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Connectivity preserving transformations for higher dimensional binary images

机译：高维二进制图像的保持连接性的转换

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An N-dimensional digital binary image (1) is a function I : Z(N) -> {0, 1}. 1 is B3N-1, W3N-1 connected if and only if its black pixels and white pixels are each (3(N) - 1)-connected. I is Only B3N-1 connected if and only if its black pixels are (3(N) - 1)-connected. For a 3-D binary image, the respective connectivity models are B-26, W-26, and B-26. A pair of (3(N) - 1)-neighboring opposite-valued pixels is called interchangeable in a N-D binary image 1, if reversing their values preserves the original connectedness. We call such an interchange to be a (3(N) - 1)-local interchange. Under the above connectivity models, we show that given two binary images of n pixels/voxels each, we can transform one to the other using a sequence of (3(N) - 1)-local interchanges. The specific results are as follows. Any two B-26-connected 3-dimensional images I and J each having n black voxels are transformable using a sequence of O((c(1) + c(2))n(2)) 26-local interchanges. Here, c(1) and c(2) are the total number of 8-connected components in all 2-dimensional layers of I and J respectively. We also show bounds on B-26 connectivity under a different interchange model as proposed in [A. Dumitrescu, J. Pach, Pushing squares around, Graphs and Combinatorics 22 (1) (2006) 37-50]. Next, we show that any two simply connected images under the B-26, W-26 connectivity model and each having n black voxels are transformable using a sequence of O(n(2)) 26-local interchanges. We generalize this result to show that any two B3N-1, W3N-1-connected N-dimensional simply connected images each having n black pixels are transformable using a sequence of O(Nn(2))(3(N) - 1)-local interchanges, where N > 1.
机译：N维数字二进制图像（1）是函数I：Z（N）-> {0，1}。 1是B3N-1，W3N-1仅当且仅当其黑色像素和白色像素分别（3（N）-1）连接时才连接。仅当B3N-1的黑色像素连接（3（N）-1）时，我才连接。对于3-D二进制图像，相应的连接模型为B-26，W-26和B-26。在N-D二进制图像1中，一对（3（N）-1）相邻的相反值像素被称为可互换像素，如果反转它们的值可以保留原始的连通性。我们称这种交换为（3（N）-1）-局部交换。在上述连通性模型下，我们显示给定两个n像素/体素的二进制图像，我们可以使用（3（N）-1）-局部互换的序列将一个变换为另一个。具体结果如下。使用O（（c（1）+ c（2））n（2））26位局部互换的序列可变换任意两个具有B黑色26像素的B-26连接的3维图像I和J。此处，c（1）和c（2）分别是I和J的所有二维层中8个连接的组件的总数。我们还显示了在[A.]中提出的不同交换模型下B-26连接的界限。 Dumitrescu，J. Pach，推方四方，图和组合学（22）（1）（2006）37-50]。接下来，我们显示在B-26，W-26连通性模型下的任何两个简单连接的图像，每个图像都有n个黑色体素，可以使用O（n（2））26个局部互换的序列进行变换。我们对该结果进行一般化，以显示可以使用O（Nn（2））（3（N）-1）序列对任何两个具有N个黑色像素的B3N-1，W3N-1连接的N维简单连接图像进行变换。 -本地交换，其中N> 1。

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