Iage restoration for a rectangular poor-pixels detector Pengcheng Wen 1, Xiangjun Wang 1, Hong Wei 2 1 State Key Laboratory of Precision Measuring Technology and Instruents, Tianjin University, China 2 School of Systes Engineering, University of Reading, UK ABSTRACT This paper presents a unique two-stage iage restoration fraework especially for further application of a novel rectangular poor-pixels detector, which, with properties of iniature size, light weight and low power consuption, has great value in the icro vision syste. To eet the deand of fast processing, only a few easured iages shifted up to subpixel level are needed to join the fusion operation, fewer than those required in traditional approaches. By axiu likelihood estiation with a least squares ethod, a preliinary restored iage is linearly interpolated. After noise reoval via Canny operator based level set evolution, the final high-quality restored iage is achieved. Experiental results deonstrate effectiveness of the proposed fraework. It is a sensible step towards subsequent iage understanding and object identification. Keywords: Iage restoration, axiu likelihood, level set, Canny operator, poor-pixels detector 1. ITRODUCTIO Poor-pixels detector, with extreely sall quantity of photosensitive cells (pixels on its iaging surface, is designed for an ebedded coputer vision syste. It is valuable in any applications such as icro unanned aerial vehicles and icro bioietic robots. As an exaple, a rectangular 25-pixels detector is shown in Figure 1. It has properties of iniature size, light weight and low power consuption, which are helpful in a highly-integrated battery-using device. Meanwhile, the iage acquisition speed can be draatically accelerated due to only a few photoelectric signals being processed [1]. Figure 1: Pixel distribution of a rectangular 25-pixels detector (grey colour areas depict photosensitive cells Figure 2: A synthetic extreely low-resolution rectangular iage An iage captured by the rectangular 25-pixels detector, shown in Figure 2, has following features. (1 It is an extreely low-resolution iage, like a osaic iage. (2 Grey inforation represented by iage pixels is not with sooth continuity. Hence the contexture is hardly to be established. These ake the iage do not reflect the sae or siilar shape inforation as an object has in the real world. It is therefore difficult to identify an object and understand a scene based on such a single iage. To ensure further application of the rectangular poor-pixels detector in real probles, this paper presents a unique iage restoration fraework fro a series of extreely low-resolution rectangular iages shifted up to subpixel level. Different fro traditional approaches in which the aount of inforation contained in the original low-resolution iages should be larger than the aount of inforation required in the restored iage [2-4], the designed fraework only needs a few easured iages to join the fusion operation. This is a guarantee to eet the deand of fast processing. However, Sixth International Syposiu on Precision Engineering Measureents and Instruentation, edited by Jiubin Tan, Xianfang Wen, Proc. of SPIE Vol. 7544, 754436 2010 SPIE CCC code: 0277-786X/10/$18 doi: 10.1117/12.885301 Proc. of SPIE Vol. 7544 754436-1 Downloaded Fro: http://proceedings.spiedigitallibrary.org/ on 12/23/2014 Ters of Use: http://spiedl.org/ters
cutting down the data eans losing the inforation. To be still able to achieve a high-quality restored iage, a two-stage iage restoration strategy is adopted in the proposed fraework. Firstly, a preliinary restored iage is linearly interpolated by axiu likelihood estiation. Secondly, as further restoration, Canny operator based level set evolution is applied for noise reoval. The rest of the paper is organized as follows. Section 2 introduces details of the proposed fraework. Experiental results on synthetic iages are presented and discussed in Section 3. Section 4 concludes the work with identified achieveents. 2. METHODOLOGY 1.1 Matheatical odel To siply and efficiently odel the proble, extreely low-resolution iages Y k (k=1,2, are acquired by shifting a rectangular 25-pixels detector up to subpixel level. As shown in Figure 3, they are different representations of an ideal high-resolution iage X. The popular forulation is defined as [4]: Yk = DkHkFkX + Vk (k=1,2, (1 F k stands for the geoetric warp operation that exists between X and an interpolated version of Y k (interpolation is required in order to treat the iage Y k in the higher resolution grid. H k is the blur atrix, representing the detector s PSF. D k denotes the deciation process, representing the reduction of the nuber of observed pixels in the acquired iages. V k is Gaussian additive easureent noise with zero ean and auto-correlation atrix W k = σ 2 I. Figure 3: Modeling the iage restoration proble based on extreely low-resolution rectangular iages Traditionally, if the nuber of pixels in the easured iage Y k is M k M k and in the ideal iage X is L L, one intuitive rule is: L < M + K+ M + K + M 2 2 2 2 1 k which can be explained as a requireent that the aount of inforation contained in the easured low-resolution iages should be larger than the aount of inforation required in the restored high-resolution iage. To estiate X well, plenty of Y k are needed, which, however, ay increase the coputation and slow the operation. In this paper, to eet the deand of fast processing, fewer easured iages are taken into the restoration. By shifting the rectangular poor-pixels detector only along the horizontal, vertical and diagonal directions, a special series of Y k are obtained. This acquisition process is easy to realize but loses soe useful inforation which needs to be copensated in the following operations. Proc. of SPIE Vol. 7544 754436-2 Downloaded Fro: http://proceedings.spiedigitallibrary.org/ on 12/23/2014 Ters of Use: http://spiedl.org/ters
1.2 Linear interpolation The axiu likelihood estiation is eployed to estiate X by axiizing the conditional probability density function P{Y X}. { } Xˆ = ArgMax P Y X (2 ML X Where X ˆ ML denotes the axiu likelihood estiate of X and Y denotes the group of Y k. Based on the assuption that Gaussian additive noise vectors are utually independent, Eq. (2 is done through the following least squares expression: ˆ X ML = ArgMin Yk DHFX k k k Wk Yk DHFX k k k X k = 1 T 1 [ ] [ ] (3 Differentiating Eq. (3 with respect to X and letting it be zero gives the well-known classic pseudo-inverse result: where k = 1 PXˆ ML = Q (4 P = F H D W D H F (5 k = 1 T T T 1 k k k k k k k Q= F H D W Y (6 T T T 1 k k k k k For the purpose of iage soothness, locally adaptive regularization is adapted in Eq. (3 by using the Laplacian operator S and a weighting atrix V. Thus, Eq. (3 becoes to ˆ X ML = ArgMin Yk DHFX k k k Wk Yk DHFX k k k + SX V SX X k = 1 T 1 T [ ] [ ] β [ ] [ ] (7 where β is a relaxing factor. Differentiating Eq. (7 again with respect to X and letting it be zero yields the sae equation T as Eq. (4 with a new ter, β SVS, added to the atrix P as T T T 1 k k k k k k k T β (8 k = 1 P = F H D W D H F + S VS To solve the above equations, the conjugate gradient algorith is chosen due to its relatively fast convergence and high quality result [5]. Although the above linear interpolation ethod is easy to operate and workable as well, there are still a lot of noises in the preliinary restored iage. Additionally, due to the wanting inforation at the acquisition stage, the contour of the restored object is not clear either. To ake further iproveent of the restoration, Canny operator based level set evolution is then applied. 1.3 oise reoval Aong all filtering techniques, level set ethod [6-9], which is less sensitive to natural noise and ore contrast preserving, has becoe popular due to its flexibility and capability in odeling coplex structures. While Canny operator [10], which is robust to noise, is probably the ost widely used edge detector. Unless the preconditions are particularly suitable, it is hard to find an edge detector that perfors significantly better than the Canny operator. To cobine their advantages, a odified level set ethod with Canny operator is presented. Proc. of SPIE Vol. 7544 754436-3 Downloaded Fro: http://proceedings.spiedigitallibrary.org/ on 12/23/2014 Ters of Use: http://spiedl.org/ters
Level set ethod is based on the partial differential equation. Its ain idea is that the oving interface is looked as a zero level set in one higher diensional space. The evolution equation of the level set function φ can be written as [6]: ϕ + F ϕ = 0 t where F is a speed function deterining the diffusion of the oving interface opposite to its noral direction. For the proble of iage noise reoval, φ is replaced by the iage data I, and F is usually a kind of curvature flow which depends on I. Thus, Eq. (9 is rewritten as: I t + F I = 0 According to the definition of Canny operator [10], n, the noral to the direction of a detected edge, can be estiated well fro the soothed gradient direction ( G I n = (11 G I ( where G is the two diensional Gaussian function. At such an edge point, the edge strength E can be calculated by G E = I x I where x E G = I y and y. 2 2 x y (9 (10 E = E + E (12 Replacing in Eq. (10 by n, the noral direction to an edge, and selecting F in Eq. (10 with consideration of the edge strength E, the level set evolution equation with Canny operator is obtained as: ( G I ( I + Fc = 0 t G I ean flow ax ( E (13 < T Fc = (14 in/ax flow otherwise where ax(e is the axiu of edge strength in 3 3 neighborhoods of the central point and T is the low threshold for Canny optiized algorith. To penalize the local irregularity in curves, an energy ter is added to Eq. (13: where g is an edge indicator function [11], ( I ε practice, g and δ ( are defined as follows: ε I ( G I ( I + FC + μ g δ ε I = t G I ( 0 (15 δ is the univariate Dirac function and μ is an adjusting factor. In 1 g = (16 + 2 1 G I δ ε ( I 1 π I = 1 cos 2ε + ε (17 Proc. of SPIE Vol. 7544 754436-4 Downloaded Fro: http://proceedings.spiedigitallibrary.org/ on 12/23/2014 Ters of Use: http://spiedl.org/ters
For all the experients in this paper, both μ and ε take the value of 1.5. This achieves a good perforance both in the whole iage for noise reoval and in the local area for edge preservation. 3. EXPERIMETAL RESULTS In this section, testing experients on synthetic iages are presented to deonstrate the effectiveness of the proposed iage restoration fraework. All siulations are ipleented in MATLAB. There are four ideal scenes (iages containing four en with different postures, respectively, as shown in Figure 4(a. For each of the, 60 extreely low-resolution rectangular iages are generated by shifting up along the horizontal, vertical and diagonal directions, i.e. 20 in each direction. Figure 4(b gives the exaples. Fig. 4(c shows the preliinary results of linear interpolation by axiu likelihood estiation. After noise reoval via the Canny operator based level set evolution, the final outputs are illustrated in Fig. 4(d. It can be seen that the en shapes are restored to a great extent copared with the easured iages. It is sufficient to further iage understanding. (a Ideal iages (b Extreely low-resolution rectangular iages (easured iages (c Preliinary restored iages by linear interpolation (d Final restored iages after noise reoval Figure 4: Results showing the proposed fraework works to iage restoration Proc. of SPIE Vol. 7544 754436-5 Downloaded Fro: http://proceedings.spiedigitallibrary.org/ on 12/23/2014 Ters of Use: http://spiedl.org/ters
4. COCLUSIOS In this paper, a unique iage restoration fraework is presented specifically for extreely low-resolution rectangular iages, which ensures further application of the rectangular poor-pixels detector. To eet the deand of fast processing, only a few easured iages shifted up to subpixel level along the horizontal, vertical and diagonal directions are needed to join the fusion operation. In order to still achieve a high-quality restored iage, the two-stage iage restoration strategy is then adopted. Based on axiu likelihood estiation, the preliinary restored iage is linearly interpolated. By the odified level set evolution with Canny operator for noise reoval, the final restored iage is iproved further. The experiental results have shown that the proposed fraework is effectively perfored in iage restoration. It is a sensible step towards subsequent iage understanding and object identification. REFERECES 1. P. Wen and X. Wang, Multiprocessor poor pixels iage acquisition syste with low power consuption, Chinese Journal of Scientific Instruent, vol. 27, no. 6, suppl., pp. 1358 1359, 2006. 2. S. Farsiu, M. D. Robinson, M. Elad and P. Milanfar, Fast and robust ultifrae super resolution, IEEE Trans. Iage Process., vol. 13, no. 10, pp. 1327 1344, 2004. 3. M. Elad and Y. Hel-Or, A fast super-resolution reconstruction algorith for pure translational otion and coon space-invariant blur, IEEE Trans. Iage Process., vol. 10, no. 8, pp. 1187 1193, 2001. 4. M. Elad and A. Feuer, Restoration of a single superresolution iage fro several blurred, noisy, and undersapled easured iages, IEEE Trans. Iage Process., vol. 6, no. 120, pp. 1646 1658, 1997. 5. L.. Trefethen and D. Bau, uerical Linear Algebra, Philadelphia, USA: SIAM, 1997. 6. J. A. Sethian, Level set ethods and fast arching ethods: evolving interfaces in coputational geoetry, fluid echanics, coputer vision, and aterials science, 2nd ed., Cabridge University Press, 1999. 7. S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: algoriths based on Hailton- Jacobi forulations, J Coput. Phys., vol. 79, no. 12, 1988. 8. D. Gil and P. Radeva, A regularized curvature flow designed for a selective shape restoration, IEEE Trans. Iage Process., vol. 13, no. 11, pp. 1444-1458, 2004. 9. C. Li, C.-Y. K, J. C. Gore and Z. Ding, Miniization of region-scalable fitting energy for iage segentation, IEEE Trans. Iage Process., vol. 17, no. 10, pp. 1940-1949, 2008. 10. J. Canny, A coputational approach to edge detection, IEEE Trans. Pattern Anal. Mach. Intell., vol. 8, no. 6, pp. 679-698, 1986. 11. C. Li, C. Xu, C. Gui and M. D. Fox, Level set evolution without re-initialization: a new variational forulation, in Proc IEEE Conf Coputer Vision and Pattern Recognition, 2005. Proc. of SPIE Vol. 7544 754436-6 Downloaded Fro: http://proceedings.spiedigitallibrary.org/ on 12/23/2014 Ters of Use: http://spiedl.org/ters