Acta Mathematicae Applicatae Sinica, English Series Vol. 31, No. 2 (2015) 519 528 DOI: 10.1007/s10255-015-0482-4 http://www.applmath.com.cn & www.springerlink.com Acta Mathema cae Applicatae Sinica, English Series The Editorial Office of AMAS & Springer-Verlag Berlin Heidelberg 2015 Binary Image Reconstruction Based on Prescribed Numerical Information K.G. Subramanian 1, Pradeep Isawasan 1, Rahmat Budiarto 2, Ibrahim Venkat 1 1 School of Computer Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia (Email: kgsmani1948@yahoo.com, pradeep.isawasan@gmail.com, ibrahim@cs.usm.my) 2 Networked Computing Center, Surya University, Serpong, Tanggerang, 15810, Indonesia (Email: rahmat.budiarto@surya.ac.id) Abstract The problem of reconstruction of a binary image in the field of discrete tomography is a classic instance of seeking solution applying mathematical techniques. Here two such binary image reconstruction problems are considered given some numerical information on the image. Algorithms are developed for solving these problems and correctness of the algorithms are discussed. Keywords discrete tomography and binary image and reconstruction and subwords and Parikh matrix 2000 MR Subject Classification 94A08 1 Introduction Tomography is concerned with an imaging procedure that leads to reconstruction of the internal structure of an object, without causing any damage to the object itself [1]. Image reconstruction is used in many application areas, especially in medical imaging, crystallography, image processing and so on [14]. A large number of projections might be needed in order to carry out reconstruction sufficiently accurately, although it may be difficult to acquire a large number of projections in certain applications. Discrete Tomography (DT) is a promising and developing field of research investigating the problem of image reconstruction, especially of binary images, from a small number of projections. A binary image is a rectangular array of pixels, each of which is either black (value 0) or white (value 1). Although reconstruction in DT is based on a small number of projections, the reconstruction problem is still undetermined. For an excellent source on this and various other aspects of Discrete Tomography, we refer to [11,12]. This field is closely related to different applications of mathematics or more specifically, discrete mathematics (see for example, [2,25]). Ryser [20] and Gale [10] have independently given a necessary and sufficient condition for a given pair of vectors to be the projections of a binary matrix along horizontal and vertical directions, with the projections being the row and column sums of the binary matrix. Subsequently, reconstruction of binary images considered as matrices of 0 s and 1 s with some constraints or some prescribed information, has been one of the problems that has received great attention from researchers in this field (see [3 9,12,14,16]. Motivated by these investigations, here we consider two problems of reconstruction of a Manuscript received July 9, 2012. Revised October 17, 2013. The first author K.G. Subramanian and the fourth author Ibrahim Venkat gratefully acknowledge support for this research respectively from a FRGS grant No. 203/PKOMP/6711267 and an ERGS Grant No. 203/PKOMP/6730075 of the Ministry of Higher Education (MoHE), Malaysia. The second author Pradeep Isawasan would like to thank MoHE for the award of MyPhD under which this research was jointly carried out by him.
520 K.G. SUBRAMANIAN, P. ISAWASAN, R. BUDIARTO, I. VENKAT binary image, given some prescribed numerical information of the image. The first problem is on constructing a binary image given the row projection in one direction, with the row projection consisting of the number of 1 s and the number of subword (also called scattered subword) 01 in the rows of the binary image. We note that in a projection in general, the number of 1 s in the image in the direction of projection is counted but here we have included in the projection some additional constraints, namely, the number of subwords 01 in the rows of the binary image to be reconstructed. The second problem is on constructing a binary image M given its row projection as in the first Problem, so that M has as its prefix or suffix, another binary image M 1 whose row projection is also given. Correctness of the algorithms are also shown. 2 Preliminaries Let Σ = {0, 1} be a binary alphabet. A word w over Σ is a finite sequence of symbols of Σ. The set of all words over Σ is denoted by Σ which includes the empty word λ, which is an empty sequence of symbols. Given a word w = a 1 a 2 a n, a i Σ, a subword s of w is a subsequence of w of the form s = b 1 b 2... b k, k n where each b i equals some a j (1 j n) in w such that w = x 1 b 1 x 2 b 2 x k b k x k+1, x i Σ, i = 1, 2, k + 1. Fig.1. (a) Chessboard patterns; (b) Binary matrix representation of the patterns Extending the well-known concept of Parikh vector [19,21] of a word w which counts the number of each of the symbols of an ordered alphabet in w, the notion of Parikh matrix of a word was introduced in [18]. For a binary word over {0, 1}, the Parikh vector of the word counts the number of 0 s and 1 s. For example, the Parikh vector of the binary word 1011001 is (3, 4). The Parikh matrix of a word w over an ordered alphabet gives, by counting certain subwords in w, more information than a Parikh vector does. The notion of Parikh matrix is known to play a significant role in the study of subwords of a given word (See for example, [17,22,23]). In particular, for a binary word w over {0, 1} with the ordering 0 < 1, the Parikh
Binary Image Reconstruction Based on Prescribed Numerical Information 521 matrix is of size 3 3 and is given by P = 1 m 11 m 13 0 1 m 23, where m 11 = the number of 0 s, m 23 = the number of 1 s and m 13 = the number of subword 01 s, in w. A binary image M of size m n is a binary matrix or a (0, 1)-matrix with m rows and n columns. Each row (as well as column) of M is a word in Σ. where Σ = {0, 1}. For example, interpreting 0 as a white square and 1 as a black square, the binary images of the chess-board patterns in Fig. 1(a) are binary matrices as shown in Fig. 1(b). 3 Reconstruction of Binary Images In this section we consider two problems of reconstruction of a binary image with apriori numerical information on its rows. 3.1 Algorithm for Problem 1 In the Problem 1, information on the number of 1 s and the number of subword 01 in each row of an unknown binary image is given and we give an algorithm, called Algorithm A, to construct a binary image satisfying the given numerical information. Also the Algorithm A can deal with binary images of different sizes but might reconstruct only one among many binary images having the same numerical information on the number of 1 s and the number of subword 01. We now introduce a vector P(w), called row projection, that captures the information in Problem 1. Definition 1. Given a binary image M of size m n that is a (0, 1)-matrix with m rows and n columns, we associate with each row, a vector as follows: If w i is the binary word of 0 s and 1 s in the ith row of M, then P(w i ) = (q i, r i ), i = 1, 2, m where q i = the number of 1 s in w i and r i = the number of subword 01 in w i. We refer to P(w i ) as a row projection. Remark 1. i) Note that if p i is the number of 0 s in the word w i in the ith row of M, then p i = n q i. In fact, the matrix 1 p i r i 0 1 q i is the Parikh Matrix of w i with the ordering 0 < 1. ii) A binary word w of length n, with the number of 1 s = q and the number of 0 s = (n q), and for which the Parikh matrix is P with the ordering 0 < 1, need not be unique in the sense that there could be several binary words with the number of 1 s = q and the number of 0 s = (n q) and having the same Parikh matrix P. For example the words 011001011 and 101000111 with five 1 s and four 0 s have the same Parikh matrix 1 4 13 0 1 5
522 K.G. SUBRAMANIAN, P. ISAWASAN, R. BUDIARTO, I. VENKAT. We now state the Problem 1. Problem 1. Given m vectors v i = (q i, r i ), i = 1, 2, m, r i (n q i )q i where n, q i, r i are non-negative integers and 0 q i n, to construct a binary image of size m n, whose row projections are v i, i = 1, 2, m. Next we state an Algorithm A to construct a binary image given row projections as in Problem 1. Algorithm A. Input : The positive integers m and n; Vectors v i = (q i, r i ), i = 1, m, where q i, r i are non-negative integers, r i (n q i )q i and 0 q i n. Output : A binary image of size m n with row projections v i (i = 1, 2,, m). Step 1 : Form a binary image M of size m n with the ith row having q i consecutive 1 s from the left end of the row and (n q i ), 0 s after the 1 s for i = 1,,m. Step 2 : For i = 1,,m do: Set k i = 0. Shift the last 1 in the consecutive 1 s in the row i over the 0 s, one by one, incrementing k i by 1 for each shift till k i = r i or there is no zero left to do the shifting. If k i < r i, shift the next last 1 in the consecutive 1 s in row i, over the 0 s, one by one, again incrementing k i by 1 for every shifting. Repeat the process until k i = r i. If in a row i, k i = r i, then stop the shifting. Step 3 : The resulting binary image M is the required binary image. We state in the following lemma, a property [18] of the Parikh matrix P of a binary word w, which is useful in showing the correctness of the Algorithm A. Lemma 1. If the Parikh matrix of a binary word w over {0, 1} with the ordering 0 < 1, is P = 1 m 11 m 13 0 1 m 23, then we have m 13 m 11 m 23. Correctness and Complexity of the Algorithm A: A necessary condition for a binary image M having the row projections as in Problem 1, to exist, is that r i (n q i )q i for all i = 1, 2,...m. This condition is a consequence of the Lemma 1. Also Step 1 of the algorithm ensures the presence of the requisite number of 1 s in each row. Subsequent steps do not increase or decrease the number of 1 s. Shifting a 1 over a 0 in a row contributes 1 to the count of the number of subword 01 and the checking condition finally yields the requisite number of subword 01. This explains the correctness of the algorithm. The complexity of the algorithm is O(mn), where the size of the reconstructed binary matrix is m n, since in each row at the most only (n 1) 0 s need to be filled and this activity needs to be done in each of the m rows. We illustrate the working of the Algorithm A with an example. Example 1. Input : m = 4, n = 5; v 1 = (3, 6), v 2 = (3, 4), v 3 = (2, 4), v 4 = (2, 3). It can be verified that the necessary condition is satisfied for the required binary image to exist.
Binary Image Reconstruction Based on Prescribed Numerical Information 523 Step 1 : The binary matrix M = 1 1 1 0 0 1 1 1 0 0. Step 2 : k 1 = k 2 = k 3 = k 4 = 0. Shifting the 1 s in the row of M as in the Algorithm A, yields the following sequence of steps: 1 1 1 0 0 1 1 1 0 0 1 1 1 1 k 1 = 2 < r 1 = 6, k 2 = 2 < r 2 = 4, k 3 = 3 < r 3 = 4, k 4 = 3 = r 4 (No more shifting of 1 s in row 4) 2 and 3) 1 1 1 1 0 1 1 1 1 1 0 1, k 1 = 4 < r 1, k 2 = 4 = r 2, k 3 = 4 = r 3 (No more shifting of 1 s in rows, k 1 = 6 = r 1 (No more shifting of 1 s in row 1). Step 3 : The output binary matrix is 1 1 1 1 0 1. The output matrix has four rows and five columns. The row projections of this matrix are (3, 6), (3, 4), (2, 4), (2, 3). Fig.2 (a) Binary output image of Example 1, (b) A binary image with the same row projections as the image in Fig.2(a) Remark 2. i) Note that in Example 1, the Algorithm A constructs and can construct only the output matrix 1 1 1 1 0 1 which is one among a finite number of matrices having the same projections m = 4, n = 5; v 1 = (3, 6), v 2 = (3, 4), v 3 = (2, 4), v 4 = (2, 3) given as input. Fig.2(a) shows the binary output image of the output matrix of Example 1 and Fig.2(b) shows another binary image which also has the same row projections as that of the output image of Example 1. This is the problem of uniqueness that is generally present in binary image reconstruction. Depending on the constraints prescribed on the image to be reconstructed, the task of uniqueness is dealt with and we address this question in the following subsection by
524 K.G. SUBRAMANIAN, P. ISAWASAN, R. BUDIARTO, I. VENKAT exhibiting a specific class of binary images for which the Algorithm A will construct the image uniquely. ii) The reconstruction Problem 1 is in fact considered in Algorithm 1 in [?] and this Algorithm 1 reconstructs a binary image based on apriori information in terms of Parikh matrices of the words in the rows of the image. But the manner in which our Algorithm A constructs the rows of the binary image, is different yet simpler than that considered in [?]. In fact only matrices with the same number of rows and columns are reconstructed in [15]. 3.2 Uniqueness of Reconstruction We now examine a specific type of binary image in which the binary word in each row of the binary image, is an element of the set U = 0 1 + 1 0 + 0 10 + 1 01 + 0 101 + 1 010. Here 0 stands for the set of all words over {0} and + denotes set union. The correspondence between a word in this set and its Parikh matrix is known [17,22] to be one-to-one. In other words, if P is the Parikh matrix of any word w in U, then no other binary word over {0, 1} with 0 < 1, has the same binary matrix P. This implies that the binary image constructed by the Algorithm A will be unique, if the binary word in each row of the binary image, belongs to the set U which means that the row projections v i = (q i, r i ) given as input to the algorithm should satisfy the feature that r i = the number of subword 01 for some word in U. The following example illustrates this feature. Example 2. Input : m = 4, n = 5; v 1 = (2, 1), v 2 = (2, 5), v 3 = (2, 6), v 4 = (1, 3). Step 1 : The binary matrix M =. 1 0 0 0 0 Step 2 : k 1 = k 2 = k 3 = k 4 = 0. Shifting the 1 s in the row of M as in the algorithm 1 0 1 0 0 A, yields the following sequence of steps:, k 1 = 1 0 0 0 0 0 0 1 = r 1, k 2 = 3 < r 2, k 3 = 3 < r 3, k 4 = 3 = r 4 (No more shifting of 1 s in rows 1 and 4) 1 0 1 0 0 0 1 0 1, k 2 = 5 = r 2, k 3 = 6 = r 3 (No more shifting of 1 s in rows 2 and 3). 0 0 Step 3 : The output binary matrix is 1 0 1 0 0 0 1 0 1 0 0 The output matrix has four rows and five columns. The row projections of this matrix are (2, 1), (2, 5), (2, 6), (1, 3). All the binary words. 10100, 00101, 00011, 00010
Binary Image Reconstruction Based on Prescribed Numerical Information 525 in the rows belong to the set U. Fig.3. Binary Output Image of Example 2 3.3 Algorithm for Problem 2 We next consider reconstruction Problem 2. We first recall some notions needed. Given two binary images M 1 of size m n and M 2 of size m l, the column catenation of M 1 and M 2 is the binary image M = M 1 M 2 of size m (n + l). M 1 is said to be a prefix of M and M 2, a suffix of M. Problem 2. The row projections v i = (q i, r i ), i = 1, 2, m, r i (n q i )q i where n, q i, r i are non-negative integers and 0 q i n of a binary image M 1 are given. A positive number s is given. The problem is to construct a binary image M of size m (1 + s)n whose projections are ((1 + s)q i, (1 + s)r i + sp i q i ), i = 1,,m where p i = n q i and M 1 is a prefix of M or a suffix of M. 2. Next we state the Algorithm B in order to construct a binary image as required in Problem Algorithm B. Input : s, m and n are the positive integers. m vectors v i = (q i, r i ), where q i, r i are non-negative integers, r i (n q i )q i and 0 q i n. Output : A binary image M of size m n(1 + s) with row projections ((1 + s)q i, (1 + s)r i + sp i q i ), i = 1,,m such that M has a prefix or a suffix, the binary image with row projections v i = (q i, r i ), i = 1, 2, m. Step 1 : Apply Algorithm A to construct a binary image M 1 with row projections v i = (q i, r i ), i = 1, 2, m and another binary image M 2 with row projections (sq i, sr i ), i = 1, 2, m Step 2 : The binary image M p = M 1 M 2 and M s = M 2 M 1 are the required binary images, both having the size m n(1+s) and row projections ((1+s)q i, (1+s)r i +sp i q i ), i = 1,,m. The binary image M p has M 1 as a prefix and the binary image M s has M 1 as a suffix. We now state another property [23] of Parikh matrices of binary words that ensures that Algorithm B works correctly.
526 K.G. SUBRAMANIAN, P. ISAWASAN, R. BUDIARTO, I. VENKAT Lemma 2. If s > 0, and w 1 and w 2 are two words respectively having Parikh matrices P 1 = 1 p r 0 1 q and P 2 = 1 sp sr 0 1 sq, then both w 1 w 2 and w 2 w 1 have the same Parikh matrix P = 1 (1 + s)p (1 + s)r + spq 0 1 (1 + s)q Correctness of the Algorithm B: The binary word in any row of M p is of the form xy with x being a word in a corresponding row of M 1 and y being a word in a corresponding row of M 2. Also the row projection of x is of the form (q i, r i ) while the row projection of y is then of the form (sq i, sr i ), for some i, 1 i m. In view of Lemma 2, the row projection of xy is ((1+s)q i, (1+s)r i +sp i q i ) where p i = n q i. A similar argument applies to M s. It is now clear that Algorithm B yields a binary images M having the binary image M 1 with row projections v i = (q i, r i ), i = 1, 2,...m as a prefix or as a suffix. Example 3. Input : s = 2, m = 4, n = 5; v 1 = (2, 1), v 2 = (2, 5), v 3 = (2, 6), v 4 = (2, 3) An application of Algorithm B (which uses Algorithm A in Step 1) constructs the binary images 1 0 1 0 0 0 1 M 1 = 0 1. and M 2 = 1 1 1 0 0 0 0 1 1 0 0 0 1 1 0. Note that the rows of M 1 have the row projections (2, 1), (2, 5), (2, 6), (2, 3) while the rows of M 2 have the row projections (4, 2), (4, 10), (4, 12), (4, 6). The binary images M 1 M 2 and M 2 M 1 are the results of the output, given by M 1 M 2 = 1 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0
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