a new ab initio reconstruction method from unknown ... - IEEE Xplore

A NEW AB INITIO RECONSTRUCTION METHOD FROM UNKNOWN-DIRECTION PROJECTIONS OF 2D BINARY SET C´elia Fillion

Alain Daurat

Benoˆıt Naegel

Gabriel Frey

´ Etienne Baudrier

University of Strasbourg, CNRS ICube 300 Bd S´ebastien Brant - BP 10413 - F-67412 ILLKIRCH, FRANCE ABSTRACT This article focuses on the tomographic reconstruction of 2D binary images from projections whose directions are unknown. Observing that there is a mutual influence between the image reconstruction and the directions reconstruction, a new ab-initio method based on joint reconstruction of directions and image is proposed for 2D binary objects. Its implementation is done by optimizing a cost function taking as arguments the input projections, the current reconstructed image and the current directions. The optimization of the cost function is based on simulated annealing. After detailing the cost function, we test the proposed method for different resolutions and different noise levels. Finally, we present the future development of this promising method. Index Terms— tomography, ab-initio reconstruction, unknown directions, 2D, binary image 1. INTRODUCTION AND STATE OF THE ART In this paper we propose an original method to address the reconstruction problem for the 2D parallel-beam model from projections with unknown directions. This problem is encountered in various domains as medical imaging (e.g. when the patient is moving during a X-ray scanner acquisition), non-destructive testing and cryo electron microscopy. The reconstruction problem with unknown directions is usually addressed in two stages [1, 2]: first the projection directions are estimated, then the reconstruction itself is computed using the estimated directions. This approach is shared by all current methods [3] except [4]. In [4], Panaretos shows that the object density can be estimated without direction assignment under assumption on it. Nevertheless, the proposed method is noise sensitive and has not been tested on real data. We describe the two steps of the other methods below. In 3D, the first step relies on algorithms derived from common line correlation [5, 2, 6]. In 2D, the projection slice theorem [5] which is the base of common lines algorithms is no more available. The distance used in this case is generally the Euclidean distance. Consistency of the projections moments has also been proved to lead to direction assignment in 2D recovery under certain assumptions [7, 8, 9]. To cope with the

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high dimensional space of the data, the direction-assignment algorithms use dimension reduction [10, 11] or optimization [12, 13]. Once the direction estimation is done, the problem is reduced to reconstruction with known directions. The main families of reconstruction methods are [3]: the algebraic methods, iterative filtered back projection, and those using the Fourier transform. In the ab-initio case, the sequence of these two steps gives a first reconstructed image whose quality is not always sufficient, and then requires a refinement step [3, 13]. The reconstructed image is used to refine the projection directions and these directions can then be used to improve the data reconstruction. This iteration reflects the fact that if the estimated directions influence the reconstruction of the image, the estimation of the image also affects the estimation of directions. Indeed, the refined image produces projections that can be then compared to input projections (from the microscope). Then this allows to reclassify the input projections. But this does not allow the estimated image to play its full role in the estimation of directions, since it was built later. We propose a method that offers this possibility. It is based on an joint estimation of projection directions and image. The method is presented for binary images to show its feasibility with reduced cost. Moreover, binary images are a good approximation of the molecular electronic density which is highly contrasted. The rest of the paper is structured as follows: in Sec. 2, a model of the acquisition is shown and the problem is specified; in Sec. 3, the resolution method is detailed and the associated cost function is studied; in Sec. 4, the choice and the implementation of the optimization is presented then results are given in Sec. 5. A conclusion and perspectives end this paper. 2. MODEL AND PROBLEM Here is presented the mathematical model of the acquisition for simultaneous tomographical projections and the notations. We first recall the definition of the Radon transform applied here on sets modeling binary images. We call K the set of the measurable sets (for the usual measure of Lebesgues) included in the unit ball B2 of the pl R2 .

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Definition 2.1 (Radon transform) Let M an element of K (therefore finite measure) and χM its characteristic function. The Radon transform on M is given by Z 1 π RM (θ, s) = χM (s · eiθ + t · ei(θ+ 2 ) )dt (1)

is classically formed of a residual norm and a regularization term : J (M, Θ, P ) =

n X

2

kπi − π(M, θi )k2 + k∇χM k1 .

i=1

−1

where (s, θ) ∈ R2 . The measurable function s 7→ RM (θ, s) is called projection of M in the direction θ and denoted π(M, θ). π(M, θ) belongs to the set of bounded measurable functions with support in [−1, 1], denoted M B. Finally, let introduce an operator for the simultaneous acquisition of several projections. Let n ∈ N and Θ = (θ1 , . . . , θn ) ∈ (S 1 )n , we define the acquisition operator Π : ( K × (S 1 )n → (M B)n Π: (M, Θ) 7→ Π(M, Θ) = {π(M, θ), θ ∈ Θ} In order to jointly reconstruct the orientations and the object, the problematic is therefore: from P = {π1 , . . . , πn }, find (M sol , Θsol ) such that

The first term estimates an error between the given projections P and current projections {π(M crt , θ), θ ∈ Θcrt } which are two finite sets of functions of M B. As we make the assumption that the real object is compact, the second term estimates the compactness of the reconstructed object. Remark 3.1 The standard L2 norm gives the maximum likelihood estimator for the Gaussian noise that is present in the electron microscopy data. We present here a discrete version of the cost function. For 1 6 i 6 n, we denote (Pi,j )j=1...m (resp. (Qi,j )j=1...m ) a discretization of πi (resp. Π(M, θi )). We introduce the discrete binary image Im : (i, j) 7→ Im(i, j) ∈ {0, 1} such that Im(i, j) = M (i, j) for all (i, j) ∈ Z2 . We define: I(Im) =

X

|Im(i, j) − Im(i + k, j + l)|

(4)

i,j |k|+|l|=1

Π(M sol , Θsol ) = P . The reconstructed object is obtained by selecting the first component M sol of the solution. As for the classical problem, the solution is defined modulo a rotation. 3. METHOD In order to reconstruct jointly the object and the directions of projections, methods based on the inverse Radon transform are excluded since they require a priori knowledge of the directions. Our solution is to provide a current object M crt and current directions Θcrt = {θ1crt , . . . , θncrt } and evaluate them through a function J depending on the input projections P = {π1 , . . . πn }, taking as arguments the current directions and object (M crt , Θcrt ) with positive values: ( K × (S 1 )n → R+ JP : (2) (M, Θ) 7→ J P (M sol , Θsol ) such that J P (M sol , Θsol ) ≈ 0 The cost function J having positive values, the solutions can be expressed as a minimum: (M sol , Θsol ) = argminM,Θ J P (M, Θ)

X

(3)

Implementation of (3) relies on the definition of the cost function J and an optimization method. The cost function is based on three elements which are the current image, current directions and the given projections. It

I(Im) is a discretization of the regularization term k∇χM k1 . This function penalizes each contour point of the object (i.e. an object point adjacent to a background one for the 4-adjacency) hence favoring compact discrete objects. Moreover, during the optimization process this term can be updated incrementally. Finally a discretization J d of J is defined as: J d (P , Im, Θ) =

m n X X (Pi,j − Qi,j )2 + I(Im) i=1 j=1

4. OPTIMIZATION The cost function is not convex in general. Indeed, some projections may be similar and then give low value to the cost function without having close directions. Deterministic optimization methods are not suitable for non-convex cost function since they give likely local minima. Thus we choose heuristic optimization methods. Among these non-deterministic methods, the simulated annealing gave relevant results to minimize the cost function J . Simulated annealing (SA) is an iterative optimization method based on the Metropolis algorithm [14]. The algorithm starts from an initial state of the system and an initial temperature T0 . At each iteration, a modification of the system is proposed and the temperature T is decreased. This modification causes a variation of the cost ∆J . If the variation is negative, the proposed modification is accepted. Other1 wise, it is accepted with a probability e− T (Metropolis rule).

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The temperature T starts from T0 and decreases at each iteration toward zero. At each iteration the modification of the system (Θ, Im) is based on: (i) an elementary modification of the image Im by picking randomly a pixel and changing its value, (ii) an elementary modification of the set Θ by selecting randomly a direction and assigning to it a random value. In order to gain efficiency, the dimensions search space is discretized due to its high dimensionality. Choice of parameters The error produced on the object reconstruction has been studied empirically by computing the mean reconstruction error on 100 reconstructions of the same 20 × 20 image for a sampling step varying between 2 and 18. The result tends to show that the mean error depends linearly on the sampling step (see Figure 1). Then a small

128×128 and 256×256), referred to as phantoms. The phantoms are generated randomly by a Matlab program1 . The importance of the different parameters (initial temperature, temperature decay, acceptance probability, length of of each reconstruction phase) has been evaluated extensively, but the study can not be shown here due to the lack of place. The parameters used for the simulation come from a compromise between the quality of the reconstruction and the process time. A set of projections uniformly distributed have been extracted from the images. A random direction have been assigned to each projections. For each phantom, an initial image with a mass similar to the phantom (the mass is obtained from the projections) have been created. An illustration of image reconstructions by our method is presented Figure 2 for a single image at all the proposed resolutions. We also present image reconstructions by the spherical Local Linear Embedding (sLLE) method presented in [11]. The proposed method obtains better results than sLLE method. The reconstruction errors are presented in the results table 4. For each phantom, 10 reconstructions have been carOur method

sLLE

32 × 32

16 × 16

Original

Fig. 1. Reconstruction error in function of the sampling error for the projection directions

Fig. 2. A database image at different resolutions and its reconstructions by our method and by the method sLLE of [11].

5. RESULTS

ried out. The image reconstruction error Eim is estimated by the proportion of the mismatching pixels (in the disk in-

The proposed method have been applied to a set of 50 binary images of different resolutions (16 × 16, 32 × 32, 64 × 64,

1 Images and program are available at http://dpt-info. u-strasbg.fr/˜g.frey/ICIP_2013/

256 × 256

128 × 128

64 × 64

sampling step produces a small reconstruction error that can be accepted if the sampling allows a better optimization. The parameters of the simulated annealing have been estimated for each resolution and projection number so as to minimize the criterion of the average reconstruction error on a set of 50 images. These parameters are the initial temperature T0 , the function of temperature decay f , the initial state (Im0 , Θ0 ) and the stopping criterion σ. Two types of temperature decay functions have been tested: piecewise constant functions and exponential decreasing functions. The latter gives the better results. Several initial images have been tested and the best results are given by a rectangle having the same surface as the object. Θ0 is chosen as a uniformly randomized sample. The initial temperature has been tested between 0 and twice the maximum value of J . The best values depend on the projection number and the resolution and cannot be detailed here.

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scribed in the image support) between the phantom and the reconstruction. Figure 4 displays Eim for the different image sizes. For all the tested resolutions, the error remains under 8% of wrong pixels. The result is less than 1% of wrong pixels for 16 × 16 and 32 × 32 sizes. The error increases from the 64 × 64 size. The slight decreases observed are not significant in view of the standard deviation. An explanation for the increase is that 10 reconstructions are not sufficient to find the global minimum for the higher research spaces. Instead of increasing the reconstruction number, we will take advantage of the low-resolution good results by integrating them in a multi-resolution process in our futur works. Reconstruction

Fig. 4. Mean value and standard deviation of the wrong-pixels proportion in the reconstructed images in function of the image size.

Directions

σ = 50

σ = 15

σ=5

Sinogram

Fig. 3. For the same image as in Fig. 2, the sinogram is noised with Gaussian noise of standard deviation σ = 5, 15, 50. The reconstruction (image and projections direction) by our method is shown for each noise level. The robustness of the method against white noise has been evaluated. A Gaussian noise with zero mean and standard deviation σ has been added to the projections. The reconstruction errors Eim of phantoms of size 64 × 64 are shown for each noise level (Figure 5). The method noise robustness is good for low noise levels and and the visual image quality decreases quickly from the 15-variance Gaussian noise. This behavior is logical as angular assignation is difficult from noisy projections. Moreover image reconstruction needs a larger number of projections with noise than without noise. This shows that our method need to preprocess the noisy projections to achieved a better robustness. 6. CONCLUSION In this paper a new ab initio reconstruction method for 2D binary images is presented in the case of unknown projection

Fig. 5. Mean value and standard deviation of the wrong-pixels proportion in the reconstructed images in function of the additive Gaussian noise level. directions. This method simultaneously assigns the projection orientations and reconstructs the image. This is done by finding the minimum argument of a cost function including a residual norm between sinograms and a regularization term. The minimum search is made by simulated annealing optimization in a sampled search space. Our method have been experimented on binary images at different resolutions and its robustness have been evaluated against different noise levels. The results are promising for an ab-initio reconstruction method. Our tests show that projection preprocessing and introduction of multiresolution in the SA can improved the reconstruction. With these improvements, the method will be extended to the 3D and gray-level cases in further works. Another perspective is to study and improve the cost function by introducing other prior information (e.g. exploiting the Ludwig Helgason property). Acknowledgements: Alain Daurat, co-author of this article, died on June the 25th, 2010. This article is dedicated to his memory.

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