FIXED POINTS AND FREEZING SETS IN DIGITAL TOPOLOGY L aurence B oxer 1 2 I ntro to digital topology In computer memory a digital image is not a set of continuous bodies Rather a digital image is a set of discrete ID: 767265
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FIXED POINTS AND FREEZING SETS IN DIGITAL TOPOLOGY Laurence Boxer 1
2 I ntro to digital topology In computer memory, a digital image is not a set of “continuous bodies.” Rather, a digital image is a set of discrete pixels or voxels . Usual approach to digital topology: Pixels as points in ; close pairs as edges of a graph. is a continuous function between digital images if close points have close images – that is, ( adjacent to ) implies either or .Notation: {) | is continuous}, the adjacency of . This approach yields some success in recognizing connectedness, contractibility, fundamental groups. However, fixed point properties of Euclidean objects and of their digital images can be very different.
3 Intro to digital topology – adjacencies The most commonly used adjacencies are the adjacencies. Given , in , , we have if f or at least 1 and at most distinct indices , , andfor all other indices , . adjacencies are also known by the number of pixels neighboring a given pixel in the adjacency. For example, adjacency in is 8–adjacency. adjacency in is 4–adjacency.
4 Intro to digital topology - connectedness Digital image is connected if given , there exists a path such that . -connected and -connected -connected; not -connected Theorem [9, 1]: is digitally continuous if and only if preserves digital connectedness.
5 Motivating tools - 1 If is a continuous map of a subset of a path-connected Euclidean space, with 2 fixed points, the image under of a path between the fixed points can be wildly distorted by . However, the analogous situation for digital images yields the following. Proposition 2 [7]. Let be a digital image and let ). Suppose such that there is a unique (graphically) shortest -path from to . Then . Sketched proof: Since , the uniqueness of implies . And if is not a subset of , it is easy to see that for some in , and would not be adjacent, contrary to the continuity of .
6 Motivating tools - 2Let be the projection to the coordinate: . Lemma 3. [4] Let be a digital image, , , in .If , then . Similarly, if , then .
7 Freezing sets - definition and basic propertiesDefinition 2 [4]: Let be a digital image. is a freezing set for if for every isomorphism , if is -continuous and , then . Theorem 4 [4]: Let be a digital image. is a freezing set for if and only if given ), implies . Trivially, is a freezing set for . How do we find proper subsets of that are freezing sets for ? How small can a freezing set be? Theorem 5 [4]: Let be a digital image. Let be a proper subset of that is a retract of . Then does not contain a freezing set for .
8 Proposition 7 [4]: Let be finite. Let . Let . Let . If , then . Sketch proof : Suffice to show Given , for each index there is a digital line segment containing with endpoints in such that is one-to-one. By Lemma 3, for all , . Hence . Boundaries
9 Example: Minimal freezing set for digital rectangle The 4 corner pixels make up a minimal freezing set . Proof : Let such that . By Proposition 2, the top and bottom rows of pixels must be fixed points. Similarly, Proposition 2 can now be applied to each column of pixels. Thus, , so is a freezing set.Minimality: if we remove a member of , the resulting set yields such that - indeed, .
10 Example: Minimal freezing set for digital rectangle In contrast to previous example, adjacency doesn’t lend itself to use of Proposition 2. We know from Proposition 7 that is a freezing set. That is minimal follows from: Proposition 10 [4]: If is a proper subset of then is not a freezing set for .Proof: Let such that . Want to obtain such that . If a corner point is in we can construct as in previous example. Otherwise, consider such that is not a corner. We obtain such that , as shown.
11 Other simple examples For a digital simple closed curve, a minimal freezing set has 3 points. For a wedge of 2 digital simple closed curves, a minimal freezing set has 4 points. For a tree, the set of leaf vertices is a minimal freezing set.
12 References[1] L. Boxer, A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision 10 (1999), 51-62.[2] L. Boxer, Remarks on fixed point assertions in digital topology, 2, Applied General Topology 20, (1) (2019), 155-175. [3] L. Boxer, Remarks on fixed point assertions in digital topology, 3, submitted. Available at https://arxiv.org/abs/1808.09903 [4] L. Boxer, Fixed point sets in digital topology, 2, submitted. Available at https :// arxiv.org/abs/1904.00534 [5] L. Boxer, O. Ege, I. Karaca, J. Lopez, and J. Louwsma, Digital fixed points, approximate fixed points, and universal functions, Applied General Topology 17(2), 2016, 159-172.[6] L. Boxer and P.C. Staecker, Remarks on fixed point assertions in digital topology, Applied General Topology 20 (1) (2019), 135-153.[7] L. Boxer and P.C. Staecker, Fixed point sets in digital topology, 1, submitted. Available athttps://arxiv.org/pdf/1901.11093.pdf [8] A. Rosenfeld, Digital topology, The American Mathematical Monthly 86 (8) (1979), 621-630.[9] A. Rosenfeld, ‘Continuous’ functions on digital pictures, Pattern Recognition Letters 4, pp. 177-184, 1986.