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Delta-sigma conversion for graph signals *
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Available online at www.sciencedirect.com The International Federation of Control The International Federation of Automatic Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 9303–9307
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N. Imoda∗ , S. Azuma∗ ,∗∗ , T.Kitao∗ ,∗∗∗ , T. Sugie ∗ N. Imoda∗∗ ,, S. Azuma∗∗ ,,∗∗ , T.Kitao∗∗ ,,∗∗∗ , T. Sugie ∗ N. N. Imoda Imoda , S. S. Azuma Azuma ,∗∗ ,, T.Kitao T.Kitao ,∗∗∗ ,, T. T. Sugie Sugie ∗ ∗ Graduate School of Informatics, Kyoto University, Kyoto, JAPAN ∗ School Kyoto, ∗ (e-mail: [email protected], [email protected]) ∗ Graduate Graduate School of of Informatics, Informatics, Kyoto Kyoto University, University, Kyoto, JAPAN JAPAN Graduate School of Informatics, Kyoto University, Kyoto, JAPAN ∗∗ (e-mail: [email protected], [email protected]) Currently, Graduate School of Engineering, Nagoya University, (e-mail: [email protected], [email protected]) (e-mail: [email protected], [email protected]) ∗∗ Currently, Graduate of Engineering, Nagoya University, ∗∗ Aichi, JAPAN (e-mail:School [email protected]) ∗∗ Currently, Graduate School of Currently, School of Engineering, Engineering, Nagoya Nagoya University, University, ∗∗∗ Graduate Aichi, JAPAN (e-mail: [email protected]) Currently, Murata Machinery, Ltd. (e-mail: Aichi, (e-mail: [email protected]) Aichi, JAPAN JAPAN (e-mail: [email protected]) ∗∗∗ Murata ∗∗∗ [email protected]) ∗∗∗ Currently, Currently, Murata Machinery, Machinery, Ltd. Ltd. (e-mail: (e-mail: Currently, Murata Machinery, Ltd. (e-mail: [email protected]) [email protected]) [email protected]) Abstract: In this paper, we establish a framework of delta-sigma conversion of graph signals. Abstract: In thisdelta-sigma paper, we establish aa framework delta-sigma conversion of graph signals. First, we define conversion for graph of signals by considering Abstract: In paper, of delta-sigma conversion of graph Abstract: In this thisdelta-sigma paper, we we establish establish a framework framework of delta-sigma conversionthe of corresponding graph signals. signals. First, we define conversion for graph signals by considering the corresponding conversion for standard time conversion series signals. We next propose aconsidering conversion the method based on First, we define delta-sigma for graph signals by corresponding First, we define delta-sigma conversion for graph signals by considering the corresponding conversion for standard time signals. We next propose conversion method based on spanning in-tree, for which it isseries proven that the method satisfiesaaa the conditions for delta-sigma conversion for time series signals. We next conversion method based conversion for standard standard time series signals. Wemethod next propose propose conversion method based on on spanning in-tree, for which it is proven that the satisfies the conditions for delta-sigma conversion. The proposed method is demonstrated by numerical examples and application to spanning in-tree, in-tree, for for which which it it is is proven proven that that the the method method satisfies satisfies the the conditions conditions for for delta-sigma delta-sigma spanning conversion. The proposed method is demonstrated by numerical examples and application to halftone image processing. conversion. The proposed method is demonstrated by numerical examples and application conversion. Theprocessing. proposed method is demonstrated by numerical examples and application to to halftone image halftone image processing. halftone image processing. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: graph signal, delta-sigma conversion, quantization, image processing Keywords: graph signal, signal, delta-sigma conversion, conversion, quantization, image image processing Keywords: Keywords: graph graph signal, delta-sigma delta-sigma conversion, quantization, quantization, image processing processing 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION Graph signal processing is signal processing on graphs [D. Graph signal processing isassignal signal processing onwhich graphsis [D. [D. I. Shuman et processing al. (2013)]is shown in Fig. 1,on an Graph signal processing graphs Graph signal processing isassignal processing onwhich graphsis [D. I. Shuman et al. (2013)] shown in Fig. 1, an extension of the conventional signal processing of discreteI. Shuman et al. (2013)] as shown in Fig. 1, which is an I. Shumanofetthe al.conventional (2013)] as shown in Fig. 1, which is an extension signal processing of discretetime signals. It conventional has become signal one ofprocessing the majorof topics in extension of discreteextension of the the signal discretetimeengineering signals. It conventional has become one as ofprocessing the majorof topics topics in the field,become especially a frequency-domain time signals. It has one of the major in time signals. It has become one of the major topics in the engineering engineering field, especially especially as aaas frequency-domain frequency-domain method for “networked signals” such digital image and the field, as the engineering field, especially as a frequency-domain for “networked “networked signals” such as digital digital image and and amethod collection of measurements of asuch sensor network. method for signals” as image method for “networked signals” such as digital image and a collection of measurements of a sensor network. aa collection of of network. of measurements measurements of aa sensor sensor network. Socollection far, several results have been obtained for the graph So far, several results have been obtained for[I.the the graph signal processing, such as sampling theorem Pesenson So far, several results have been obtained for graph So far,processing, several results have been obtained for[I.the graph signal such as sampling theorem Pesenson (2008)], up-/down-sampling [D. I. Shuman (2013)], signal processing, such theorem [I. Pesenson signal processing, such as as sampling sampling theoremet [I.al. Pesenson (2008)], up-/down-sampling [D. I. Shuman et al. (2013)], Fourier transformation [D. [D. I. I. Shuman (2008)], up-/down-sampling Shuman et al. (2008)], up-/down-sampling [D. I. Shumanet et al. al. (2013)], (2013)], Fourier transformation transformation [D. [D. K. I. Hammond Shuman etet al. (2013)], wavelet al. (2011)], Fourier transformation [D. I. Shuman et al. (2013)], Fourier transformation [D. I. Shuman et al. (2013)], wavelet transformation [D. K. Hammond et al. (2011)], and reconstruction [S. K.[D. Narang et al. (2013)]. Moreover, wavelet transformation K. Hammond et al. (2011)], wavelet transformation [D. K. Hammond et al. (2011)], and graph reconstruction [S. K. K. Narang et al. al. (2013)]. (2013)].toMoreover, Moreover, the signal processing has been analysis and reconstruction [S. et and reconstruction [S. K. Narang Narang et al.applied (2013)].toMoreover, the graph graph signal processing has been been applied analysis and design of multi-agent dynamical systems [S. Izumi the signal processing has applied to analysis the graph signal processing dynamical has been applied to[S.analysis and design of multi-agent systems Izumi et al. (2015)]. and design of multi-agent dynamical systems [S. Izumi and design of multi-agent dynamical systems [S. Izumi et al. (2015)]. et (2015)]. et al. al. (2015)]. On the other hand, analogue-digital conversion is a fairly On the other hand, hand, analogue-digital conversion is are fairly basic technique in signal processing,conversion but thereis few On the other analogue-digital aaa fairly On thetechnique other hand, analogue-digital conversion is are fairly basic in signal processing, but there few studies for graph signals. If it is established for graph sigbasic technique in signal processing, but there are few basic technique in signalIfprocessing, but there are sigfew studies for graph signals. it is established for graph nals, a wide range of applications is expected. For example, studies for signals. If for graph sigstudies for graph graph signals. If it it is is established established for graph signals, a wide range of applications is expected. For example, the unit commitment problem (start-stop planning of gennals, aa wide range is For nals, wide range of of applications applications is expected. expected. For example, example, the unit unit commitment problem (start-stop planning of gengenerators) in a power system is considered as analogue-digital the commitment problem (start-stop planning of the unit in commitment problem (start-stop planning of generators) a power system is considered as analogue-digital conversion the graph on the graph representing erators) in power system is as erators) in aaof systemsignal is considered considered as analogue-digital analogue-digital conversion ofpower the graph signal on the the graph graph representing the generator network. conversion of the graph signal on representing conversion of the graph signal on the graph representing the generator generator network. network. the the generator network. This paper thus addresses sigma-delta conversion, which This paper thus thus addresses addresses sigma-delta sigma-delta conversion, which is fundamental conversion, for graph sigThis paper conversion, which This paper thusanalogue-digital addresses sigma-delta conversion, which is fundamental fundamental analogue-digital conversion, for graph graph signals. By extending the conventional sigma-delta converis analogue-digital conversion, for sigis fundamental analogue-digital conversion, for graph signals. By extending the conventional sigma-delta conversion [H. Inose and Y.the Yasuda (1963)], i.e., for time signals, nals. By extending conventional sigma-delta convernals. By extending the conventional sigma-delta conversionfirst [H. define Inose and and Y. Yasuda Yasuda (1963)], conversion i.e., for for time signals, we a notion of sigma-delta forsignals, graph sion [H. Inose Y. (1963)], i.e., sion [H. define Inose and Y. Yasuda (1963)], conversion i.e., for time time signals, we first a notion of sigma-delta for graph signals. Next, a method of sigma-delta conversion isgraph prewe first define notion of sigma-delta conversion for we first define a notion of sigma-delta conversion for signals. Next, Next, aa method method of of sigma-delta sigma-delta conversion conversion is isgraph presignals. presignals. Next, method of sigma-delta conversion is pre⋆ This work was apartly supported by Grant-in-Aid for Challenging
Fig. 1. Image of graph signal processing Fig. graph Fig. 1. 1. Image Image of of graph signal signal processing processing Fig. 1. Image of graph signal processing sented. This method is based on the diffusion of quantisented. This method is on of zation a spanning the corresponding sented.error This over method is based basedin-tree on the theofdiffusion diffusion of quantiquantisented. This method is based on the diffusion of quantization error over aaapply spanning in-tree of the corresponding graph. Finally, we the proposed method to halftone zation error over spanning in-tree of the corresponding zation error over a spanning in-tree of the corresponding graph. Finally, we apply the proposed method halftone image processing, that is, of ato graph. Finally, the proposed to halftone graph. Finally, we we apply apply thetransformation proposed method method tograyscale halftone image processing, that is, transformation of a grayscale image into a binary image with keeping the visual quality. processing, that is, transformation of a grayscale image processing, that is, transformation of a grayscale image into binary with keeping quality. The effectiveness the proposed imageresult into aaademonstrates binary image imagethe with keeping the theofvisual visual quality. image into binary image with keeping the visual quality. The result demonstrates the effectiveness of the proposed method. The result demonstrates the effectiveness of the proposed The result demonstrates the effectiveness of the proposed method. method. method. Note that this paper is based on our earlier preliminary Note that paper based on our earlier preliminary version [N.this Imoda et is (2016)], published a 1-page Note that this paper is based on earlier preliminary Note that this paper isal. based on our our earlier as preliminary version [N. Imoda et al. (2016)], published as a 1-page position paper. The previous paper has shown just idea, version [N. Imoda et al. (2016)], published as 1-page version [N. Imoda et al. (2016)], published as aaan 1-page position paper. The previous paper has shown just idea, while this paper presents a complete framework of sigmaposition paper. paper. The The previous previous paper paper has has shown shown just just an an idea, position an idea, while this paper presents aasignals. complete framework of sigmadelta conversion for graph while this paper presents complete framework of sigmawhile this paper presents a complete framework of sigmadelta conversion for graph signals. delta delta conversion conversion for for graph graph signals. signals. Notation: Notation: 1) General Mathematical Notation: Let R be the set Notation: Notation: General Mathematical Notation: the set of1) real numbers. For the discrete setLet Q, R thebe 1) General Mathematical Notation: Let R be the 1)real General Mathematical Notation: Let R bequantizer the set set of numbers. For the discrete set Q, the quantizer is defined as the For mapping q : R→ Q Q, whose output is of real numbers. the discrete set the quantizer of real numbers. For the discrete set Q, the quantizer is defined as the qq in :: R → Q output given by the number the is defined as smallest the mapping mapping R →numbers Q whose whoseminimizing output is is is defined as the mapping q : R → Q whose output is given the smallest the numbers |x − y|by with respect tonumber y ∈ Qin where x is the minimizing input. For given by the smallest number in the numbers minimizing given by the smallest number in the numbers minimizing |x − y| with respect to yy = ∈ Q where is the input. For example, q(0.5) = 0 for ±1, ±2,x Moreover, the |x − respect to ∈ Q x is input. |x − y| y| with with respect to Q y= ∈ {0, Q where where x...}. is the the input. For For example, q(0.5) = 0 for Q {0, ±1, ±2, ...}. Moreover, the maximum quantization error, denoted by δ , is defined as example, q(0.5) = 0 for Q = {0, ±1, ±2, ...}. Moreover, the q example, q(0.5) = 0 for Q = {0, ±1, ±2,by ...}. Moreover, the maximum quantization error, denoted δ , is defined q sup |x − q(x)|. In the above example, δ = 0.5. maximum quantization error, error, denoted denoted by by δδqqq ,, is is defined defined as as x∈R maximum quantization as sup |x − q(x)|. In the above example, δδq = 0.5. 2) x∈R Notation for graphs: Consider the digraph G = (V, E) sup |x − q(x)|. In the above example, 0.5. q = x∈R sup |x − q(x)|. In the above example, δ = 0.5. q x∈R 2) Notation for graphs: Consider the digraph E) for node set and the edge set If (j, i) G ∈= E (V, is held 2) Notation for graphs: Consider the digraph G = (V, E) 2) the Notation for V graphs: Consider theE. digraph G = (V, E) for the node set V and the edge set E. If (j, i) ∈ E is held for any (i, j) ∈ E, then G is said to be undirected. The the node set V and the edge set E. If (j, i) ∈ E is held for the node set V and the edge set E. If (j, i) ∈ E is held for ∈ G be The digraph G j) is called the in-tree if 1)to exists a unique for any any (i, (i, j) ∈ E, E, then then G is is said said tothere be undirected. undirected. The for any (i, j) ∈ E, then G is said to be undirected. The digraph G is called the in-tree if 1) there exists aa unique node whose outdegree is zero, 2) the outdegree of all nodes digraph G is called the in-tree if 1) there exists unique digraph G is called the in-tree if 1) there exists a unique node whose outdegree is zero, 2) of all nodes except for the node specified in the 1) isoutdegree one, 3) there exists node whose outdegree is the outdegree of node whose outdegree is zero, zero, 2) 2) the outdegree of all all nodes nodes except for the node specified in 1) is one, 3) there exists except for the node specified in 1) is one, 3) there except for the node specified in 1) is one, 3) there exists exists
⋆ This work was partly supported by Grant-in-Aid ⋆ Exploratory Research #16K14283 from the Ministryfor ofChallenging Education, was supported by for ⋆ This This work work Research was partly partly supported from by Grant-in-Aid Grant-in-Aid forofChallenging Challenging Exploratory #16K14283 the Ministry Education, Culture, Sports, Science, and Technology of Japan. Exploratory Research #16K14283 from the Ministry of Exploratory Research #16K14283 from the Ministry of Education, Education, Culture, Sports, Science, and Technology of Japan. Culture, Culture, Sports, Sports, Science, Science, and and Technology Technology of of Japan. Japan. Copyright © 2017, 2017 IFAC 9711Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 9711 Copyright © 2017 IFAC 9711 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 9711Control. 10.1016/j.ifacol.2017.08.1177
Proceedings of the 20th IFAC World Congress 9304 N. Imoda et al. / IFAC PapersOnLine 50-1 (2017) 9303–9307 Toulouse, France, July 9-14, 2017
Fig. 2. Example of graph signal
a directed path reaching to the node specified in 1) from ¯ is a spanning subgraph of G any nodes. If the digraph G ¯ is called the spanning in-tree of G. and in-tree, G
(a) Example of grayscale digital image [S. Ueyama. (1998)]
2. GRAPH SIGNALS AND DELTA-SIGMA CONVERSION In this section, we first introduce the concept of graph signals and the path integral on graph signals. Next, we define the delta-sigma conversion for graph signals.
City A 1207
2.1 Graph signals Consider a digraph G = (V, E) for the node set V := {1, 2, ..., n} and the edge set E ⊆ V × V. Moreover, consider the mapping x : V → R. Then, the graph signal ˆ is defined as the pair of G and x. An example of G ˆ = graph signal is illustrated in Fig. 2. In this case, G (G, x) for the digraph G with V = {1, 2, 3, 4, 5} and E = {(1, 3), (2, 1), (3, 1), (3, 2), (4, 3), (5, 2), (5, 3)}, and the mapping x satisfying x(1) = 2, x(2) = 1, x(3) = 1, x(4) = 2, and x(5) = 3. We can mathematically represent various engineering objects as a graph signal. Examples are given as follows.
City C
3016
City B 147
City D
6521
(b) Example of population distribution map Fig. 3. Examples of engineering objects modelled as a graph signal
Example. 1 A grayscale image as shown in Fig. 3(a) can be modelled as a graph signal (G, x). For example, V corresponds to the indices of pixels and E expresses the adjacency among pixels. On the other hand, x is of the pixel values. Example. 2 The population distribution map as in Fig. 3(b) can be regarded as a graph signal. In this case, V is the cities and E is the adjacency of cities. Moreover, the mapping x corresponds to the population of cities. In the following sections, we use the accent mark “ ˆ ” ˆ is a to express graph signals (not graphs). For example, G graph signal for the graph G.
� and domain of integration Fig. 4. Spanning subgraph G V(3) The set V(j) is called the domain of integration.
2.2 Path integral on graph signals We next define an integral for graph signals. ˆ = (G, x) is Definition. 1 Suppose that a graph signal G given, and assume that the graph G contains a connected � Let V(j) := {j} ∪ {i ∈ V| there spanning subgraph G. � Then exists a directed path reaching to j from i in G}. � and node j is defined as the integral for the subgraph G ∑ x(i). (1) i∈V(j)
ˆ = (G, x) in Example. 3 Consider the graph signal G � Fig. 2. For example, we choose the spanning subgraph G and the domain of integration V(3) as shown in Fig. 4. � Then, ∑ the integral for the subgraph G and node 3 is given by i∈V(3) x(i) = x(3) + x(4) + x(5) = 1 + 2 + 3 = 6. 2.3 Delta-sigma conversion for graph signals
Now, we introduce the notion of delta-sigma conversion for graph signals.
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For the conventional sigma-delta conversion, i.e., for time signals, the following property is well-known [S. R. Norsworthy et al. (1996)]: if a low-pass filter is applied to the original continuous-valued signal and to the discretevalued signal given by the delta-sigma conversion, then the resulting signals are almost the same. A typical low-pass filter is the integrator. By considering this property, we define delta-sigma conversion for graph signals as follows. Definition. 2 Suppose that a discrete set Q ⊆ R and a ˆ 1 = (G1 , x1 ) to a graph mapping F from a graph signal G ˆ signal G2 = (G2 , x2 ) are given, and assume that G1 and G2 are strongly connected and undirected. The mapping F is said to be delta-sigma conversion if the following three conditions hold: (i) G1 = G2 , (ii) Let V2 be the node set of the graph G2 . For any i ∈ V2 , x2 (i) ∈ Q. (iii) Let V1 be the node set of the graph G1 . In the � such that graph G1 , there exists a spanning subgraph G it is connected and contains no cycle, and the following inequality holds for any j ∈ V1 : � � � � ∑ ∑ � � � x1 (i) − x2 (i)�� ≤ δq (2) � � �i∈V(j) i∈V(j) where V(j) := {j} ∪ {i ∈ V1 | there exists a directed � and δq is maximum quantization path from i to j in G}, error.
The condition (i) means that the graph topologies are the same in the input and output, and (ii) expresses that the output is discrete-valued. The condition (iii) implies that the aforementioned property of the conventional deltaˆ 1 and G ˆ 2 . That is, the results sigma conversion holds for G of a low-pass filter (the integration in this case) are similar in the sense that the difference is not more than δq . ˆ 1 and G ˆ 2 satisfying the three Let us give an example of G conditions in Definition 2. ˆ 1 = (G1 , x1 ) and G ˆ 2 = (G2 , x2 ) Example. 4 Consider G in Fig. 5(a), (b) for Q := {0, ±1, ±2, ...}. It is clear that G1 = G2 , which implies that the condition (i) holds. Next, it is obvious that x2 (i) ∈ Q for all i ∈ {1, 2, 3, 4}. Namely, (ii) holds. Finally, let us consider the satisfaction of (iii). � is given as Fig. 5(c). It is connected and Suppose that G has no cycle. Then, the left-hand side of (2) is equal to 0.3 for j = 1, 0.3 for j = 2, 0.1 for j = 3, and 0.4 for j = 4. On the other hand, δq = 0.5 for the discrete set Q. Hence, (iii) is satisfied. 3. DELTA-SIGMA CONVERSION BASED ON SPANNING IN-TREE
� of graph G1 (c) Spanning subgraph G
Fig. 5. Examples of input and output of delta-sigma � of G ˆ1 conversion and spanning in-tree G where V0 := {i ∈ V| the indegree of node i is zero in the ¯ and N(i) := {j ∈ V| (j, i) ∈ E} ¯ for a spanning graph G} ¯ ¯ in-tree G = (V, E) of G1 . Then, we consider the following ˆ 1 = (G1 , x1 ) mapping, denoted by F , from a graph signal G ˆ to a graph signal G2 = (G2 , x2 ): {
Suppose that a discrete set Q is given. Let y : V → R be the mapping defined as if i ∈ V0 , x1 (i) ∑ y(i) = x1 (i) − (q(y(j)) − y(j)) otherwise, j∈N(i)
(3)
For the mapping F defined in (3), we obtain the following result. Theorem 1. For the mapping F defined in (3), the conditions (i)–(iii) in Definition 2 hold. Proof. It is obvious that the mapping F satisfies the conditions (i) and (ii). Furthermore, the following relation holds for any j ∈ V: ∑ ∑ x2 (i) − x1 (i) = q(y(j)) − y(j). (4) i∈V(j)
In this section, we propose a method of delta-sigma conversion for graph signals.
G2 = G1 , x2 (i) = q(y(i)) (i ∈ V).
i∈V(j)
From (4) and the definition of the maximum quantization error( |q(x) − x| ≤ δq for any x ∈ R), it turns out that the mapping F satisfies the condition (iii). This completes the proof. Example. 5 Consider the proposed delta-sigma conversion F for Q = {0, ±1, ±2, ...}, and the input shown in Fig. 6(a). Then the output is given as Fig. 6(b), where the
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Table 1. Values of left-hand of (2) in Example 1 V(j) V(1) V(2) V(3) V(4) V(5)
value 0.4456 0.0227 0.1385 0.0791 0.3474
V(j) V(6) V(7) V(8) V(9) V(10)
value 0.4076 0.0213 0.0312 0.1639 0.4408
Fig. 7. Grayscale image [SIDBA. (2007)]
4. APPLICATION TO HALFTONE IMAGE PROCESSING Halftoning is one of image processing techniques which transforms a grayscale image to a binary image preserving the original visual quality. We apply the proposed sigma-delta conversion to halftone image processing. We use the standard grayscale digital image with 256 × 256 pixels [SIDBA. (2007)] shown in Fig. 7. We regard the image as a graph signal with the graph G = (V, E) shown in Fig. 8 and the mapping expressing the pixel values. Then, we apply the proposed delta-sigma conversion to the graph signal with Q = {0, 255}.
Fig. 9 shows the result of the delta-sigma conversion for the spanning in-tree shown in Fig. 10. We see that the proposed method provides a binary image whose appearance is similar to the original image.
¯ of graph G1 (c) Spanning in-tree G Fig. 6. Example in Section 4.1
5. CONCLUSION
spanning in-tree is defined as Fig. 6(c). In this case, the left-hand side of (2) is calculated as Table 1 and is not more than δq = 0.5.
This paper has established a framework of delta-sigma conversion for graph signals. We first has extended the conventional sigma-delta conversion to that for graph signals. Second, we have proposed a concrete method based on spanning in-tree. Finally we have demonstrated the proposed method by a numerical example and applying to halftone image processing.
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Acknowledgements: This work was partly supported by Grant-in-Aid for Challenging Exploratory Research 16K14283 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
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D. I. Shuman et al. The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Process. Mag, volume 30, no. 3, pages 83–98, 2013. S. R. Norsworthy et al. Delta-sigma data converters: theory, design, and simulation. IEEE Circuits and System Society, pages 1–6, 1996. S. K. Narang et al. Signal processing techniques for interpolation in graph structured data. In Proc. ICASSP’13, pages 5445–5449, 2013. D. K. Hammond et al. Wavelets on graphs via spectral graph theory. Appl. Comput. Harmon. Anal., volume 30, no. 2, pages 129–150, 2011. I. Pesenson. Sampling in Paley-Wiener spaces on combinatorial graphs. Trans. Am. Math. Soc., volume 360, no. 10, pages 5603–5627, 2008. S. Izumi, S. Azuma, and T. Sugie. On a Relation between Graph Signal Processing and Multi-Agent Consensus. The 2nd Multi-symposium on Control Systems, pages 723–2, 2015. H. Inose and Y. Yasuda. A unity coding method by negative feedback. In Proc. IEEE, volume 51, no. 11, pages 1524–1535, 1963. N. Imoda, S. Azuma, T. Kitao and T. Sugie. Noise-shaping A/D Conversion of Graph Signals. 22nd International Symposium on Mathematical Theory of Networks and Systems, pages 575–576 , 2016. S. Ueyama. Pixel. http://www.infonet.co.jp/ueyama/ ip/glossary/pixel.html SIDBA. Standard Image Data-Base. http://www.ess.i c.kanagawa-it.ac.jp/app images j.html
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Fig. 9. Result of proposed delta-sigma conversion
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… ¯ of graph in Fig. 8 Fig. 10. Spanning in-tree G
We believe that the proposed framework will be a useful solution to engineering problems modelled by graph signals. 9715
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Delta-sigma Delta-sigma Delta-sigma Delta-sigma ∗
conversion conversion conversion conversion ∗ ∗∗
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N. Imoda∗ , S. Azuma∗ ,∗∗ , T.Kitao∗ ,∗∗∗ , T. Sugie ∗ N. Imoda∗∗ ,, S. Azuma∗∗ ,,∗∗ , T.Kitao∗∗ ,,∗∗∗ , T. Sugie ∗ N. N. Imoda Imoda , S. S. Azuma Azuma ,∗∗ ,, T.Kitao T.Kitao ,∗∗∗ ,, T. T. Sugie Sugie ∗ ∗ Graduate School of Informatics, Kyoto University, Kyoto, JAPAN ∗ School Kyoto, ∗ (e-mail: [email protected], [email protected]) ∗ Graduate Graduate School of of Informatics, Informatics, Kyoto Kyoto University, University, Kyoto, JAPAN JAPAN Graduate School of Informatics, Kyoto University, Kyoto, JAPAN ∗∗ (e-mail: [email protected], [email protected]) Currently, Graduate School of Engineering, Nagoya University, (e-mail: [email protected], [email protected]) (e-mail: [email protected], [email protected]) ∗∗ Currently, Graduate of Engineering, Nagoya University, ∗∗ Aichi, JAPAN (e-mail:School [email protected]) ∗∗ Currently, Graduate School of Currently, School of Engineering, Engineering, Nagoya Nagoya University, University, ∗∗∗ Graduate Aichi, JAPAN (e-mail: [email protected]) Currently, Murata Machinery, Ltd. (e-mail: Aichi, (e-mail: [email protected]) Aichi, JAPAN JAPAN (e-mail: [email protected]) ∗∗∗ Murata ∗∗∗ [email protected]) ∗∗∗ Currently, Currently, Murata Machinery, Machinery, Ltd. Ltd. (e-mail: (e-mail: Currently, Murata Machinery, Ltd. (e-mail: [email protected]) [email protected]) [email protected]) Abstract: In this paper, we establish a framework of delta-sigma conversion of graph signals. Abstract: In thisdelta-sigma paper, we establish aa framework delta-sigma conversion of graph signals. First, we define conversion for graph of signals by considering Abstract: In paper, of delta-sigma conversion of graph Abstract: In this thisdelta-sigma paper, we we establish establish a framework framework of delta-sigma conversionthe of corresponding graph signals. signals. First, we define conversion for graph signals by considering the corresponding conversion for standard time conversion series signals. We next propose aconsidering conversion the method based on First, we define delta-sigma for graph signals by corresponding First, we define delta-sigma conversion for graph signals by considering the corresponding conversion for standard time signals. We next propose conversion method based on spanning in-tree, for which it isseries proven that the method satisfiesaaa the conditions for delta-sigma conversion for time series signals. We next conversion method based conversion for standard standard time series signals. Wemethod next propose propose conversion method based on on spanning in-tree, for which it is proven that the satisfies the conditions for delta-sigma conversion. The proposed method is demonstrated by numerical examples and application to spanning in-tree, in-tree, for for which which it it is is proven proven that that the the method method satisfies satisfies the the conditions conditions for for delta-sigma delta-sigma spanning conversion. The proposed method is demonstrated by numerical examples and application to halftone image processing. conversion. The proposed method is demonstrated by numerical examples and application conversion. Theprocessing. proposed method is demonstrated by numerical examples and application to to halftone image halftone image processing. halftone image processing. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: graph signal, delta-sigma conversion, quantization, image processing Keywords: graph signal, signal, delta-sigma conversion, conversion, quantization, image image processing Keywords: Keywords: graph graph signal, delta-sigma delta-sigma conversion, quantization, quantization, image processing processing 1. INTRODUCTION 1. INTRODUCTION INTRODUCTION 1. 1. INTRODUCTION Graph signal processing is signal processing on graphs [D. Graph signal processing isassignal signal processing onwhich graphsis [D. [D. I. Shuman et processing al. (2013)]is shown in Fig. 1,on an Graph signal processing graphs Graph signal processing isassignal processing onwhich graphsis [D. I. Shuman et al. (2013)] shown in Fig. 1, an extension of the conventional signal processing of discreteI. Shuman et al. (2013)] as shown in Fig. 1, which is an I. Shumanofetthe al.conventional (2013)] as shown in Fig. 1, which is an extension signal processing of discretetime signals. It conventional has become signal one ofprocessing the majorof topics in extension of discreteextension of the the signal discretetimeengineering signals. It conventional has become one as ofprocessing the majorof topics topics in the field,become especially a frequency-domain time signals. It has one of the major in time signals. It has become one of the major topics in the engineering engineering field, especially especially as aaas frequency-domain frequency-domain method for “networked signals” such digital image and the field, as the engineering field, especially as a frequency-domain for “networked “networked signals” such as digital digital image and and amethod collection of measurements of asuch sensor network. method for signals” as image method for “networked signals” such as digital image and a collection of measurements of a sensor network. aa collection of of network. of measurements measurements of aa sensor sensor network. Socollection far, several results have been obtained for the graph So far, several results have been obtained for[I.the the graph signal processing, such as sampling theorem Pesenson So far, several results have been obtained for graph So far,processing, several results have been obtained for[I.the graph signal such as sampling theorem Pesenson (2008)], up-/down-sampling [D. I. Shuman (2013)], signal processing, such theorem [I. Pesenson signal processing, such as as sampling sampling theoremet [I.al. Pesenson (2008)], up-/down-sampling [D. I. Shuman et al. (2013)], Fourier transformation [D. [D. I. I. Shuman (2008)], up-/down-sampling Shuman et al. (2008)], up-/down-sampling [D. I. Shumanet et al. al. (2013)], (2013)], Fourier transformation transformation [D. [D. K. I. Hammond Shuman etet al. (2013)], wavelet al. (2011)], Fourier transformation [D. I. Shuman et al. (2013)], Fourier transformation [D. I. Shuman et al. (2013)], wavelet transformation [D. K. Hammond et al. (2011)], and reconstruction [S. K.[D. Narang et al. (2013)]. Moreover, wavelet transformation K. Hammond et al. (2011)], wavelet transformation [D. K. Hammond et al. (2011)], and graph reconstruction [S. K. K. Narang et al. al. (2013)]. (2013)].toMoreover, Moreover, the signal processing has been analysis and reconstruction [S. et and reconstruction [S. K. Narang Narang et al.applied (2013)].toMoreover, the graph graph signal processing has been been applied analysis and design of multi-agent dynamical systems [S. Izumi the signal processing has applied to analysis the graph signal processing dynamical has been applied to[S.analysis and design of multi-agent systems Izumi et al. (2015)]. and design of multi-agent dynamical systems [S. Izumi and design of multi-agent dynamical systems [S. Izumi et al. (2015)]. et (2015)]. et al. al. (2015)]. On the other hand, analogue-digital conversion is a fairly On the other hand, hand, analogue-digital conversion is are fairly basic technique in signal processing,conversion but thereis few On the other analogue-digital aaa fairly On thetechnique other hand, analogue-digital conversion is are fairly basic in signal processing, but there few studies for graph signals. If it is established for graph sigbasic technique in signal processing, but there are few basic technique in signalIfprocessing, but there are sigfew studies for graph signals. it is established for graph nals, a wide range of applications is expected. For example, studies for signals. If for graph sigstudies for graph graph signals. If it it is is established established for graph signals, a wide range of applications is expected. For example, the unit commitment problem (start-stop planning of gennals, aa wide range is For nals, wide range of of applications applications is expected. expected. For example, example, the unit unit commitment problem (start-stop planning of gengenerators) in a power system is considered as analogue-digital the commitment problem (start-stop planning of the unit in commitment problem (start-stop planning of generators) a power system is considered as analogue-digital conversion the graph on the graph representing erators) in power system is as erators) in aaof systemsignal is considered considered as analogue-digital analogue-digital conversion ofpower the graph signal on the the graph graph representing the generator network. conversion of the graph signal on representing conversion of the graph signal on the graph representing the generator generator network. network. the the generator network. This paper thus addresses sigma-delta conversion, which This paper thus thus addresses addresses sigma-delta sigma-delta conversion, which is fundamental conversion, for graph sigThis paper conversion, which This paper thusanalogue-digital addresses sigma-delta conversion, which is fundamental fundamental analogue-digital conversion, for graph graph signals. By extending the conventional sigma-delta converis analogue-digital conversion, for sigis fundamental analogue-digital conversion, for graph signals. By extending the conventional sigma-delta conversion [H. Inose and Y.the Yasuda (1963)], i.e., for time signals, nals. By extending conventional sigma-delta convernals. By extending the conventional sigma-delta conversionfirst [H. define Inose and and Y. Yasuda Yasuda (1963)], conversion i.e., for for time signals, we a notion of sigma-delta forsignals, graph sion [H. Inose Y. (1963)], i.e., sion [H. define Inose and Y. Yasuda (1963)], conversion i.e., for time time signals, we first a notion of sigma-delta for graph signals. Next, a method of sigma-delta conversion isgraph prewe first define notion of sigma-delta conversion for we first define a notion of sigma-delta conversion for signals. Next, Next, aa method method of of sigma-delta sigma-delta conversion conversion is isgraph presignals. presignals. Next, method of sigma-delta conversion is pre⋆ This work was apartly supported by Grant-in-Aid for Challenging
Fig. 1. Image of graph signal processing Fig. graph Fig. 1. 1. Image Image of of graph signal signal processing processing Fig. 1. Image of graph signal processing sented. This method is based on the diffusion of quantisented. This method is on of zation a spanning the corresponding sented.error This over method is based basedin-tree on the theofdiffusion diffusion of quantiquantisented. This method is based on the diffusion of quantization error over aaapply spanning in-tree of the corresponding graph. Finally, we the proposed method to halftone zation error over spanning in-tree of the corresponding zation error over a spanning in-tree of the corresponding graph. Finally, we apply the proposed method halftone image processing, that is, of ato graph. Finally, the proposed to halftone graph. Finally, we we apply apply thetransformation proposed method method tograyscale halftone image processing, that is, transformation of a grayscale image into a binary image with keeping the visual quality. processing, that is, transformation of a grayscale image processing, that is, transformation of a grayscale image into binary with keeping quality. The effectiveness the proposed imageresult into aaademonstrates binary image imagethe with keeping the theofvisual visual quality. image into binary image with keeping the visual quality. The result demonstrates the effectiveness of the proposed method. The result demonstrates the effectiveness of the proposed The result demonstrates the effectiveness of the proposed method. method. method. Note that this paper is based on our earlier preliminary Note that paper based on our earlier preliminary version [N.this Imoda et is (2016)], published a 1-page Note that this paper is based on earlier preliminary Note that this paper isal. based on our our earlier as preliminary version [N. Imoda et al. (2016)], published as a 1-page position paper. The previous paper has shown just idea, version [N. Imoda et al. (2016)], published as 1-page version [N. Imoda et al. (2016)], published as aaan 1-page position paper. The previous paper has shown just idea, while this paper presents a complete framework of sigmaposition paper. paper. The The previous previous paper paper has has shown shown just just an an idea, position an idea, while this paper presents aasignals. complete framework of sigmadelta conversion for graph while this paper presents complete framework of sigmawhile this paper presents a complete framework of sigmadelta conversion for graph signals. delta delta conversion conversion for for graph graph signals. signals. Notation: Notation: 1) General Mathematical Notation: Let R be the set Notation: Notation: General Mathematical Notation: the set of1) real numbers. For the discrete setLet Q, R thebe 1) General Mathematical Notation: Let R be the 1)real General Mathematical Notation: Let R bequantizer the set set of numbers. For the discrete set Q, the quantizer is defined as the For mapping q : R→ Q Q, whose output is of real numbers. the discrete set the quantizer of real numbers. For the discrete set Q, the quantizer is defined as the qq in :: R → Q output given by the number the is defined as smallest the mapping mapping R →numbers Q whose whoseminimizing output is is is defined as the mapping q : R → Q whose output is given the smallest the numbers |x − y|by with respect tonumber y ∈ Qin where x is the minimizing input. For given by the smallest number in the numbers minimizing given by the smallest number in the numbers minimizing |x − y| with respect to yy = ∈ Q where is the input. For example, q(0.5) = 0 for ±1, ±2,x Moreover, the |x − respect to ∈ Q x is input. |x − y| y| with with respect to Q y= ∈ {0, Q where where x...}. is the the input. For For example, q(0.5) = 0 for Q {0, ±1, ±2, ...}. Moreover, the maximum quantization error, denoted by δ , is defined as example, q(0.5) = 0 for Q = {0, ±1, ±2, ...}. Moreover, the q example, q(0.5) = 0 for Q = {0, ±1, ±2,by ...}. Moreover, the maximum quantization error, denoted δ , is defined q sup |x − q(x)|. In the above example, δ = 0.5. maximum quantization error, error, denoted denoted by by δδqqq ,, is is defined defined as as x∈R maximum quantization as sup |x − q(x)|. In the above example, δδq = 0.5. 2) x∈R Notation for graphs: Consider the digraph G = (V, E) sup |x − q(x)|. In the above example, 0.5. q = x∈R sup |x − q(x)|. In the above example, δ = 0.5. q x∈R 2) Notation for graphs: Consider the digraph E) for node set and the edge set If (j, i) G ∈= E (V, is held 2) Notation for graphs: Consider the digraph G = (V, E) 2) the Notation for V graphs: Consider theE. digraph G = (V, E) for the node set V and the edge set E. If (j, i) ∈ E is held for any (i, j) ∈ E, then G is said to be undirected. The the node set V and the edge set E. If (j, i) ∈ E is held for the node set V and the edge set E. If (j, i) ∈ E is held for ∈ G be The digraph G j) is called the in-tree if 1)to exists a unique for any any (i, (i, j) ∈ E, E, then then G is is said said tothere be undirected. undirected. The for any (i, j) ∈ E, then G is said to be undirected. The digraph G is called the in-tree if 1) there exists aa unique node whose outdegree is zero, 2) the outdegree of all nodes digraph G is called the in-tree if 1) there exists unique digraph G is called the in-tree if 1) there exists a unique node whose outdegree is zero, 2) of all nodes except for the node specified in the 1) isoutdegree one, 3) there exists node whose outdegree is the outdegree of node whose outdegree is zero, zero, 2) 2) the outdegree of all all nodes nodes except for the node specified in 1) is one, 3) there exists except for the node specified in 1) is one, 3) there except for the node specified in 1) is one, 3) there exists exists
⋆ This work was partly supported by Grant-in-Aid ⋆ Exploratory Research #16K14283 from the Ministryfor ofChallenging Education, was supported by for ⋆ This This work work Research was partly partly supported from by Grant-in-Aid Grant-in-Aid forofChallenging Challenging Exploratory #16K14283 the Ministry Education, Culture, Sports, Science, and Technology of Japan. Exploratory Research #16K14283 from the Ministry of Exploratory Research #16K14283 from the Ministry of Education, Education, Culture, Sports, Science, and Technology of Japan. Culture, Culture, Sports, Sports, Science, Science, and and Technology Technology of of Japan. Japan. Copyright © 2017, 2017 IFAC 9711Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 9711 Copyright © 2017 IFAC 9711 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 9711Control. 10.1016/j.ifacol.2017.08.1177
Proceedings of the 20th IFAC World Congress 9304 N. Imoda et al. / IFAC PapersOnLine 50-1 (2017) 9303–9307 Toulouse, France, July 9-14, 2017
Fig. 2. Example of graph signal
a directed path reaching to the node specified in 1) from ¯ is a spanning subgraph of G any nodes. If the digraph G ¯ is called the spanning in-tree of G. and in-tree, G
(a) Example of grayscale digital image [S. Ueyama. (1998)]
2. GRAPH SIGNALS AND DELTA-SIGMA CONVERSION In this section, we first introduce the concept of graph signals and the path integral on graph signals. Next, we define the delta-sigma conversion for graph signals.
City A 1207
2.1 Graph signals Consider a digraph G = (V, E) for the node set V := {1, 2, ..., n} and the edge set E ⊆ V × V. Moreover, consider the mapping x : V → R. Then, the graph signal ˆ is defined as the pair of G and x. An example of G ˆ = graph signal is illustrated in Fig. 2. In this case, G (G, x) for the digraph G with V = {1, 2, 3, 4, 5} and E = {(1, 3), (2, 1), (3, 1), (3, 2), (4, 3), (5, 2), (5, 3)}, and the mapping x satisfying x(1) = 2, x(2) = 1, x(3) = 1, x(4) = 2, and x(5) = 3. We can mathematically represent various engineering objects as a graph signal. Examples are given as follows.
City C
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City B 147
City D
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(b) Example of population distribution map Fig. 3. Examples of engineering objects modelled as a graph signal
Example. 1 A grayscale image as shown in Fig. 3(a) can be modelled as a graph signal (G, x). For example, V corresponds to the indices of pixels and E expresses the adjacency among pixels. On the other hand, x is of the pixel values. Example. 2 The population distribution map as in Fig. 3(b) can be regarded as a graph signal. In this case, V is the cities and E is the adjacency of cities. Moreover, the mapping x corresponds to the population of cities. In the following sections, we use the accent mark “ ˆ ” ˆ is a to express graph signals (not graphs). For example, G graph signal for the graph G.
� and domain of integration Fig. 4. Spanning subgraph G V(3) The set V(j) is called the domain of integration.
2.2 Path integral on graph signals We next define an integral for graph signals. ˆ = (G, x) is Definition. 1 Suppose that a graph signal G given, and assume that the graph G contains a connected � Let V(j) := {j} ∪ {i ∈ V| there spanning subgraph G. � Then exists a directed path reaching to j from i in G}. � and node j is defined as the integral for the subgraph G ∑ x(i). (1) i∈V(j)
ˆ = (G, x) in Example. 3 Consider the graph signal G � Fig. 2. For example, we choose the spanning subgraph G and the domain of integration V(3) as shown in Fig. 4. � Then, ∑ the integral for the subgraph G and node 3 is given by i∈V(3) x(i) = x(3) + x(4) + x(5) = 1 + 2 + 3 = 6. 2.3 Delta-sigma conversion for graph signals
Now, we introduce the notion of delta-sigma conversion for graph signals.
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For the conventional sigma-delta conversion, i.e., for time signals, the following property is well-known [S. R. Norsworthy et al. (1996)]: if a low-pass filter is applied to the original continuous-valued signal and to the discretevalued signal given by the delta-sigma conversion, then the resulting signals are almost the same. A typical low-pass filter is the integrator. By considering this property, we define delta-sigma conversion for graph signals as follows. Definition. 2 Suppose that a discrete set Q ⊆ R and a ˆ 1 = (G1 , x1 ) to a graph mapping F from a graph signal G ˆ signal G2 = (G2 , x2 ) are given, and assume that G1 and G2 are strongly connected and undirected. The mapping F is said to be delta-sigma conversion if the following three conditions hold: (i) G1 = G2 , (ii) Let V2 be the node set of the graph G2 . For any i ∈ V2 , x2 (i) ∈ Q. (iii) Let V1 be the node set of the graph G1 . In the � such that graph G1 , there exists a spanning subgraph G it is connected and contains no cycle, and the following inequality holds for any j ∈ V1 : � � � � ∑ ∑ � � � x1 (i) − x2 (i)�� ≤ δq (2) � � �i∈V(j) i∈V(j) where V(j) := {j} ∪ {i ∈ V1 | there exists a directed � and δq is maximum quantization path from i to j in G}, error.
The condition (i) means that the graph topologies are the same in the input and output, and (ii) expresses that the output is discrete-valued. The condition (iii) implies that the aforementioned property of the conventional deltaˆ 1 and G ˆ 2 . That is, the results sigma conversion holds for G of a low-pass filter (the integration in this case) are similar in the sense that the difference is not more than δq . ˆ 1 and G ˆ 2 satisfying the three Let us give an example of G conditions in Definition 2. ˆ 1 = (G1 , x1 ) and G ˆ 2 = (G2 , x2 ) Example. 4 Consider G in Fig. 5(a), (b) for Q := {0, ±1, ±2, ...}. It is clear that G1 = G2 , which implies that the condition (i) holds. Next, it is obvious that x2 (i) ∈ Q for all i ∈ {1, 2, 3, 4}. Namely, (ii) holds. Finally, let us consider the satisfaction of (iii). � is given as Fig. 5(c). It is connected and Suppose that G has no cycle. Then, the left-hand side of (2) is equal to 0.3 for j = 1, 0.3 for j = 2, 0.1 for j = 3, and 0.4 for j = 4. On the other hand, δq = 0.5 for the discrete set Q. Hence, (iii) is satisfied. 3. DELTA-SIGMA CONVERSION BASED ON SPANNING IN-TREE
� of graph G1 (c) Spanning subgraph G
Fig. 5. Examples of input and output of delta-sigma � of G ˆ1 conversion and spanning in-tree G where V0 := {i ∈ V| the indegree of node i is zero in the ¯ and N(i) := {j ∈ V| (j, i) ∈ E} ¯ for a spanning graph G} ¯ ¯ in-tree G = (V, E) of G1 . Then, we consider the following ˆ 1 = (G1 , x1 ) mapping, denoted by F , from a graph signal G ˆ to a graph signal G2 = (G2 , x2 ): {
Suppose that a discrete set Q is given. Let y : V → R be the mapping defined as if i ∈ V0 , x1 (i) ∑ y(i) = x1 (i) − (q(y(j)) − y(j)) otherwise, j∈N(i)
(3)
For the mapping F defined in (3), we obtain the following result. Theorem 1. For the mapping F defined in (3), the conditions (i)–(iii) in Definition 2 hold. Proof. It is obvious that the mapping F satisfies the conditions (i) and (ii). Furthermore, the following relation holds for any j ∈ V: ∑ ∑ x2 (i) − x1 (i) = q(y(j)) − y(j). (4) i∈V(j)
In this section, we propose a method of delta-sigma conversion for graph signals.
G2 = G1 , x2 (i) = q(y(i)) (i ∈ V).
i∈V(j)
From (4) and the definition of the maximum quantization error( |q(x) − x| ≤ δq for any x ∈ R), it turns out that the mapping F satisfies the condition (iii). This completes the proof. Example. 5 Consider the proposed delta-sigma conversion F for Q = {0, ±1, ±2, ...}, and the input shown in Fig. 6(a). Then the output is given as Fig. 6(b), where the
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Table 1. Values of left-hand of (2) in Example 1 V(j) V(1) V(2) V(3) V(4) V(5)
value 0.4456 0.0227 0.1385 0.0791 0.3474
V(j) V(6) V(7) V(8) V(9) V(10)
value 0.4076 0.0213 0.0312 0.1639 0.4408
Fig. 7. Grayscale image [SIDBA. (2007)]
4. APPLICATION TO HALFTONE IMAGE PROCESSING Halftoning is one of image processing techniques which transforms a grayscale image to a binary image preserving the original visual quality. We apply the proposed sigma-delta conversion to halftone image processing. We use the standard grayscale digital image with 256 × 256 pixels [SIDBA. (2007)] shown in Fig. 7. We regard the image as a graph signal with the graph G = (V, E) shown in Fig. 8 and the mapping expressing the pixel values. Then, we apply the proposed delta-sigma conversion to the graph signal with Q = {0, 255}.
Fig. 9 shows the result of the delta-sigma conversion for the spanning in-tree shown in Fig. 10. We see that the proposed method provides a binary image whose appearance is similar to the original image.
¯ of graph G1 (c) Spanning in-tree G Fig. 6. Example in Section 4.1
5. CONCLUSION
spanning in-tree is defined as Fig. 6(c). In this case, the left-hand side of (2) is calculated as Table 1 and is not more than δq = 0.5.
This paper has established a framework of delta-sigma conversion for graph signals. We first has extended the conventional sigma-delta conversion to that for graph signals. Second, we have proposed a concrete method based on spanning in-tree. Finally we have demonstrated the proposed method by a numerical example and applying to halftone image processing.
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Acknowledgements: This work was partly supported by Grant-in-Aid for Challenging Exploratory Research 16K14283 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.
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D. I. Shuman et al. The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Process. Mag, volume 30, no. 3, pages 83–98, 2013. S. R. Norsworthy et al. Delta-sigma data converters: theory, design, and simulation. IEEE Circuits and System Society, pages 1–6, 1996. S. K. Narang et al. Signal processing techniques for interpolation in graph structured data. In Proc. ICASSP’13, pages 5445–5449, 2013. D. K. Hammond et al. Wavelets on graphs via spectral graph theory. Appl. Comput. Harmon. Anal., volume 30, no. 2, pages 129–150, 2011. I. Pesenson. Sampling in Paley-Wiener spaces on combinatorial graphs. Trans. Am. Math. Soc., volume 360, no. 10, pages 5603–5627, 2008. S. Izumi, S. Azuma, and T. Sugie. On a Relation between Graph Signal Processing and Multi-Agent Consensus. The 2nd Multi-symposium on Control Systems, pages 723–2, 2015. H. Inose and Y. Yasuda. A unity coding method by negative feedback. In Proc. IEEE, volume 51, no. 11, pages 1524–1535, 1963. N. Imoda, S. Azuma, T. Kitao and T. Sugie. Noise-shaping A/D Conversion of Graph Signals. 22nd International Symposium on Mathematical Theory of Networks and Systems, pages 575–576 , 2016. S. Ueyama. Pixel. http://www.infonet.co.jp/ueyama/ ip/glossary/pixel.html SIDBA. Standard Image Data-Base. http://www.ess.i c.kanagawa-it.ac.jp/app images j.html
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Fig. 9. Result of proposed delta-sigma conversion
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… ¯ of graph in Fig. 8 Fig. 10. Spanning in-tree G
We believe that the proposed framework will be a useful solution to engineering problems modelled by graph signals. 9715