Learning an Artificial Neural Network for Discovering Combinations of Bit-Quads to Compute the Euler Characteristic of a 2-D Binary Image

Arce, Fernando; Sossa, Humberto; Gómez-Flores, Wilfrido; Lira, Laura; Arce, Fernando; Sossa, Humberto; Gómez-Flores, Wilfrido; Lira, Laura

doi:10.13053/cys-26-1-4021

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Computación y Sistemas

versión On-line ISSN 2007-9737versión impresa ISSN 1405-5546

Comp. y Sist. vol.26 no.1 Ciudad de México ene./mar. 2022 Epub 08-Ago-2022

https://doi.org/10.13053/cys-26-1-4021

Articles

Learning an Artificial Neural Network for Discovering Combinations of Bit-Quads to Compute the Euler Characteristic of a 2-D Binary Image

Fernando Arce¹

Humberto Sossa²³^*

Wilfrido Gómez-Flores⁴

Laura Lira²

¹ Centro de Investigaciones en Óptica A. C., Mexico. fernando.arce.vega@gmail.com.

² Instituto Politécnico Nacional, Centro de Investigación en Computación, Mexico. laugi2808@hotmail.com.

³Tecnológico de Monterrey, Mexico.

⁴ Centro de Investigación y de Estudios Avanzados del IPN, Mexico. wgomez@cinvestav.mx.

Abstract:

The Image Analysis community has widely used so-called bit-quads to propose formulations for computing the Euler characteristic of a 2-D binary image. Reported works have manually proposed different combinations of bit-quads to provide one or more formulations to calculate this important topological feature. This paper empirically shows how an Artificial Neural Network can be trained to find an optimal combination of bit-quads to compute the Euler characteristic of any binary image. We present results with binary images of different complexities and sizes and compare them with state-of-the-art machine learning algorithms.

Keywords: Euler characteristic; bit-quads; holes; objects; artificial neural network

1 Introduction

The Euler characteristic is a topological feature commonly used to describe the connectivity properties of an isolated object or a binary image.

Let S∈ℝ2 be a digital 0-1-image, where S is related to the foreground pixels, and its complement Sc is associated with the background pixels. In the plane, the Euler characteristic can be written as an alternating sum of Betti numbers as [²⁰]:

E(S)=Nc(S)−Nh(S), (1)

where Nc(S) is the number of connected components of S and Nh(S) is the number of holes; that is, bounded connected components of Sc.

The Euler characteristic of an image I(x,y) has been widely used since the eighties in many applications. In [⁴²], for example, the Euler number has been used to recognize industrial parts. In [³³], the same topological feature has been used for real-time image thresholding. It has also been applied in object number counting in [²¹], real-time Malayan license plate recognition in [¹⁰], digit recognition from pressure sensor data in [²⁸], gender recognition from offline handwritten signature in [²³], image description in [⁴⁹], gender discrimination from offline Hindi signature in [²⁷]. In short, in [²⁴], the Euler characteristic has been used for character recognition.

Several methods to compute the Euler characteristic of a binary image I(x,y) have been reported in the literature. Refer, for example, to [², ¹, ⁴, ⁶, ⁵, ⁸, ⁹, ¹¹, ¹³, ¹⁴, ¹², ¹⁵, ¹⁷, ¹⁶, ¹⁸, ²⁰, ²¹, ²⁵, ²⁶, ³⁰, ³⁹, ³⁸, ⁴¹, ⁴⁰, ³⁶, ³⁵, ³⁷, ³⁴, ¹⁹, ²², ⁴⁷, ⁴⁸, ⁴⁶, ³, ⁴³, ⁴⁵, ⁴⁴, ⁵⁰, ⁵¹].

So-called bit-quads have been widely used since they were proposed in the image processing area to count the Euler characteristic of a 2-D binary image. Table 1 depicts the 16 possible bit-quad matrices that people have proposed to obtain different formulations to compute the Euler characteristic of a 2-D binary image.

Table 1 Possible bit-quads that can be used to compute the Euler characteristic of a 2-D binary image

Q1:	Q11=[0000]
Q2:	Q21=[1000]	Q22=[0100]	Q23=[0001]	Q24=[0010]
Q3:	Q31=[1100]	Q32=[0101]	Q33=[0011]	Q34=[1010]
Q4:	Q41=[1101]	Q42=[0111]	Q43=[1011]	Q44=[1110]
Q5:	Q51=[1111]
Q6:	Q61=[1001]	Q62=[0110]

The set of papers that we have found in the literature that uses a combination of bit-quads to compute Euler characteristic of a binary image are the following: [¹¹, ¹³, ¹⁴, ¹², ¹⁷, ¹⁶, ²¹, ³⁶, ³⁵, ³⁷, ³⁴, ⁴⁷, ⁴⁸, ⁴⁶, ³, ⁴³, ⁴⁵, ⁴⁴].

In [¹¹, ¹³, ¹⁴, ¹², ¹⁷, ¹⁶, ²¹, ⁴⁷, ⁴⁶, ³, ⁴⁵], authors used ten bit-quads to compute the Euler characteristic of a binary image I(x,y) as follows:

4−connected case:E=#Q2−#Q4+2#Q64, (2)

and

8−connected case:E=#Q2−#Q4−2#Q64, (3)

where:

— #Q2 is the occurrences of bit-quads Q21, Q22, Q23, and Q24.
— #Q4 is the occurrences of bit-quads Q41, Q42, Q43, and Q44.
— #Q6 is the occurrences of bit-quads Q61, and Q62.

Equations 2 and 3 are also known as Gray’s formulas.

As we have seen before, Equation (2) needs to perform ten comparisons on each image pixel. The time complexity for this method is O(N)2for a N×N image.

In [²¹], authors also propose using bit-quads Q23, Q42, Q61, and Q62 to compute the Euler characteristic of image an I(x,y) as follows:

4−connected case:E=#Q23−#Q42+#Q61, (4)

8−connected case:E=#Q23−#Q42−#Q62, (5)

where #Q23, #Q42, #Q61, and #Q62 are the occurrences of bit-quads Q23, Q42, Q61 and Q62, respectively.

In [³⁴], authors propose combining bit-quads Q21, Q44, Q61, and Q62 to compute the Euler characteristic of a 2-D binary image as follows:

4−connected case:E=#Q21−#Q44+#Q61, (6)

8−connected case:E=#Q21−#Q44−#Q62, (7)

where #Q21, #Q44, #Q61, and #Q62 are the occurrences of bit-quads Q21, Q44, Q61 and Q62, respectively.

In [⁴⁸, ⁴³, ⁴⁴], authors describe three different improvements for computing the Euler characteristic based on counting previous bit-quads. This procedure allows reducing the time and the number of required pixels for processing a bit-quad.

In [³²], the authors discuss how to use bit-quads in the so-called tree of shapes by using the depth of the node as the value of pixels in a larger and interpolated image representation. As a result, they propose an algorithm that uses larger image representations. In [³¹], the authors present a review of previously published works to compute attributes such as area, perimeter, and the Euler characteristic by incrementally counting patterns while traversing nodes of a component tree.

In short, in [³⁶, ³⁵, ³⁷], the authors propose using the 16 bit-quads shown in Table 1 to train distinct kinds of learning machines for estimating the Euler characteristic of a 2-D binary image: a multilayered perceptron, a morphological neural network, and a support vector machine.

Inspired in the previous discussion, in this paper we propose solving the following problem: automatically derive equations for computing the Euler characteristic of a binary image. For this, we suggest training an Artificial Neural Network (ANN) to automatically find optimal combinations of bit-quads to compute the Euler characteristic of a binary image.

The rest of the paper is organized as follows. In the Section 2, we explain the methodology followed. We show how two ANNs, one for the 4-connected and the other for the 8-connected case, can be trained to find optimal combinations of bit-quads to calculate the Euler characteristic of a binary image. Section 3 is oriented to present the results after training both ANNs. Besides, a set of experimental results are presented, where the performance of the ANNs is tested when computing the Euler characteristic over different images with distinct resolutions and complexities and compared them with state-of-the-art machine learning algorithms. Section 4 is finally focused on concluding remarks and future research.

2 Methodology

From the material presented in Section 1, for about 47 years, only three combinations of bit-quads and three improvements have been proposed. Probably, some more can be found. As we have seen, the most popular uses ten bit-quads, the one initially proposed in [¹¹]. The other two, as explained in [²¹, ³⁴], combine three bit-quads. Apparently, three is the minimum number of bit-quads that can be used to compute the Euler characteristic of a 2-D binary image in 4(8) connectivity.

Henceforward, the 16 bit-quads shown in Table 1 will be ordered and represented as depicted in Table 2.

Table 2 New representation of the 16 bit-quads from Table 1

Bit-quad number	Bit-quad (matrix form)	Bit-quad (binary form)
Q1:	[0000]	0000
Q2:	[0001]	0001
Q3:	[0010]	0010
Q4:	[0011]	0011
Q5:	[0100]	0100
Q6:	[0101]	0101
Q7:	[0110]	0110
Q8:	[0111]	0111
Q9:	[1000]	1000
Q10:	[1001]	1001
Q11:	[1010]	1010
Q11:	[1011]	1011
Q13:	[1100]	1100
Q14:	[1101]	1101
Q15:	[1110]	1110
Q16:	[1111]	1111

2.1 Finding an Optimal set of Bit-Quads to Compute the Euler Characteristic of a 2-D Binary Image

Finding an appropriate combination C of bit-quads to compute the Euler characteristic of a 2-D binary image can be manually obtained by testing the performance of such combinations from a set of images SI={I1,I2,…,Ip} and their corresponding Euler characteristics SEN={E1,E2,…,Ep}. It is clear that such combination C is one of those that can be obtained from the set:

C116+C216,…,C1616. (8)

One could test with all those combinations to find a convenient formulation. Doing this certainly could take a lot of time. Using common sense, we can appreciate that some bit-quads do not contribute to solving the problem. This is the case of bit-quads: Q11=[0000] and Q51=[1111] that describe the background of the image and the interior of the objects, respectively.

In what follows, we demonstrate how an ANN can be used to find an optimal combination of C bit-quads from the set of 16 bit-quads depicted in Table 2 to efficiently compute the Euler characteristic of a 2-D binary image I(x,y). For this, we propose using the following two-step procedure. Given a set of 2-D binary images SI={I1,I2,…,Ip} and their corresponding Euler characteristics: SEN={E1,E2,…,Ep}:

Map the set of images SI={I1,I2,…,Ip} to a matrix V={v1,v2,…,vp}, with vi=(#Q1,#Q2,…,#Q16)T, where #Q1,#Q2,…,#Q16 represent the occurrences of bit-quads Q1,Q2,…,Q16, respectively. As an example, refer to Table 3, where nine bit-quads can be found.
Present each pair (vi,Ei) to an untrained neural network and iteratively adjust its weights until convergence. Algorithm 1 shows the pseudo-code of the training process.

Table 3 Binary encoding example

4×4 binary image	Vector v
[1110101011100000]	v=[0100020021111000]T

Algorithm 1 Training process

We used 400 4×4 2-D images for the 4-connected case and 400 4×4 2-D images for the 8-connected case. All random images were generated using a discrete and uniform distribution and zero-padded. Furthermore, both linear perceptrons were trained with a learning rate of 0.001, using the RMSprop optimizer [²⁹] and 250 epochs [⁷]. Besides, the weights were initialized using Xavier normal initializer.

2.2 Performance of Four Different Machine Learning (ML) Algorithms to Forecast the Euler Characteristic

We also test the performance of four different machine learning (ML) algorithms to accurately forecast the Euler characteristic: the proposed single-layer ANN, a multi-layer perceptron (MLP) with one hidden layer, a support vector regression (SVR) with Gaussian kernel, and one dimensional convolutional neural network (CNN-1D). A benchmark with 400 binary images of 10×10, 15×15, 20×20 and 25×25 pixels are used for training the models, considering 4- and 8-connectivity cases. With k=10, the k-fold cross-validation technique is used to generate disjoint training and test sets.

For all the algorithms, the input vectors correspond to the 16 bit-quad patterns calculated from binary images. Furthermore, a tuning process is performed to find the hyperparameters that reduce the mean squared error.

For MLP, the number of hidden units is searched in the range [2,16,32] with ReLU activation functions and with a linear neuron in the output. For SVR, the penalization cost C and the kernel bandwidth γ are in the ranges [25,26,…,29] and [2−4,2−3,…,24], respectively. For CNN-1D, the number filters are examined in the range [8,16,32] with hyperbolic tangent activation functions and with a kernel size in range [1,2,3,4], with a hidden neurons and a linear neuron in the output.

The performance is tested by the mean squared error (MSE), mean absolute error (MAE), Pearson correlation coefficient (PCC), and discrepancy (D). The mean value over the 10-folds of cross-validation is calculated.

3 Results

In this section, we present the results at which we arrived after training an artificial neural network to estimate the Euler characteristic in 2-D binary images. We experimentally began testing with the simplest artificial neural network. We used a linear perceptron with 17 inputs (i.e., 16 bit-quads plus the bias value) with a linear activation function, as illustrated in Fig. 1, trained with Algorithm 1. As we will see, we found that this very simple learning machine was enough to find an optimal combination of bit-quads.

Fig. 1 Proposed artificial neural network to find a good combination of bit-quads to estimate the Euler characteristic E of a 2-D binary image

3.1 Learned ANN Weight Values

For the 4-connected case, we end up with a vector of weights, as shown in Table 4. For the 8-connected case, on the other hand, we finalized with a vector of weights depicted in Table 5.

Table 4 Weights obtained when training a linear perceptron using 4×4 2-D binary images for the 4-connected case

Weight number	Weight value	Rounded values	Bit-quad
1	-0.016	0	0000
2	-0.022	0	0001
3	-0.011	0	0010
4	-0.017	0	0011
5	-0.011	0	0100
6	-0.017	0	0101
7	-0.007	0	0110
8	-0.013	0	0111
9	0.978	1	1000
10	0.976	1	1001
11	-0.015	0	1010
12	-0.021	0	1011
13	-0.015	0	1100
14	-0.020	0	1101
15	-1.000	-1	1110
16	-0.017	0	1111
bias	0.145

Bold values indicate the most representative weights and their corresponding bit-quads

Table 5 Weights obtained when training a linear perceptron using 4×4 2-D binary images for the 8-connected case

Weight number	Weight value	Rounded values	Bit-quad
1	0.000	0	0000
2	-0.003	0	0001
3	0.004	0	0010
4	0.001	0	0011
5	0.005	0	0100
6	0.000	0	0101
7	-0.991	-1	0110
8	0.004	0	0111
9	0.997	1	1000
10	-0.006	0	1001
11	0.001	0	1010
12	-0.002	0	1011
13	0.001	0	1100
14	-0.003	0	1101
15	-0.994	-1	1110
16	0.001	0	1111
bias	0.108

Bold values indicate the most representative weights and their corresponding bit-quads

In both cases, note that most of the weight values are almost zero. This finding indicates that the ANN found that the corresponding bit-quads are not representative for estimating the Euler characteristic of a 2-D image. On the other hand, note also that in both 4(8)-connected cases, three weights approach +1 and −1 values. To establish the validity of the obtained results, we re-trained the linear perceptron 100 times over sets of 400 4×4 2-D binary images and their corresponding Euler characteristics for both the 4(8) connected cases. Tables 4 and 5 show the average of the perceptron weighs after training it 100 times:

For the 4-connected case, note that three weights approach to +1 and −1 values, corresponding to bit-quads 9, 10, and 15, respectively. Both weights 9 and 10 approach to +1, while weight 15 approaches to −1. Notice that the highest weights exactly correspond to the bit-quads used in Equation (5) to calculate the Euler characteristic of a 4-connected 2-D binary image.
For the 8-connected case, note that three weights approach to +1 and −1, corresponding to bit-quads 9, 7, and 15, respectively. Weight 9 approaches to +1, while both weights 7 and 15 approach to −1. Notice that the highest weights exactly correspond to the bit-quads used in Equation (6) to calculate the Euler characteristic of a 8-connected 2-D binary image.

Furthermore, we changed the weight initialization method and retraining 100 times the linear perceptron. We averaged and rounded all the learned weights by the ANNs to the nearest integer to reduce the influence of undesired bit-quads. As a result, we obtained the following new equations never reported in the literature for the 4-connected case:

E=#Q5+#Q7−#Q14, (9)

and for the 8-connected case:

E=#Q5−#Q10−#Q14. (10)

Based on the Stochastic Gradient Descent (SGD) training algorithm [²⁹], the linear perceptron has learned to select an optimal combination of bit-quads C to estimate the Euler number of a 2-D binary image. The perceptron arrived precisely at the same results that a human being has empirically done before, in this case, Equations (5) and (6). Moreover, when changing the weight initialization method, the linear perceptron discovered new formulations for computing the Euler characteristic. This relevant finding suggests the ability of the ANN to automatically provide new expressions that would take a human being a long time to discover.

3.2 Assessment of Discovered Expressions

Next, we tested the new expressions of the Euler characteristics shown in Equations 9 and 10 in images with distinct sizes and complexities. Three experiments were performed. In the first case, we used the small resolution images of 10 logos shown in Fig. 2. In the second case, we used medium resolution images with different objects with and without holes, as shown in Fig. 3. Lastly, in the third experiment, we used medium resolution images with one object subjected to different image transformations, Fig. 4. The idea is to assess invariance to different variations of the same object.

Fig. 2 Binary images of 128×128 pixels of ten different logos used to test the performance of the trained perceptron to estimate the Euler characteristic of a 2-D binary image

Fig. 3 Binary images of 320×240 pixels with a different number of objects of diverse complexity used to test the performance of the trained perceptron to estimate the Euler characteristic of a 2-D binary image

Fig. 4 Binary images of 320×240 pixels with one object subjected to different image transformation to test the performance of the trained perceptron to estimate the Euler characteristic of a 2-D binary image

Tables 6, 7 and 8 summarize the results. As the reader can rapidly appreciate, the desired Euler characteristic for all images has been obtained with perfect precision in all cases.

Table 6 Estimated Euler characteristics by the trained perceptron over images of Fig. 2

Image
Computed Euler characteristic	+3	−1	+4	+10	−6
Desired Euler characteristic	+3	−1	+4	+10	−6
Image
Computed Euler characteristic	+1	+5	+3	+12	+5
Desired Euler characteristic	+1	+5	+3	+12	+5

Table 7 Estimated Euler characteristics by the trained perceptron over images of Fig. 3

Image
Computed Euler characteristic	−6	+1	−9	−18
Desired Euler characteristic	−6	+1	−9	−18

Table 8 Estimated Euler characteristics by the trained perceptron over images of Fig. 4

Image
Computed Euler characteristic	+1	+1	+1	+1
Desired Euler characteristic	+1	+1	+1	+1
Image
Computed Euler characteristic	−4	−4	−4	−4
Desired Euler characteristic	−4	−4	−4	−4
Image
Computed Euler characteristic	−6	−6	−6	−6
Desired Euler characteristic	−6	−6	−6	−6

The trained neuron is applied to each of the three sequences of images shown in Fig. 4, and it obtains the same output. This results could be explained as follows. Suppose we have a binary image I1 with O objects and H holes. Then, this image is transformed into another image by applying to I1 an image transformation T as follows I2=T(I1). T could be any image transformation: translation, rotation, scale change, affine or projection, even a combination of several of them. The corresponding bit-quad representations of these two images are v1 and v2, respectively. Of course, due T, v1≠v2. However, as depicted in Table 6, the neuron’s output in the three examples is the same.

Coming back to our example of transforming I1 to I2 by T, when we feed v1 and v2 to the trained neuron, we obtain s1=w1x1,1+w2x2,1+…+w17 and s2=w1x1,2+w2x2,2+…+w17. If we take out the elements that do not contribute to the computation, as illustrated in Table 4, for the 4-connected case, we have s1=x9,1+x10,1−x15,1 and s2=x9,2+x10,2−x15,2. It is clear that, in order to the trained neuron output the same result, it is necessary that x9,1+x10,1−x15,1=x9,2+x10,2−x15,2.

As an example, let us take the first sequence of four images shown in Fig. 4, with the three most representative bit-quads. For these four images we have that: v1=[69,0,68]T, v2=[8,0,7]T, v3[140,0,139]T and v4=[66,0,65]T, where vi=[#Q9,#Q10,#Q15]T. This discussion can be formally stated as follows:

Proposition 1. The Euler characteristic E of any binary image is the same after applying any type of image transformation by counting the most representative bit-quads found by the linear perceptron.

Proof. Basis: For the first and fourth images of the third row of Table 8, the most representative bit-quads from each of these two images are v1=[328,0,334]T and v2=[6,0,12]T, respectively. By using the trained neuron, we obtain that 328+0−334=−6 and 6+0−12=−6, which is true because both images have the same Euler characteristic E of −6.

Induction step: Let I1 be a binary image with O objects and H holes, I2→T(I1) its transformed version trough image transformation T, and v1 and v2 their two corresponding bit-quads representations. So s1=x9,1+x10,1−x15,1 and s2=x9,2+x10,2−x15,2, the two outputs of the trained neuron, respectively. If s1=s2, then x9,1+x10,1−x15,1=x9,2+x10,2−x15,2.

Notice that this preposition was focused on the 4-connected case shown in Table 4, but it also applies, with minimal changes, for the 8-connected case shown in Table 5, as well as for Equations 9 and 10.

3.3 Euler Characteristic Prediction

Table 9 displays the results of the proposed single-layer ANN and three ML algorithms to predict the Euler characteristic for the 4- and 8-connectivity cases. Remarkably, the single-layer ANN method shows an outstanding capacity to predict the Euler characteristic with fewer computational resources. Although the other three algorithms also achieve significant results.

Table 9 Comparison of ML algorithms to predict the Euler characteristic. The mean value of 10-fold cross validation experiments is shown

Methods	4-connectivity				8-connectivity
Methods	MSE	MAE	PCC	D	MSE	MAE	PCC	D
ANN	0.000	0.000	1.000	0.000	0.000	0.000	1.000	0.000
MLP	0.000	0.002	0.998	0.010	0.000	0.004	0.996	0.050
CNN-1D	0.000	0.003	0.999	0.037	0.000	0.003	0.999	0.037
SVR	0.000	0.002	0.999	0.007	0.000	0.001	0.999	0.001

In bold are highlighted the best results

This finding points out that encoding the input binary image to a bit-quad representation allows describing the image’s topology such that a single-layer ANN, MLP, SVR and CNN-1D methods can build a mapping function to predict the Euler characteristic accurately. Nevertheless, though the MLP, SVR and CNN-1D predict the Euler characteristic very efficiently, they lack the property of explainability; that is, a human being understands the meaning of all the parts that make up the calculation of the Euler characteristic.

4 Conclusions and Future Work

In this section, we present the conclusions we have arrived at in this research. We also talk about future trends that emanate from this investigation.

We have shown that a very simple machine can find optimal combinations of bit-quads to estimate the Euler characteristic of a 2-D binary image.

This result is relevant because the described procedure, i.e., training a linear perceptron to find the most relevant bit-quads, could be applied to solve other combinatorial problems with possibly hundreds of variables.

Moreover, the neuron arrives at the same equations found manually by humans without a priori knowledge by using a limited set of training images. Also, the linear perceptron was capable of discovering new expressions never reported previously in the literature.

Additionally, the estimated Euler numbers obtained by the discovered expressions, in both cases (4-connected and 8-connected), are accurate, regardless of the number of objects and holes in the image.

These promising results encourage us to train an artificial neural network to estimate the Euler number of binary 3-D images. In this case, instead of bit-quads, we should use bit-octos represented by 2×2×2 matrices.

Acknowledgments

Authors would like to thank the Instituto Politécnico Nacional and the Centro de Investigaciones en Óptica A.C., for the economic support under project 20200630 to undertake this research. Laura Lira thanks CONACYT for the scholarship granted to carry out her doctoral studies.

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Received: August 02, 2021; Accepted: September 17, 2021

^* Corresponding author: Humberto Sossa, e-mail: humbertosossa@gmail.com

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