图像稀疏编码算法及应用研究
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摘要
图像处理技术已成为人类生活与生产实践不可或缺的重要信息获取手段，已广泛应用于空间技术、医学影像、遥感图像处理、工业控制、文化艺术、计算机视觉、视频与多媒体系统、科学可视化、电子商务等众多领域。生物视觉系统是一种高效的图像处理系统。随着脑科学研究的不断发展，人类对生物视觉系统的研究逐步深入，从初级视皮层到高级视觉区域，从初级视觉感知到高级知觉组织等，都取得了许多重要的研究成果。
从视网膜接收到光刺激作为输入信号开始，视觉系统利用一套完整的信息处理机制对输入图像进行处理和加工。尽管我们尚未透彻了解生物视觉系统的工作机制，但视觉信息处理过程的“稀疏编码”特性已证实了其合理性及有效性。广义的视觉图像稀疏编码特性是一种符合生物进化过程能量最节约原则的视觉信息表达方式。图像处理的稀疏编码算法及应用正是基于这一生物学背景而发展起来的一种高效信息处理技术。已有的研究成果表明：有效的视觉信息稀疏编码系统一般具有多分辨、临界采样及过完备性；用于表示图像的基函数具有局部性、带通性、方向性、各向异性等特点。基于这些规律，本文以图像稀疏编码算法的应用为重点，研究了图像信息的基本稀疏编码模型、基于优化Gabor字典的图像稀疏编码算法、基于过完备稀疏表示的图像处理及应用、基于多分辨分析理论的图像稀疏表示及应用，以及视觉皮层脉冲耦合神经网络模型及应用。所取得的研究成果如下：
1.提出了一种基于优化Gabor字典的图像稀疏编码算法。图像稀疏表示的关键问题之一是如何构造有效的过完备字典。二维Gabor函数具有良好的局部性、方向选择性及空间频率选择性等特点，可有效模拟视觉皮层V1区简单细胞的感受野特性。以Gabor函数为原子生成函数构造的过完备字典能匹配图像中的边缘、纹理等几何机构，可实现图像的有效表示。但该字典仍存在原子数量巨大，匹配追踪算法计算开销大等问题。针对这些不足，提出了一种基于优化Gabor字典的图像稀疏编码算法。新算法主要采用两种策略对上述问题做一改进：其一是采用图像重叠分块的策略，以有效减少输入样本的长度；其二是采用粒子群优化算法（Particle Swarm Optimization，PSO），模拟视皮层神经元的竞争机制，以输入样本在Gabor原子上投影的模值为适用度值，以优化求解最匹配Gabor原子的自由度参数代替在大规模字典上的搜索过程，最后在优化所得的Gabor字典上采用正交匹配追踪（Orthogonal MatchingPursuit，OMP）算法完成图像的稀疏分解。实验结果验证了所提算法的有效性，算法在较低时间复杂度的前提下可获得较高的重建图像质量。
2.基于图像的过完备稀疏表示理论，针对传统变换域方法的适应性及噪声鲁棒性问题，提出了一种自适应字典学习的多传感器图像融合算法。算法首先从待融合图像中随机取块构成训练样本集，经自适应字典学习算法迭代运算获取过完备字典；然后由OMP算法完成图像块的稀疏分解；再按分解系数的显著性选择融合系数并完成图像块的重构；重构块经重新排列并取平均后获得最后的融合图像。实验结果可见，新算法具有较好的噪声抑制能力，融合图像的主观质量及客观评价指标均要好于传统算法。
3.基于图像的过完备稀疏表示理论，针对实例学习图像超分辨方法中低分辨图像块与高分辨图像块特征映射不一致问题，提出一种在线字典学习的图像超分辨重建算法。在学习阶段，算法首先获取一组高分辨图像，并经降质获得对应的低分辨图像，以该两组实例图像构建相应的高分辨及低分辨特征训练集，在低分辨训练集上经在线字典学习算法迭代运算获得低分辨字典，然后采用OMP算法获得输入样本在低分辨字典上的稀疏编码矩阵，通过共享稀疏编码系数求解高分辨字典；在超分辨重建阶段，对输入的低分辨图像块首先在低分辨字典上采用OMP算法实现稀疏编码，同样依据高分辨图像块与低分辨图像块共享稀疏编码系数的原则，以高分辨字典实现待估计高分辨图像块的稀疏逼近，最后经块重新排序并取平均实现高分辨图像的重建。实验结果可见，所提方法可取得优于传统方法的图像超分辨质量，重建图像的细节及纹理保持能力较好，且能有效抑制图像边缘的伪影现象。
4.基于图像稀疏表示的多尺度几何分析（Multiscale Geometric Analysis，MGA）理论，提出了一种改进的非下采样Contourlet变换（Nonsubsampled ContourletTransform，NSCT）结合高斯比例混合模型（Gaussian Scale Mixtures Model，GSM）的图像去噪算法。MGA是近年来发展起来的一种新的高维函数多尺度多分辨分析方法，可实现较小波变换更优的图像“稀疏表示”能力。NSCT是一种有效的多尺度几何分析工具，具有多尺度、多方向性及平移不变性等特点。NSCT结合GSM的图像去噪算法能获得较满意的去噪效果，算法具有一定的通用性，但算法耗时较大。针对这一问题，提出一种改进的快速NSCT变换，以拓展其在算法时间要求较高场合下的应用。由于方向滤波器组主要影响NSCT的性能，故采用一种具有提升结构并经优化处理的方向滤波器改进了NSCT变换中的非下采样方向滤波器组，同时将改进后的NSCT结合GSM应用于图像去噪。实验结果可见，改进算法保持了原NSCT结合GSM算法的图像去噪效果，同时将算法速度提高了近11倍。
5.基于图像稀疏表示的多分辨分析理论，模拟视觉系统同步振荡机制及视皮层神经元分层分级信息处理机制，提出了一种基于改进拉普拉斯金字塔（Laplacian Pyramid，LP）变换结合脉冲耦合神经网络（Pulse Coupled NeuralNetworks，PCNN）的抗噪声多聚焦图像融合算法。算法采用改进的LP变换构造图像的多分辨数据结构，分解系数按照分层多尺度的方式激励PCNN，经迭代运算生成其对应的神经元点火映射图，并以此为依据经判决算子完成数据融合，再采用一种新的伪逆重构算法（又称对偶框架算子）生成融合图像。实验结果证实新算法具有抗加性噪声及JPEG压缩系统噪声的能力，融合图像的客观评价指标较高，其中PSNR比原LP方法平均提升19.4%，且能有效减弱原LP算法融合图像边缘的伪影现象。
Image processing technology has been become one of very important methods forinformation acquisition in human society, and has been widely used in space technology,medical imaging, remote sensing image processing, industrial control, computer vision,video and multimedia systems, etc. Thus, the study on high efficient image processingsystem has deep significance. Biological visual system has been proved to be veryefficient for image processing. Accompanied by the deep study on brain science, manyresearch results on biological visual system are reported, from the primary propertyextraction to high level perceptual organization. Currently, the perception mechanismand related core technologies of visual system have become a very important researchfield of cognitive science.
From the start input stimulated by the scene light on retina, visual system uses acomplete set of information processing mechanism to deal with the input image.Although, we have not understood the whole mechanism of visual system, the sparserepresentation ways for input features are accepted by many researches and proved bymany neurophysiology experiments. An efficient sparse coding method must be withmultiscale, critical sampling, and over-complete properties, and the basis function mustbe with local, directional, and anisotropic properties. Base on these principles, thispaper studied the basic image sparse coding model, multiscale and multiresolutiontheory for image processing, over-complete representation methods for imageprocessing, and visual cortex neuron model (the pulse coupled neural networks). Themain research work and contributions are summarized as follows:
1. An image sparse representation algorithm over optimal Gabor dictionary isproposed. For image sparse representation, one of the key factors is to constructan efficient over-complete dictionary.2-D Gabor function has perfect local,directional, and space-frequency selective properties; this is very similar to thereceptive field properties of simple cells in primary visual cortex. The dictionaryelements generated by Gabor function can well match the geometric structure(e.g., edge, texture) in natural image, and can achieve well sparse representationfor image. However, because of the over-complete demands, the elements of thisdictionary are very bulky, and lead to an intensive computational efficiency inorthogonal pursuit matching procedure. To overcome this problem, two mainstrategies are adopted:(1) divide the original image into overlapped patches to reduce the length of input sample;(2) adopt Particle Swarm Optimization (PSO)algorithm to imitate the competition activity of neurons in visual cortex. In theoptimization procedure of PSO, the norm of the projection of input signal on thedictionary element is used as fitness value, to optimize a set of optimal Gaborparameters is used to replace the pursuit procedure on the whole dictionary. At thefinal stage of the algorithm, the Orthogonal Matching Pursuit (OMP) algorithm isused for image coding. Experimental results demonstrate that our algorithm canachieve high reconstructed image quality under a low algorithm complexity.
2. The traditional algorithm can not adapt to noise environments in transformdomain. To alleviate this problem and improve the adaptability and the noiserobustness of them, a novel multi-sensor image fusion algorithm based on imagesparse representation theory is introduced. For the new algorithm, the relatedimage is partitioned into image patches, and the patches are decomposed by theOMP algorithm, then the sparse coefficients are selected demanding theirprominent property. The selected coefficients are used for the reconstructed imagepatches, and the patches are realigned according to the partition order. Finally, theoverlapped patches are averaged to get the fused image. Experimental resultsshow that the proposed algorithm is with anti-noise property, and outperforms thetypical algorithms in term of objective criteria and visual appearance.
3. These methods based on sample learning are more efficient for imagesuper-resolution (SR). However, there are always one-to-many mappings betweenlow-resolution (LR) image patches and high-resolution (HR) image patches. Toovercome or at least reduce the problem induced by the disagreement of LR andHR patches, an algorithm based on image sparse representation and onlinedictionary leaning method is proposed for single image SR problem. Thealgorithm is processed by two stages:1) Training stage. A set of HR images areused as HR samples. They are blurred, subsampled and then upsampled toconstructed the LR samples which are with same size with the HR samples, thenthese two sample sets are filtered to get the feature sample set. The LR featuresample set is used to obtain LR dictionary and the sparse code by onlinedictionary learning algorithm and OMP algorithm. Because LR samples and HRsamples can share the sparse code, the HR dictionary can be calculated;2)Reconstruction stage. When a LR image is inputted, it is also filtered to constructthe feature sample set, and then be processed by the OMP algorithm to get itssparse code. These codes are shared by the HR dictionary to estimate the HR patches. Finally, all the HR patches are arranged according their partitioning order,and the overlapped patches are averaged to reconstruct the HR image. Theexperimental results show that the proposed algorithm outperforms in bothobjective and subjective evaluation compared with several conventional methods.Moreover, the proposed method performs better for detail and texture recover, andalso can reduce the artifacts around edge in the reconstructed HR image.
4. A fast image denoising algorithm based on a modified Nonsubsampled ContourletTransform (NSCT) combined with Gaussian Scale Mixtures Model (GSM) isproposed. The Multiscale Geometric Analysis (MGA) theory can achieve optimalsparse representation for high dimension function and can lead to highperformance over wavelets. As one of the effective tools of MGA, the NSCT iswith fully shift-invariant, multiscale, and multidirection properties. For imagedenoising, the algorithm using NSCT combined with GSM seems to be one of theexcellent options. However, the computational efficiency of it is lower. Aiming atthis problem, a modified NSCT is proposed to be applicable to wide applicationsin speed demanding environments. Since the Nonsubsampled Directional FilterBank (NSDFB) mainly affects the performance of NSCT, an optimizedDirectional Filter Bank (DFB) with lifting scheme is adopted to modify theNSDFB for a fast NSCT. Moreover, combined with GSM, the modified NSCT isused for image denoising. The numerical experiments show that the processingspeed of the proposed algorithm is increased by11times than that of theconventional one, while keeping good visual quality of the denoised image.
5. In view of the applied peculiarity of ordinary optics, based on the new LaplacianPyramid (LP) transform and the Pulse Coupled Neural Networks (PCNN), a novelmultifocus image fusion algorithm which is with antinoise properties is proposed.For the algorithm, the new LP transform is adopted to construct the pyramidal dataof the related images; the data is then inputted into PCNN. Through the iterativeoperation by the PCNN, the firing map of neurons can be obtained, and then thesemaps are used by the decision operator for data fusion. The fused pyramidal dataare reconstructed by the new pseudo inverse (also named dual frame operator) toobtain the fused image. The experimental results show that new algorithmoutperforms the traditional algorithm, both in visual appearance and objectiveevaluation criteria. The PSNR value increases19.4percent than that of the old LPalgorithm. Moreover, the proposed algorithm is with antinoise properties foradditive noise and system noise from JPEG compression, and can efficiently reduce the “pseudo Gibbs” phenomenon which is inevitable in the fused image bythe old LP algorithm.
从视网膜接收到光刺激作为输入信号开始，视觉系统利用一套完整的信息处理机制对输入图像进行处理和加工。尽管我们尚未透彻了解生物视觉系统的工作机制，但视觉信息处理过程的“稀疏编码”特性已证实了其合理性及有效性。广义的视觉图像稀疏编码特性是一种符合生物进化过程能量最节约原则的视觉信息表达方式。图像处理的稀疏编码算法及应用正是基于这一生物学背景而发展起来的一种高效信息处理技术。已有的研究成果表明：有效的视觉信息稀疏编码系统一般具有多分辨、临界采样及过完备性；用于表示图像的基函数具有局部性、带通性、方向性、各向异性等特点。基于这些规律，本文以图像稀疏编码算法的应用为重点，研究了图像信息的基本稀疏编码模型、基于优化Gabor字典的图像稀疏编码算法、基于过完备稀疏表示的图像处理及应用、基于多分辨分析理论的图像稀疏表示及应用，以及视觉皮层脉冲耦合神经网络模型及应用。所取得的研究成果如下：
1.提出了一种基于优化Gabor字典的图像稀疏编码算法。图像稀疏表示的关键问题之一是如何构造有效的过完备字典。二维Gabor函数具有良好的局部性、方向选择性及空间频率选择性等特点，可有效模拟视觉皮层V1区简单细胞的感受野特性。以Gabor函数为原子生成函数构造的过完备字典能匹配图像中的边缘、纹理等几何机构，可实现图像的有效表示。但该字典仍存在原子数量巨大，匹配追踪算法计算开销大等问题。针对这些不足，提出了一种基于优化Gabor字典的图像稀疏编码算法。新算法主要采用两种策略对上述问题做一改进：其一是采用图像重叠分块的策略，以有效减少输入样本的长度；其二是采用粒子群优化算法（Particle Swarm Optimization，PSO），模拟视皮层神经元的竞争机制，以输入样本在Gabor原子上投影的模值为适用度值，以优化求解最匹配Gabor原子的自由度参数代替在大规模字典上的搜索过程，最后在优化所得的Gabor字典上采用正交匹配追踪（Orthogonal MatchingPursuit，OMP）算法完成图像的稀疏分解。实验结果验证了所提算法的有效性，算法在较低时间复杂度的前提下可获得较高的重建图像质量。
2.基于图像的过完备稀疏表示理论，针对传统变换域方法的适应性及噪声鲁棒性问题，提出了一种自适应字典学习的多传感器图像融合算法。算法首先从待融合图像中随机取块构成训练样本集，经自适应字典学习算法迭代运算获取过完备字典；然后由OMP算法完成图像块的稀疏分解；再按分解系数的显著性选择融合系数并完成图像块的重构；重构块经重新排列并取平均后获得最后的融合图像。实验结果可见，新算法具有较好的噪声抑制能力，融合图像的主观质量及客观评价指标均要好于传统算法。
3.基于图像的过完备稀疏表示理论，针对实例学习图像超分辨方法中低分辨图像块与高分辨图像块特征映射不一致问题，提出一种在线字典学习的图像超分辨重建算法。在学习阶段，算法首先获取一组高分辨图像，并经降质获得对应的低分辨图像，以该两组实例图像构建相应的高分辨及低分辨特征训练集，在低分辨训练集上经在线字典学习算法迭代运算获得低分辨字典，然后采用OMP算法获得输入样本在低分辨字典上的稀疏编码矩阵，通过共享稀疏编码系数求解高分辨字典；在超分辨重建阶段，对输入的低分辨图像块首先在低分辨字典上采用OMP算法实现稀疏编码，同样依据高分辨图像块与低分辨图像块共享稀疏编码系数的原则，以高分辨字典实现待估计高分辨图像块的稀疏逼近，最后经块重新排序并取平均实现高分辨图像的重建。实验结果可见，所提方法可取得优于传统方法的图像超分辨质量，重建图像的细节及纹理保持能力较好，且能有效抑制图像边缘的伪影现象。
4.基于图像稀疏表示的多尺度几何分析（Multiscale Geometric Analysis，MGA）理论，提出了一种改进的非下采样Contourlet变换（Nonsubsampled ContourletTransform，NSCT）结合高斯比例混合模型（Gaussian Scale Mixtures Model，GSM）的图像去噪算法。MGA是近年来发展起来的一种新的高维函数多尺度多分辨分析方法，可实现较小波变换更优的图像“稀疏表示”能力。NSCT是一种有效的多尺度几何分析工具，具有多尺度、多方向性及平移不变性等特点。NSCT结合GSM的图像去噪算法能获得较满意的去噪效果，算法具有一定的通用性，但算法耗时较大。针对这一问题，提出一种改进的快速NSCT变换，以拓展其在算法时间要求较高场合下的应用。由于方向滤波器组主要影响NSCT的性能，故采用一种具有提升结构并经优化处理的方向滤波器改进了NSCT变换中的非下采样方向滤波器组，同时将改进后的NSCT结合GSM应用于图像去噪。实验结果可见，改进算法保持了原NSCT结合GSM算法的图像去噪效果，同时将算法速度提高了近11倍。
5.基于图像稀疏表示的多分辨分析理论，模拟视觉系统同步振荡机制及视皮层神经元分层分级信息处理机制，提出了一种基于改进拉普拉斯金字塔（Laplacian Pyramid，LP）变换结合脉冲耦合神经网络（Pulse Coupled NeuralNetworks，PCNN）的抗噪声多聚焦图像融合算法。算法采用改进的LP变换构造图像的多分辨数据结构，分解系数按照分层多尺度的方式激励PCNN，经迭代运算生成其对应的神经元点火映射图，并以此为依据经判决算子完成数据融合，再采用一种新的伪逆重构算法（又称对偶框架算子）生成融合图像。实验结果证实新算法具有抗加性噪声及JPEG压缩系统噪声的能力，融合图像的客观评价指标较高，其中PSNR比原LP方法平均提升19.4%，且能有效减弱原LP算法融合图像边缘的伪影现象。
Image processing technology has been become one of very important methods forinformation acquisition in human society, and has been widely used in space technology,medical imaging, remote sensing image processing, industrial control, computer vision,video and multimedia systems, etc. Thus, the study on high efficient image processingsystem has deep significance. Biological visual system has been proved to be veryefficient for image processing. Accompanied by the deep study on brain science, manyresearch results on biological visual system are reported, from the primary propertyextraction to high level perceptual organization. Currently, the perception mechanismand related core technologies of visual system have become a very important researchfield of cognitive science.
From the start input stimulated by the scene light on retina, visual system uses acomplete set of information processing mechanism to deal with the input image.Although, we have not understood the whole mechanism of visual system, the sparserepresentation ways for input features are accepted by many researches and proved bymany neurophysiology experiments. An efficient sparse coding method must be withmultiscale, critical sampling, and over-complete properties, and the basis function mustbe with local, directional, and anisotropic properties. Base on these principles, thispaper studied the basic image sparse coding model, multiscale and multiresolutiontheory for image processing, over-complete representation methods for imageprocessing, and visual cortex neuron model (the pulse coupled neural networks). Themain research work and contributions are summarized as follows:
1. An image sparse representation algorithm over optimal Gabor dictionary isproposed. For image sparse representation, one of the key factors is to constructan efficient over-complete dictionary.2-D Gabor function has perfect local,directional, and space-frequency selective properties; this is very similar to thereceptive field properties of simple cells in primary visual cortex. The dictionaryelements generated by Gabor function can well match the geometric structure(e.g., edge, texture) in natural image, and can achieve well sparse representationfor image. However, because of the over-complete demands, the elements of thisdictionary are very bulky, and lead to an intensive computational efficiency inorthogonal pursuit matching procedure. To overcome this problem, two mainstrategies are adopted:(1) divide the original image into overlapped patches to reduce the length of input sample;(2) adopt Particle Swarm Optimization (PSO)algorithm to imitate the competition activity of neurons in visual cortex. In theoptimization procedure of PSO, the norm of the projection of input signal on thedictionary element is used as fitness value, to optimize a set of optimal Gaborparameters is used to replace the pursuit procedure on the whole dictionary. At thefinal stage of the algorithm, the Orthogonal Matching Pursuit (OMP) algorithm isused for image coding. Experimental results demonstrate that our algorithm canachieve high reconstructed image quality under a low algorithm complexity.
2. The traditional algorithm can not adapt to noise environments in transformdomain. To alleviate this problem and improve the adaptability and the noiserobustness of them, a novel multi-sensor image fusion algorithm based on imagesparse representation theory is introduced. For the new algorithm, the relatedimage is partitioned into image patches, and the patches are decomposed by theOMP algorithm, then the sparse coefficients are selected demanding theirprominent property. The selected coefficients are used for the reconstructed imagepatches, and the patches are realigned according to the partition order. Finally, theoverlapped patches are averaged to get the fused image. Experimental resultsshow that the proposed algorithm is with anti-noise property, and outperforms thetypical algorithms in term of objective criteria and visual appearance.
3. These methods based on sample learning are more efficient for imagesuper-resolution (SR). However, there are always one-to-many mappings betweenlow-resolution (LR) image patches and high-resolution (HR) image patches. Toovercome or at least reduce the problem induced by the disagreement of LR andHR patches, an algorithm based on image sparse representation and onlinedictionary leaning method is proposed for single image SR problem. Thealgorithm is processed by two stages:1) Training stage. A set of HR images areused as HR samples. They are blurred, subsampled and then upsampled toconstructed the LR samples which are with same size with the HR samples, thenthese two sample sets are filtered to get the feature sample set. The LR featuresample set is used to obtain LR dictionary and the sparse code by onlinedictionary learning algorithm and OMP algorithm. Because LR samples and HRsamples can share the sparse code, the HR dictionary can be calculated;2)Reconstruction stage. When a LR image is inputted, it is also filtered to constructthe feature sample set, and then be processed by the OMP algorithm to get itssparse code. These codes are shared by the HR dictionary to estimate the HR patches. Finally, all the HR patches are arranged according their partitioning order,and the overlapped patches are averaged to reconstruct the HR image. Theexperimental results show that the proposed algorithm outperforms in bothobjective and subjective evaluation compared with several conventional methods.Moreover, the proposed method performs better for detail and texture recover, andalso can reduce the artifacts around edge in the reconstructed HR image.
4. A fast image denoising algorithm based on a modified Nonsubsampled ContourletTransform (NSCT) combined with Gaussian Scale Mixtures Model (GSM) isproposed. The Multiscale Geometric Analysis (MGA) theory can achieve optimalsparse representation for high dimension function and can lead to highperformance over wavelets. As one of the effective tools of MGA, the NSCT iswith fully shift-invariant, multiscale, and multidirection properties. For imagedenoising, the algorithm using NSCT combined with GSM seems to be one of theexcellent options. However, the computational efficiency of it is lower. Aiming atthis problem, a modified NSCT is proposed to be applicable to wide applicationsin speed demanding environments. Since the Nonsubsampled Directional FilterBank (NSDFB) mainly affects the performance of NSCT, an optimizedDirectional Filter Bank (DFB) with lifting scheme is adopted to modify theNSDFB for a fast NSCT. Moreover, combined with GSM, the modified NSCT isused for image denoising. The numerical experiments show that the processingspeed of the proposed algorithm is increased by11times than that of theconventional one, while keeping good visual quality of the denoised image.
5. In view of the applied peculiarity of ordinary optics, based on the new LaplacianPyramid (LP) transform and the Pulse Coupled Neural Networks (PCNN), a novelmultifocus image fusion algorithm which is with antinoise properties is proposed.For the algorithm, the new LP transform is adopted to construct the pyramidal dataof the related images; the data is then inputted into PCNN. Through the iterativeoperation by the PCNN, the firing map of neurons can be obtained, and then thesemaps are used by the decision operator for data fusion. The fused pyramidal dataare reconstructed by the new pseudo inverse (also named dual frame operator) toobtain the fused image. The experimental results show that new algorithmoutperforms the traditional algorithm, both in visual appearance and objectiveevaluation criteria. The PSNR value increases19.4percent than that of the old LPalgorithm. Moreover, the proposed algorithm is with antinoise properties foradditive noise and system noise from JPEG compression, and can efficiently reduce the “pseudo Gibbs” phenomenon which is inevitable in the fused image bythe old LP algorithm.
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