338-353, 1965. [8] B. Kosko. “Fuzzy entropy and conditioning”. Information Sciences, Vol. 40, No. 2, pp. 165-174. 1986. [9] S. Al-Sharhan, F. Karray, W Gueaieb, and O. Basir, “Fuzzy entropy: a brief survey”, Proceedings of the 10th IEEE International Conference on Fuzzy Systems, pp. 1135–1139, 2001. [10] A. D. Luca and S. Termini. “A definition of non probabilistic entropy in the setting of fuzzy set theory”. Information and Control, vol. 20, pp. 301–312, 1972. [11] B. Liu, ”A survey of entropy of fuzzy variables.” Journal of Uncertain Systems, vol.1 no.1, pp 4–13, 2007. [12] D. Li, and C. Cheng, “New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions”. Pattern Recognition Letters, vol. 23, no. 1-3, pp. 221-225, 2002. [13] S. Balci, P. Golland, M. Shenton, and W. Wells, “Free-form B-spline deformation model for groupwise registration”. Proceedings of MICCAI ’07, pp. 23-30, 2007. [14] Q. Hu, D. Yu, and Z. Xie, “Information-preserving hybrid data reduction based on fuzzy-rough techniques”, Pattern Recognition Letters, vol. 27, no.5, pp. 414–423, 2006. [15] A. Frank and A. Asuncion, “UCI Machine Learning Repository” [http://archive.ics.uci.edu/ml]. Irvine, CA: University of California, School of Information and Computer Science, 2010. [16] F. Jolesz, BWH MRI Prostate data. 10 datasets, including a derived segmentation series with labelmaps. [http://bit.ly/TxINcm).] 2010.

Fig. 7. The top set of images shows the segmentations of the middle prostate image across 7 different patients with the last image depicting the fuzzysimilarity image prior to congealing. The second set shows the same segmentations after fuzzy congealing has been applied with the final image again showing the fuzzy-similarity. The fuzzy-similarity image shows that there is a much more compact ‘mean’ image whilst the individual shape characeteristics are also preserved.