4 materials scene reconstructions from line Mojette projections

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8th International Conference on Advances in Computing and Communication (ICACC-2018) 8th International Conference on Advances in Computing and Communication (ICACC-2018)

44 materials materials scene scene reconstructions reconstructions from from line line Mojette Mojette projections projections a b a a a Chuanlin LIUa,∗ a,∗, Amit Yadava , Asif Khanb , Yi ZHANGa , Yu BAOa , Mei XUEa Chuanlin LIU , Amit Yadav , Asif Khan , Yi ZHANG , Yu BAO , Mei XUE a Chengdu Neusoft University,No.1 Dongruan Road, Qingchengshan, Dujiangyan, Chengdu, Sichuan 611844,P.R.China

b Univeristyaof Chengdu Neusoft University,No.1 Dongruan Road, Qingchengshan, Dujiangyan, Chengdu, Sichuan 611844,P.R.China Electronic Science and Technology of China, Qingshuihe Campus:No.2006, Xiyuan Ave, West Hi-Tech Zone, 611731, b Univeristy of Electronic Science and Technology of China, Qingshuihe Campus:No.2006, Xiyuan Ave, West Hi-Tech Zone, 611731,

Sichuan, P.R.China Sichuan, P.R.China

Chengdu, Chengdu,

Abstract Abstract The binary image reconstruction has been adressed for many years by discrete tomography.The Dirac Mojette transform allows The image reconstruction hasany been adressed for and many by discrete tomography.The Dirac Mojette allows for abinary more general framework with kind of values anyyears number of projections. This paper mainly adresstransform the 4 materials for a more general framework any algorithm.The kind of valuesline andMojette any number Thisthat paper adressreconstruction the 4 materials scene reconstrction by the line with Mojette uses of theprojections. characteristics the mainly scene under is scene reconstrction by the line Mojette Mojette uses the characteristics that then the scene underofreconstruction is composed of 4 material values. Since a algorithm.The projection lineline might contain a unique material value, the pixels this line can be composed of 4 directly materialwith values. line might containline a unique value, thenmore the pixels of this line be back-projected the Since uniquea projection material value. The inverse Mojettematerial can always solve than one pixel at can a time back-projected with theinversion. unique material value. The inverse linesecond Mojette can solve more than one at a time contrarily to thedirectly Dirac Mojette The constraint solver as the part of always the algorithm can make the pixel reconstruction contrarily to thethere DiracisMojette inversion. The solver as the part of series the algorithm make the tree reconstruction continue while no correspondence binconstraint can be used. Under thesecond Farey-Haros and the can Stern-Brocot projection continue while there isresults no correspondence bin line can Mojette be used.algorithm Under thecan Farey-Haros series and the scene Stern-Brocot tree projection sets, the experimental demonstrated that complete the 4 materials reconstruction with the sets, the experimental results demonstrated that line Mojette algorithm can complete the 4 materials scene reconstruction with the limited projection sets, which the Katz’s criterion is far to be checked. limited projection sets, which the Katz’s criterion is far to be checked. c 2018  2018 The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. B.V. © c 2018  The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND This is an open access article under the CC BY-NC-ND license license (https://creativecommons.org/licenses/by-nc-nd/4.0/) (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) committee of the 8th International Conference on Advances in Selection and peer-review under responsibility of the scientific Selection and under(ICACC-2018). responsibility of the scientific committee of the 8th International Conference on Advances in Computing andpeer-review Communication (ICACC-2018). Computing and Communication (ICACC-2018). Keywords: Discrete tomography; Mojette transform; line Mojette transform; 4 materials secene reconstruction Keywords: Discrete tomography; Mojette transform; line Mojette transform; 4 materials secene reconstruction

1. Introduction 1. Introduction The Mojette transform is a discrete Radon transform [3] [7][2]. The line Mojette was developed on the same basis Mojettea transform a discrete [3] the [7][2]. The lineinverse Mojetteline wasMojette developed the sameinbasis butThe assuming very smallisnumber of Radon discretetransform values into image.The wasonpresented the but assuming a very small number of discrete values into the image.The inverse line Mojette was presented in the case of sparse data (highly reduced number of projections)[5]. The 3 materials reconstruction problem was already case of sparse reduced number of projections)[5]. 3 materials reconstruction problem was already adressed in [5]data [6].(highly The dental scenes we want to separate air,The blood and bone which are the three incriminated adressed in [5] [6]. The dental scenes we want to separate air, blood and bone which are the three incriminated materials with very different Hounsfield numbers (used in CT)[1]. The Hounsfield number of a tissue is given by the materials very different Hounsfield numbers (used in CT)[1]. The Hounsfield number of a tissue is given by the following with formulate: following formulate: HN = 1000x(mtissue − mwater)/mwater. (1) HN = 1000x(mtissue − mwater)/mwater. (1) ∗ ∗

Chuanlin LIU. Tel.: +86-137-9577-2592; fax: +86-28-6488-8088. Chuanlin LIU. Tel.: +86-137-9577-2592; fax: +86-28-6488-8088. E-mail address: [email protected] E-mail address: [email protected] c 2018 1877-0509  The Authors. Authors. Published Published by by Elsevier Elsevier B.V. B.V. 1877-0509 © 2018 The c 2018 1877-0509  Thearticle Authors. Published byBY-NC-ND Elsevier B.V. This is access under thethe CCCC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an anopen open access article under license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an and openpeer-review access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection under responsibility ofofthe committee ofofthethe 8th8th International Conference on Advances in Computing and Selection and peer-review under responsibility thescientific scientific committee International Conference on Advances in Computing Selection and peer-review under responsibility of the scientific committee of the 8th International Conference on Advances in Computing and and Communication (ICACC-2018). Communication (ICACC-2018). Communication (ICACC-2018). 10.1016/j.procs.2018.10.379

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According to the equation 1, HN = 0 for the water, HNair is around -1000 for the air, whereas HN value oscillates between 50 to 300 for spongious bone and from 200 to 1000 for compact bone. Therefore, the four cases of the dental scene are highly discriminated onto this large scale. To identify the spongious bone and the compact bone, the 4 materials scene reconstrucion problem need to adress. We first recall the basic properties of the Mojette transform, the details can refer to [3]. The Mojette definition is given by Equation 2:  M f (k, l) = pro j(b, p, q) = M f (k, l) = f (k, l)∆(b + qk − pl), with∆(b) = 1i f b = 0and0elsewhere. (2) k

l

For a rectangular image P × Q, the number of bins onto projection (p, q) is computed as Equation 3: #bins(P, Q, p, q) = P(IqI − 1) + Q(I pI − 1) + 1.

(3)

The measure of redundancy is an important core idea of the Mojette transform. Since the first method to choose a specific set of projections is motivated by minimizing the redundancy of the transform, and giving a discrete optimization problem. This redundancy is defined by Equation 4: Red =

nbbins −1 nb pixels

(4)

The unicity reconstruction conditions from a set S = (p1, q1), (p2, q2), ..., (pI, qI) was first given by Myron Katz in 1978 [8] as the equivalence of the two following propositions: • i) the set S reconstruct the image P × Q;   • ii) ( i I pi I > P)or( i Iqi I > Q).

The exact inverse Mojette algorithm (see [3] chapter 4) has been derived by N. Normand and only use subtractions. The algorithm starts from a list of one to one pixel bin are computed according to the projections of the characteristic function[4]. Then, any bin in the list can be reprojected onto its pixel and the correspondence bin of each projection is updated consequently. The line Mojette is derived on the basis of Mojette transform to address the limited materials scene reconstruction problem [5]. The line Mojette shares the same direct algorithms as the classical (Dirac) Mojette producing the discrete projections. Its inverse algorithm in ternary case was presented in the case of sparse data (reduced number of projections) [6]. Based on the basics of Mojette transform and line Mojette transform, section 2.1 derived the line Mojette transform for the 4 materials scene case. After that, the 4 materials value choice was discussed in section 2.2. The experimental results in section 3 demonstrated the 4 line Mojette algorithm’s power before conclusion in section 4. 2. 4 materials image reconstructed from line Mojette algorithm In this section, the line Mojette algorithm is extended to 4 materials image reconstruction. Some properties are inherited from the 3 materials line Mojette, some properties are specifically derived for the 4 materials cases. We assumed that the 4 materials values in the image are (0, h, d, c). 2.1. 4 materials line Mojette reconstruction The first part of the algorithm still start recover an image from the correspondence bins. When all the pixels in the line of projection b = qi k − pi l have the same material value, these pixels can be back-projected directly. An extra check of the reconstructability of the correspondence bins must be performed to exclude the possible mistakes. The second part of the 4 materials inverse line Mojette still introduce the constraint solver to continue the reconstruction, but the constraint situations have some differences from the 3 materials problem. The same as 3 materials problem, two problems are to solve with this algorithm. First, we need to locate the reconstructible bins. Second, we have to determine which one of the pixels in the line of projection b = qi k − pi l is yet to be reconstructed.



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Fig. 1. Transforms of the 4 materials image and corresponding unitary, index images

One unitary image f1 is projected in the same way as the unknown image, one index image f s is projected with a stack S in each projection. They are all projected with the same set of projection angles and reconstructed simultaneously with the unknown image. These are all demonstrated in Figure 1. When the projection checks the equation [M pi ,qi f ](b) ≡ [M pi ,qi f1 ](b)×Vk whereVk ∈ (0, h, d, c), then it is referred as the correspondence bin, all the pixels in the bin can be back-projected directly with the value Vk . The reconstruction start from the initial correspondence bins, the newly discovered correspondence bins during the reconstruction process are pushed on top of the correspondence bin stack. When the stack is empty, no correspondence bin can be used, and the image has not been completely reconstructed, then the algorithm step into the constraint solver part[11][12]. In Figure 2, the incomplete reconstructed image is the reconstruction result of the initial correspondence bins (here, (0,h,d,c)=(0,1,5,7)). No newly discovered correspondence bins can be used, so the reconstruction stepped into the constraint solver part of the algorithm.

Fig. 2. 4 materials image reconstruction from the initial correspondence bins

There are two different cases of the positive constraint solver for the 4 materials line Mojette algorithm. The first case is the same as the constraint solver used in the three materials case, in this case each projection bin in the solver only contain two different material values. However, there are more possible intersected situations than in 3 materials image case.These intesected cases illustrated in Table 1 The second case is different from the 3 materials problem. In this case, the first projection bin still contain only two different material values. The projection bin intersected with this bin have three different material values. And the two projection bin have one and only one intersection material value. All the possible cases are illustrated in Table 2. In these two situations: there must be one projection bin which contain only two material values, so we loop over all the left projection bins, find a bin which only contain two different material values. Then, loop over the pixels in this bin orderly, get all the projection bins including this pixel. If there is one projection that contains 2 or 3 material values and have only one intersection with this bin, then the pixel value can be decided with this intersection material

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Table 1. Constraint solver of the two material value projection bin. Two material values projection bin (0,h) (0,d) (0,c) (h,d) (h,c) (d,c)

Intersected two material value projection bin

Intersected material value

(0,d),(0,c) (h,d),(h,c) (0,h),(0,c) (h,d),(d,c) (0,h),(0,d) (h,c),(d,c) (0,h),(h,c) (0,d),(d,c) (0,h),(h,d) (0,c),(d,c) (0,d),(h,d) (0,c),(h,c)

0 h 0 d 0 c h d h c 0 c

Table 2. Constraint solver of the three material value projection bin Two material values projection bin (0,h) (0,d) (0,c) (h,d) (h,c) (d,c)

Intersected three material value projection bin

Intersected material value

(0,d,c) (h,d,c) (0,h,c) (h,d,c) (0,h,d) (h,d,c) (0,h,c) (0,d,c) (0,h,d) (0,d,c) (0,h,d) (0,h,c)

0 h 0 d 0 c h d h c 0 c

value. In the constraint solver part, the first important thing is to decide the material values existing in the projection bin. If the bin contains only two different materials, we have Equation 5: [M pi ,qi f ](b) ≡ v1 × k1 + v2 × k2 , [M pi ,qi f1 ](b) ≡ k1 + k2 , wherev1 , v2 ∈ (0, h, d, c)

(5)

From Equation 5, we can get Equation 6 k1 ≡

[M pi ,qi f ](b) − v2 × [M pi ,qi f1 ](b) v1 − v2

(6)

Then, we check the value of k1 , while k1 is an integer and 0 < k1 < [M pi ,qi f1 ](b), the material value in the bin can be decided as (v1 , v2 ). To decide whether the projection bin contains 3 materials (v1 , v3 , v4 )(wherev1 , v2 , v4 ∈ (0, h, d, c)), some new constraint solver are derived. We suppose the number of the pixels with the value v1 is k1 and set k1 ∈ [M p ,q ](b) (1, [ i+ jv1i+ j ). Then according to the computation of the transform we have Equation 7: v3 × k3 + v4 × k4 = [M pi+ j ,qi+ j f ](b) − v1 × k1 k3 + k4 = [M pi+ j ,qi+ j f1 ](b) − k1

(7)

From Equation 7 gives Equation 8 k3 =

[M pi+ j ,qi+ j f ](b) − k1 × v1 − v4 × ([M pi+ j ,qi+ j f1 ](b) − k1 ) [M pi+ j ,qi+ j ](b) , wherek1 ∈ (1, [ ) v3 − v 4 v1

(8)

If there exists for a k1 value an integer k3 which satisfies 0 < k3 < [M( pi , qi ) f1 ](b), then the material values existing in the projection must be(v1 , v3 , v4 ). If there exists no integer k3 for all the possible k1 , then the material values of all the



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pixels aligned on the line of the projection were not (v1 , v3 , v4 ). After the material values in the intersected projection bins are decided, according to the Table 1 and Table 2 the only intersection material value is back-projected into the intersection pixel. In Figure 3(b), the corresponding unitary projection of projection M1,1 f (3) = 6 is M1,1 f1 (3) = 2, so we have M1,1 f (3) = 6 = 1 + 5. The first pixel in this projection with coordinates (1,0) also on the projection M−1,1 f (−3) = 1 = 1 + 0 + 0. These two projections have only one intersection material value 1, so this pixel’s value can be deceided as 1. This is the first constraint solver situation. In Figure 3(a), according to the Equation 6, we have projection M1,1 f (0) = 9 = 1 + 1 + 7. Its intersected projection is M−1,1 f (−2) = 12. According to the Equation 8, we have M−1,1 f (−2) = 12 = 0 + 7 + 5. Then the intersected pixel’s value is 7. This is the second case of the constraint sovler.

Fig. 3. Positive constraint solver in the 4 material image

2.2. Choice of the 4 materials value The different materials values choice might lead to different performances of reconstructions. In this section, we will discuss the influences of the material values to the reconstructions. A criterion is derived to ensure no mistakes exist in the reconstructions for any materials values. In order to demonstrate this problem clearly, we suppose a 4x4 image with the 4 materials values (0, 3, 5, 7). Take a projection bin of value 10 summing up 2 pixels in it. According to the correspondence bin definition, all the pixels in this bin can be back projected with the value 5. If the 2 pixels value are (3, 7) the bin value is also 10. If we directly back-project to pixels the bin value, it will cause mistakes as illustrated by Fig. 4. A rule is derived to exclude the possible mistakes. When the bin is a correspondence bin, we assume

Fig. 4. Positive constraint solver in the 4 material image [M

f ](b)

[M

f ](b)−n ×c

nc ∈ (1, pi ,qci ).We check whether there is a integer value of the formula0 < pi ,qi b c < [M( pi , qi ) f1 ](b), if so the correspondence bin is not reconstructible, if not, the correspondence bin is reconstructible and push on top of the correspondence bins stack. 3. Experiments In the experiments, three square medical image sizes were considered: 64×64,128×128,256×256. According to the Farey-Haros series and Stern-Brocot trees, different projection sets were computed to obtain complete reconstruction results.The Katz’s criterion can not be reached for all these projection sets. 4 materials images were bulit from original images. The 4 materials value in the 4 materials image are 0, 85, 160 and 255 respectively. And we introduce the FareyHaros series and Stern-Brocot tree firstly.These two old concepts both use a couple of integers to represent a ratio or even some approximation of a real. For the practical situations and the experiments, we need to have a principle to

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produce the sets of projections. According to such a principle, a series sets of projections can be produced, and the corresponding projection data will be used into direct projection algorithms. Let θi be a real number expressing an angle. As for the famous Bolzano Weirstrass theorem, there exists a discrete vector(pi , qi )which is a rational (angle)pi /qi very close to θi [9]. The pi /qi needs to be irreducible. All the terms of the Farey-Haros series and Stern-Brocot tree sequences are irreducible ratios built from 2 integers. Definition: The Farey-Haros seriesFm of order m is the ascending series of irreducible fractions between 0 and 1 whose denominators do not exceed m. Let take the Farey-Haros series of order 5 ’F5 ’ as an example:  F5 = { 01 , 15 , 14 , 13 , 25 , 12 , 35 , 23 , 34 , 45 , 11 , }. Such a series has the following two important properties: if hk and hk are two  successive terms inFm (with hk < hk ), then kh − hk = 1;       h if hk , hk and hk are three successive terms in Fm (with hk < hk < hk ), then hk = h+h k+k (the fractions k called the  mediant of hk and hk ). As soon as we have the Farey-Haros series Fm , the series Fm+1 can be computed by adding the mediant with denominator less than or equal to m + 1 of each two successive fractions in Fm [9]. The Stern-Brocot tree was first discovered by 19 century German mathematician Moritz Abraham Stern and French clock-maker Achille Brocot[10]. The Stern-Brocot tree shares some properties with the Farey-Haros series. As the Stern-Brocot tree in Fig.5 illustrated clearly, the left part of the tree is exactly the Farey-Haros series. We introduce the Farey-Haros series into the discrete projection.While the Farey-Haros series without the symmetric terms, all the projections belongs to a limited angle range of (0-45). We deonted the projection set from the Farey-Haros series Fn as S Fn . In order to get more projections, we use the Farey-Haros series and its symmetric terms (the symmetric terms according to the Y-axis (Fig.5) ) as projection angles. Then, the former projection set is enlarged as S FHn . According to the Stern-brocot tree sequence S tn , all the projections in the projection set S S tn are in the angle range 0-90. In order to get comprehensive projections, we use the Stern-brocot tree sequence and its symmetric terms (the symmetric terms according to the Y-axis) as projection angles. The projection set denoted as S S tS n .

(a) (b) Fig. 5. The Farey-Haros series F4 and Stern-Brocot tree and their symmetric points.

Based on that, three 4 materials scene are introduced to do many experiments. Figure 6 shows the original images and the 4 materials versions. In these experiments, the projection sets were set up according to the Farey-Haros series and the Stern-Brocot series. After that, the line Mojette projection data produced from these projection sets. The inverse line Mojette used these projection data to do the reconstruction and give out the reconstruction result. Table ?? summarizes some results obtained from Farey-Haros series projection sets.Table ?? summarizes some results obtained from Stern-Brocot series projection sets.



Chuanlin et al. /Computer Procedia Science Computer 143 (2018) 203–210 Chuanlin LIU /Liu Procedia 00 Science (2018) 000–000

(a)

(b)

(c)

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(d)

(e) (f) Fig. 6. (a) MRI 64x64 image and its 4 materials version (b); (c) 128x128 X-rays scan image and its 4 materials version (d); (e) lungs 256x256 X-rays scan image and its 4 materials version (f).

Table 3. Approximate results for Farey-Haros series reconstructions according to the image size and the equivalent redundancy. Image size

Proj set

Red:P:Q

Proj set

Red:P:Q

64 × 64 128 × 128 256 × 256

S FH4 S FH6 S FH6

-0.18:35:18 0.16:99:50 -0.42:99:50

S F5 S F7 S F8

-0.12:38:19 0.07:85:42 -0.46:101:50

Table 4. Approximate results for Stern-Brocot series reconstructions according to the image size and the equivalent redundancy. Image size

Proj set

Red:P:Q

Proj set

Red:P:Q

64 × 64 128 × 128 256 × 256

S S tS 3 S S tS 5 S S tS 5

-0.17:27:26 0.88:82:81 -0.06:82:81

S S t5 S S t5 S S t6

0.71:56:55 -0.14:56:55 -0.43:74:73

4. Conclusion The line Mojette algorithm proposed in this paper is to adress the 4 materials scene reconstruction problem. The algorithm mainly included the correspondence bin and constraint solver two parts. The reconstruction can be completed with sparse data(the Katz’s criterion of all the projection sets is far to be checked). The experiments starts from the Farey-Haros series projection sets and the Stern-Brocot tree projections sets. All the projection sets were computed with the 3 different size images. Then, all the projection data computed from these projection sets which can complete the reconstructions with a negative redundancy. When a Farey-Haros projection set allows a unique reconstruction result, then any larger Farey-Haros projection sets will also do it. In these experiments, we found all the Farey-Haros series ( as well as Stern-Brocot tree sequence) projection sets which can complete the reconstructions to the 3 different size ternary images both in the angle limited or not situations. Finally, we began to decrease the number of projections, and images can still be reconstructed successfully for projection sets only composed of few projections. These experimental results demonstrated the power of the line Mojette algorithm in the 4 materials scene reconstructions. Then the 4 different materials can be effectively descriminated from few projections.

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