MORPHOLOGICAL IMAGE INTERPOLATION A study and a proposal

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1 MORPHOLOGICAL IMAGE INTERPOLATION A study and a proposal Alumno : Javier Vidal Valenzuela 1 Tutor: Jose Crespo del Arco 1 1 Facultad de Informática Universidad Politécnica de Madrid Boadilla del Monte (Madrid), SPAIN jvidal@infomed.dia.fi.upm.es, jcrespo@fi.upm.es Abstract This research concern to morphological image interpolation. Interpolation is a very important step in many application of image processing. This report presents a list of morphological interpolation methods and describes briefly a proposal of a morphological interpolation technique for binary images. Keywords: Mathematical Morphology, Imaging, Interpolation 1. Introduction In many application of imaging, data are composed of different slices. In particular, this situation arises when we process volumetric medical images, as well as in video images. In the first case slices corresponds to spacial dimension, whereas in the second case slices correspond to a temporal dimension. It is often the case that the distance between adjacent elements in adjacent slices is much larger (until 10 times) than the distance between adjacent image elements in a slice. Therefore, it is often useful to be able to interpolate data between adjacent slices, and many interpolation techniques have been developed [10]. The objective is normally to produce a set of intermediary slices between two known ones. Particularly, there exists a more recent category of interpolation techniques, called shape-based interpolation [4], which attempt to incorporate knowledge about the image contents to the interpolation process. Morphological interpolation [1, 3, 6, 15] is a kind of the shape-based interpolation techniques.

2 2 This research project concern to the use of mathematical morphology [3, 13, 14] for image interpolation. In particular, the overall study was based on binary images. 2. Bibliographical notes Application of mathematical morphology into image interpolation is a recent subject. This area of image processing was opened to mathematical morphology in 1994 by J. Serra [15], F. Meyer [11] and S. Beucher [1] at the Centre de Morphologie Mathematique at Fontainebleau (Paris), France. Serra proposed an interpolation method for binary images using Hausdorff distance. On the other hand, Meyer introduced a concept of interpolation functions. Finally, the interpolation using median sets was presented by Beucher. In 1995 Guo et al.[5] reported a morphological interpolation using, by they called, weighted dilation and weighted erosion. In 1996, F. Meyer published the first paper related with mosaic image interpolation [12]. After that, in 1999 Iwanowski and Serra [8] used the concept of median sets to produce the first method to interpolate color images. Morphological skeletons was part of an interpolation method developed by V. Chatzis and I. Pitas [2] in An important contribution to this subject was the Iwanowski s PhD Thesis [6], which summarizes a group of method (some of them mentioned here) and proposed a new techniques to interpolate binary, mosaic, graytone and color images. After that, he published an article [7] at International Conference on Computer Vision and Graphics related with using of a convex mask to improve the interpolation using Hausdorff distance. Finally, Lee and Wang [9] proposed an interpolation technique based on dilation but, whose main contribution is their concern at homotopy aspects and the procedure exposed. 3. A proposal The analysis of methods mentioned above led us to a proposal of a binary image interpolation technique based on the notion of median sets. This method is motivated by the following. If we denote as A 1 and B 1 two sets of input slice 1, such as B 1 A 1 ; and C 2 and D 2 two sets of input slice 2, such as D 2 C 2. If there is a correspondence between these two pairs (i.e., we want to interpolate A 1 with C 2, and B 1 with D 2 ), then the following condition should be satisfied: Inter[A 1 \ B 1, C 2 \ D 2 ] = Inter[A 1, C 2 ] \ Inter[B 1, D 2 ] (1) where Inter denotes our interpolation operator, We consider equation (1) as fundamental in order to maintain inclusion relationships between interpolated structures. Our technique can be viewed as a

3 Experimental Results 3 generalization of that expression, where the connected components (CCs) of the slices are treated recursively. In our method we reduce each CC to a point, using the minimal skeleton by pruning (MSP) [16]. The original skeleton is reduced by pruning until a final point is reached (in the case of a CC with no holes). This minimal reference of a set is better for our purposes than others (such as, for example the centroid), because the MSP will always be part of the CC (necessary in general for the interpolation using median sets). 4. Experimental Results Finally, in order to show the results of the proposed methods we offers a set of examples considering firstly, the most simple case, the interpolation of CCs with matching between them and then, we extend this case to handle two more complicated cases. Interpolation of CCs with matching Figure 1 shows an example of interpolation between a circle-shape connected component and a square-shape connected component. The resulting interpolated slices are at the center of the image. Figure 1. Interpolation of CCs with matching Interpolation of CCs without matching The second example illustrates the case where CCs do not match, which produce interpolated images with two CCs, one corresponding to the CC in slice 1 and the second corresponding to the CC in the slice 2 (see Figure 2). Figure 2. Interpolation of CCs without matching Interpolation of holed sets Finally, we illustrates the interpolation of the holed sets. Figure 3 shows at the extreme positions the input slices and the 7 slices between them are the interpolated ones.

4 4 Figure 3. Holed sets interpolation

5 References [1] Serge Beucher. Interpolations d ensembles, de partitions et de fonctions. Technical Report N-18/94/MM, Centre de Morphologie Mathématique, Mai [2] Vassilios Chatzis and Ioannis Pitas. Interpolation of 3d binary images based on morphological skeletonizations. In Proceedings IEEE International Conference on Multimedia Computing Systems, Florence, Italy, volume II, pages , June [3] E.R. Dougherty and R.A. Lotufo. Hands-on Morphological Image Processingods in Imaging. SPIE Press, Bellingham, WA, [4] J. Zheng G.T.Herman and C.A.Bucholtz. Shape-based interpolation. IEEE Computer Graphics and Applications, 12(3):69 79, May [5] Jun-Feng Guo, Yuan-Long Cai, and Yu-Ping Wang. Morphology-based interpolation for 3D medical image reconstruction. Computarized Medical Imaging and Graphics, 19(3): , May-June [6] Marcin Iwanowski. Application of Mathematical Morphology to Image Interpolation. PhD thesis, School of Mines of Paris - Warsaw University of Technology, [7] Marcin Iwanowski. Morphological binary interpolation with convex mask. In Proceedings International Conference on Computer Vision and Graphics, Zakopane, Poland, September [8] Marcin Iwanowski and Jean Serra. Morphological interpolation and color images. In Proceedings of International Conference on Image Processing, Vennice, Italy, [9] Tong-Yee Lee and Wen-Hsiu Wang. Morphology-based three-dimensional interpolation. IEEE Transactions on Medical Imaging, 19(7): , July [10] Erik Meijering. A chronology of interpolation. from ancient astronomy to modern signal and image processing. Proceedinig of the IEEE, 90(3): , March [11] Fernand Meyer. Interpolations. Technical Report N-16/94/MM, Centre de Morphologie Mathématique, Mai [12] Fernand Meyer. A morphological interpolation method for mosaic images. In Mathematical Morphology and its Applications to Image and Signal Processing. Kluwer Academics Publishers, [13] Jean Serra. Mathematical Morphology, volume I. London : Academic Press, 1982.

6 6 [14] Jean Serra. Mathematical Morphology, volume II. London : Academic Press, [15] Jean Serra. Interpolations et distances of hausdorff. Technical Report N-15/94/MM, Centre de Morphologie Mathématique, Mai [16] Pierre Soille. Morphological Image Analysis: Principles and Applications. Springer- Verlag, 2nd edition edition, 2003.