A Novel Approach for Plant Embryonic Cells Serial ... - IEEE Xplore

A Novel Approach for Plant Embryonic Cells Serial Section Images Registration* Dong Liang

Zhaobin Wang

Yide Ma†

School of Information Science and Engineering Lanzhou University Lanzhou, China [email protected]



Abstract—The three dimensional reconstruction of the serial section images is an effective approach for plant embryo cells quantitative analysis. However, because of the complexity of the section images, the registration problem of the serial section images is difficult to be solved. In this paper, a novel approach based on Radon transform and phase correlation, which is used for the plant embryonic cells serial section images registration, is introduced. After the analysis of these image’s features, we use the gravity center of the image to compute the quantity of parallel translation. Compared with the previous image registration algorithm using Radon transform and phase correlation, this approach observably improves both the efficiency and robustness. Keywords-image registration; three dimensional reconstruction; plant embryonic cells serial section; Radon transform; phase correlation; gravity center

I.

INTRODUCTION

Research on the growth of plant embryonic cells requires calculating the change of micro-molecules such as protein, RNA and starch, and the elements, such as calcium during their different growing stages. According to the information, their growth can be quantitatively analyzed [1]. We hope to use the three dimensional reconstruction technology of plant embryonic cells serial section to achieve quantitative analysis. Namely, make the serial section of the ultrastructure of plant embryonic cells, and reconstruct the three dimensional morph of the ultrastructure [2, 9]. Image registration is the first step of the three dimensional reconstruction. The task of registration algorithm is to robustly detect the angle of rotation and the quantity of parallel translation between two plant embryonic cells serial section under the noise environment. Radon transform and phase correlation have been introduced to image registration algorithm to detect the rotation and parallel translation [7, 8, 12]. It is able to estimate large rotations and translations, but experiments show that it requires much computation time and its robustness is not good enough for section images registration. In this paper, we will introduce Radon transform and its properties, and then after investigating the features of plant

embryonic cells serial section images, we develop algorithm to detect the angle of rotation and the quantity of parallel translation for section images registration. Our approach cannot only be applied to the registration of plant embryonic cells serial section images, but also to other image registrations, when the effective information is in the interior region of the image. II.

RADON TRANSFORM AND PHASE CORRELATION

A. Radon transform Let x-y be a coordinate frame fixed on a plant embryonic cells serial section grayscale image. Let f(x, y) be a pixel value at a point (x, y) on the image. Let us integrate the pixel value f(x, y) along a line where the distance from the coordinate origin is given by ρ and the angle from the x-axis is specified by θ, as shown in Fig. 1. This integral depends on two parameters, ρ and θ, and is denoted as R(ρ, θ), as shown in Fig. 2. Namely, ∞

R(ρ,θ ) = ∫ f (ρ cosθ − λ sinθ , ρ sinθ + λ cosθ )dλ .

(1)

−∞

Let f1 be a rotated and translated image of the original image with rotational angle α, translational direction β, and translational distance d0. Then, the Radon transform R1(ρ, θ) of the image f3 satisfies the following equation: R 0 ( ρ , θ ) = R 1 ( ρ − d 0 sin( θ − β ), θ + α ) .

(2)

The above equation describes that the rotation of an image is given by a shift with respect to parameter θ and that its translation is given by a shift with respect to parameter ρ. The shift with respect to θ coincides to the rotational angle α while the shift with respect to ρ is given by −d0sin(θ−β), which varies according to parameter θ. Note that shifts parameter ρ must be eliminated by computing the one dimensional Fourier transform to get the one dimensional power spectrum with respect to the parameter

* This work was supported by the National Science Foundation of China under the Grant No.60572011 and No.60872109, Program for New Century Excellent Talents in University (NCET-06-0900), National Natural Science Foundation in Gansu Province under the Grant No. 0710RJZA015. † Corresponding author. Email: [email protected]

978-1-4244-2902-8/09/$25.00 ©2009 IEEE

1

[6]. The power spectrum of Radon transform R(ρ, θ) with respect to ρ is given by

F (ω , θ ) =

∞

∫ R ( ρ ,θ )e

iρ f

dρ .

(3)

−∞

B. Phase correlation Let F0(ω, θ) and F1(ω, θ) be power spectrums of Radon transforms R0(ρ, θ) and R1(ρ, θ). From (4), we have

F0 (ω ,θ ) = F1 (ω ,θ + α ) .

(4)

This implies that the rotation angle α must be computed through one-dimensional phase correlation between power spectrums F0 and F1, namely,

III.

GRAVITY CENTERS

Plant embryonic cells section image is different from other kinds of images, especially the effective information is located in the interior region of the image, even if the images have planar motion, the effective border of the cell will not beyond the scope of the image. Therefore, based on the gravity centers of the whole image, the quantity of parallel translation between the images can be detected, as shown in Figure 3. Reference to related concepts in physics, the gravity center G of the grayscale image is defined as:

G = ( x , y ), where

x=

∑ x × f ( x, y ) , ∑ f ( x, y )

( x , y )∈S

( x , y )∈S

F1 (ω) F1 (ω) F0* (ω) A1 (ω) A0 (ω)ei[ψ 0 (ω )−ψ1 (ω )] = = = eiαω .(5) F0 (ω) A1 (ω) A0 (ω) A1 (ω) A0 (ω) Note that

F − 1 ( e iαω ) = δ (θ + α ) .

(6)

So we can have the angle of rotation after finding the peak. Namely,

⎛ F (ω ) ⎞ ⎟⎟ . α rot = arg θ max F −1 ⎜⎜ 1 ⎝ F0 (ω ) ⎠

(7)

.

y=

(8)

∑ y × f ( x, y ) ∑ f ( x, y )

( x , y )∈S

( x , y )∈S

S is the set of the pixels whose grayscale value are f(x, y). As shown in Fig. 4(a), first, because of the complex structure of image, there is a bit distinction of the gray scales between different plant embryonic cells section images. Second, uneven distribution of light will cause the uneven distribution of the gray scale. Third, while dyeing, disturbance and noise are introduced, and both gray scales of the cells and the background unevenly distribute in the whole image. Even though the gray scales of some cells are lower than those of the background. Sometimes, there is no obvious distinction of gray scales between the cells and the background or between the edges and the inner elements of cells. When parts of cells are blended into the background, it is difficult to extract the gravity centers exactly. Therefore, we use Canny operator for edges extraction and then compute the gravity centers of the binary image to solve this problem. As shown in Fig. 4(b), this method can effectively prevent the gravity center planar motion caused by uneven distribution of light, and reduce the impact of other random noises. After extracting the edges, (8) is simplified as:

G = ( x , y ), where x =

Figure 1. Integral paths in Radon transform

∑x ∑y . ,y= ∑1 ∑1

( x , y )∈D

( x , y )∈D

( x , y )∈D

( x , y )∈D

(9)

D is the set of the binary pixels whose value are 1. Besides, as mentioned in (3), the Radon transform must compute Fourier transform to eliminate shift impacted on orientation detection. Detecting the quantity of parallel translation by using the gravity centers before detecting the angle of rotation, omits the step of Fourier transform and does not need to find the quantity of parallel translation by using 2D phase correlation [7]. So it reduces the amount of computing obviously.

Figure 2. Radon transform

2

translation can be determined from the location of the gravity centers. This algorithm is summarized in Algorithm 2: Step 1. Compute gravity centers Gs and Gr. Step 2. Compute Rs(ρ, θ) and Rr(ρ, θ) around individual gravity centers. Step 3. Find the angle of rotation by using 1-D phase correlation. In order to prevent the gravity centers planar motion caused by the uneven distribution of light, let Bs and Br be the binary image of these images after edge extraction. We have Algorithm 3: Step 1. Extract the edges by using Canny operator to achieve the binary images Bs and Br. Step 2. Compute gravity centers Gs and Gr of the binary images Bs and Br. Step 3. Compute Rs(ρ, θ) and Rr(ρ, θ) around gravity centers. Figure 3.

Cell image gravity centers

Step 4. Find the angle of rotation by using 1-D phase correlation. We carried out five groups of experiments, and the images are seriously polluted by the uneven distribution of light in the last two groups. The experimental results are given in Tab. ІІІІ. TABLE I.

a

1 2 3 4 5

b

TABLE II.

Figure 4. Plant embryonic cells image and its edges image

IV.

True Translation (pixel) (32,-25) (17,-19) (28,31) (-22,13) (15,-29)

ALGORITHMS EXPERIMENTAL EVALUATION

We formulate algorithms to detect the rotation and parallel translation. Let Is be a sample image and Ir be a registration of this image, and Fs(f, θ) and Fr(f, θ) are Fourier transforms of the Radon transform along parameter ρ. The algorithm introduced in [7] using Radon transform and phase correlation is summarized in Algorithm 1:

1 2 3 4 5

Step 2. Compute Fs(f, θ) and Fr(f, θ).

Step 4. Find the quantity of parallel translation by using 2-D phase correlation. Let Gs and Gr respectively be the image gravity center in a sample image and that in a registration image. The parallel

1 2 3 4 5

True rotation (deg.) -14.5 6.1 2.2 11.5 -8.9

Algorithm 2 (s) 4.533 4.092 4.416 4.291 4.116

Algorithm 3 (s) 4.524 4.090 4.417 4.292 4.114

PARALLEL TRANSLATION ESTIMATION RESULTS

Algorithm 1 (pixel) (27,-20) (13,-15) (20,35) (-18,16) (17,-25)

TABLE III.

Step 1. Compute Rs(ρ, θ) and Rr(ρ, θ). Step 3. Find the angle of rotation by using 1-D phase correlation.

Algorithm 1 (s) 4.752 4.224 4.512 4.406 4.248

COMPUTING TIME RESULTS

Algorithm 2 (pixel) (26,-22) (20,-22) (24,27) Failure Failure

Algorithm 3 (pixel) (29,-23) (19,-19) (30,29) (-20,16) (16,-28)

ROTATION ESTIMATION RESULTS

Algorithm 1 (deg.) -13.5 6.9 2.4 10.2 -9.4

Algorithm 2 (deg.) -15.3 7.4 2.4 Failure Failure

Algorithm 3 (deg.) -15.1 6.7 2.2 11.0 -9.4

3

The results given in Tab. І show that Algorithm 2 and Algorithm 3 cost fewer time than Algorithm 1, the average time is 4.428s, 4.290s and 4.287s respectively. Thus, we conclude that if we compute the gravity centers instead of Fourier transform to eliminate planar motion which impacts on orientation detection, we can reduce the registration time. The results given in Tab. ІІ show that the quantity of parallel translation error of Algorithm 1 was δ = (4.6, 4), while the error of Algorithm 3 was δ = (2, 1.6). The results given in Tab. ІІІ show that the angle of rotation error of Algorithm 1 is δ = 0.76 deg., while the error of Algorithm 3 was δ = 0.44 deg. Thus, we conclude that extracting the edges and computing the gravity centers improve the accuracy of the plant embryonic cells serial section images registration. The results that are given both in Tab. ІІ and Tab. ІІІ show that Algorithm 2 cannot achieve the correct registration parameter when the image is seriously polluted by the random noise, while Algorithm 3 can. Thus, we conclude that using Canny operator can reduce both the parallel translation and rotation registration error and observably improve the robustness. As mentioned above, we conclude that Algorithm 3 has a capability to detect the rotation and parallel translation of plant embryonic cells serial section images. The result of registration for four serial section images is shown in Fig. 5. V.

SUMMARY

In this paper, we have developed algorithms to detect the angle of rotation and the quantity of parallel translation of the plant embryonic cells serial section images. It has turned out that Algorithm 3 has a capability to detect the rotation and parallel translation of the cell section image. It cannot only be applied to the registration of plant embryonic cells serial section images, but also to other image registration when the effective information is in the interior region of the image.

Figure 5.

REFERENCES [1]

Yide Ma, Gengsheng Xing, “3D image reconstruction of plant tissue slice and its application in the plant body cell’s embryogeny research”, Letters in Biotechnology of China Vol. 11, No. 4 , Dec. 2000, pp. 303308. [2] Carlbom I., Terzopoulos D. and Harris M., “Computer assisted registration segmentation and 3D reconstruction from images of neuronal tissue sections”, IEEE Trans Med Imaging ,1994 , pp.13-351. [3] Ballard, D., “Generalizing the Hough Transform to Detect Arbitrary Shapes”, Pattern Recognition, Vol. 13, No. 2, 1981, pp.111-122. [4] Pao, D.C.W., Li, H.F. and Jayakumar, R., “Shape Recognition Using the Straight Line Hough Transform: Theory and Generalization”, IEEE. Trans. PAMI, Vol.14, No.11, 1992, pp.1076-1089. [5] Chen, Q., Defries, M. and Deconinck, F., “Symmetric Phase-Only Matched Filtering of Fourier-Mellin Transforms for Image Registration and Recognition”, IEEE Trans. PAMI, Vol.16, No.12, 1994, pp. 11561168. [6] Onishi, H. and Suzuki, H., “Detection of Rotationand Parallel Translation Using Hough and Fourier Transforms”, Proc. 1996 Int. Conf. on Image Processing, Vol.3, 1996 pp. 827-830. [7] Li Zhongke, Wu lenan, “Image registration on Hough transform and phase correlation”, IEEE ICNNSP. 2003. pp. 956-959. [8] T. Tsuboi, A. Masubuchi, S. Hirai, S. Yamamoto, K. Ohnishi, and S. Arimoto, “Video-frame Rate Detection of Position and Orientation of Planar Motion Objects using One-sided Radon Transform”, Proc. IEEE Int. Conf. on Robotics and Automation, Vol.2, Seoul, May 2001, pp. 1233-1238. [9] Otten E. and Van Lecuwen JL., “A 3D reconstruction package from serial sections images”, Eur J Cell biol, 1989, pp. 48 -25. [10] Y. Keller, A. Averbuch, “A projection-based extension to phase correlation image alignment”, Signal Processing 87 (2007), pp. 124-133. [11] S. Reddy, B.N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration”, IEEE Trans. Image Process. 3 (8) (1996), pp. 1266-1270. [12] You Jiangsheng, Lu Weiguo, Li Jian, Gindi Gene, Liang Zhengrong, "Image matching for translation, rotation and uniform scaling by the Radon transform", International Conference on Image Processing, Volume 1, 1998, pp. 847-852.

The result of registration for four plant embryonic cells serial section images

4