Abstract
The quench function of a binary image is the distance transform of the image sampled on its skeleton. In principle the original image can be reconstructed from the quench function by drawing a disk at each point on the skeleton with radius given by the corresponding quench function value. This reconstruction process is of more than theoretical interest. One possible use is in coding of binary images, but our interest is in an applied image analysis context where the skeleton has been (1) reduced by, for example, deletion of barbs or other segments, and/or (2) labelled so that segments, or indeed individual pixels, have identifying labels. A useful reconstruction, or partial reconstruction, in such a case would be a labelled image, with labels propagated from the skeleton in some intuitive fashion, and the support of this labelled output would be the theoretical union of disks.
An algorithm which directly draws disks would, in many situations, be very inefficient. Moreover the label value for each pixel in the reconstruction is highly ambiguous in most cases where disks are highly overlapping. We propose a vector propagation algorithm based on Ragnelmalm’s Euclidean distance transform algorithm which is both efficient and provides a natural label value for each pixel in the reconstruction. The algorithm is based on near-exact Euclidean distances in the sense that the reconstruction from a single-pixel skeleton is, to a very good approximation, a Euclidean digital disk. The method is illustrated using a biological example of neurite masks originating from images of neurons in culture.
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References
P. Brigger, M. Kunt, and F. Meyer. The geodesic morphological skeleton and fast transformation algorithms. In J. Serra and P Soille, editors, Mathematical morphology and its applications to image processing, pages 133–140. Kluwer Academic Publishers, 1994.
O. Cuisenaire. Distance Transformations: Fast Algorithms and Applications to Medical Image Processing. PhD thesis, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, October 1999. http://www.tele.ucl.ac.be/PEOPLE/OC/these/these.html.
P.-E. Danielsson. Euclidean distance mapping. Computer Graphics and Image Processing, 14:227–248, 1980.
C.T. Huang and O.R. Mitchell. A Euclidean distance transform using grayscale morphology decomposition. IEEE Trans. Pattern Analysis and Machine Intelligence, 16(4):443–448, 1994.
Ch. Lantuéjoul. La Squelettisation et son Application aux Mesures Topologiques des Mosaiques Polycrystalline. PhD thesis, Ecole de Mines de Paris, 1978.
Andrew J.H. Mehnert and Paul T. Jackway. On computing the exact Euclidean distance transform on rectangular and hexagonal grids. J. Mathematical Imaging and Vision, 11:223–230, 1999.
A. Meijster, J.B.T.M. Roerdink, and W.H. Hesselink. A general algorithm for computing distance transforms in linear time. In J. Goutsias, L. Vincent, and D.S. Bloomberg, editors, Mathematical Morphology and its Applications to Image and Signal Processing, pages 331–340. Kluwer, 2000.
Tun-Wen Pai and John H. L. Hansen. Boundary-constrained morphological skeleton minimization and skeleton reconstruction. IEEE Trans. Pattern Analysis and Machine Intelligence, pages 201–208, 1994.
I. Ragnemalm. Fast erosion and dilation by contour processing and thresholding of distance maps. Pattern Recognition Letters, 13:161–166, 1992.
F. Shih and O.R. Mitchell. A mathematical morphology approach to Euclidean distance transformation. IEEE Trans. on Image Processing, 2(1):197–204, 1992.
P. Soille. Morphological Image Analysis. Springer-Verlag, 2003.
R. van den Boomgaard, L. Dorst, S. Makram-Ebeid, and J. Schavemaker. Quadratic structuring functions in mathematical morphology. In P Maragos, R. W. Schafer, and M. A. Butt, editors, Mathematical Morphology and its Application to Image and Signal Processing, pages 147–154. Kluwer Academic Publishers, Boston, 1996.
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Buckley, M., Lagerstrom, R. (2005). Labelled Reconstruction of Binary Objects: A Vector Propagation Algorithm. In: Ronse, C., Najman, L., Decencière, E. (eds) Mathematical Morphology: 40 Years On. Computational Imaging and Vision, vol 30. Springer, Dordrecht. https://doi.org/10.1007/1-4020-3443-1_12
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DOI: https://doi.org/10.1007/1-4020-3443-1_12
Publisher Name: Springer, Dordrecht
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