Learning of a high-dimensional simplex from uniformly-sampled observations, generally known as the "unmixing problem", is a long-studied task in computer science. More recently, a significant interest is focused on this problem from other areas, such as computational biology and remote sensing. In this paper, we have studied the Probably Approximately Correct (PAC)-learnability of simplices in terms of sample complexity. Our analysis shows that a sufficient sample size for reliable learning of a @math -simplex is only @math , which yields a significant improvement over the current bound, i.e. @math . In particular, we propose a continuous-relaxation for Maximum Likelihood estimator which comes with a certificate of PAC-learnability, and is also followed by a scalable algorithm whose performance is extensively tested on synthetic and real-world datasets. Experimental results show that not only being comparable to rival methods on noiseless samples, our framework is superior to the state-of-the-art in noisy cases.
From a more applied perspective, learning of simplices is of practical interest in hyper-spectral imaging @cite_13 . Hyper-spectral cameras capture electromagnetic energy scatters from a scene in multiple wavelengths, where the spectrum of each pixel is a combination of patterns related to the reflection (or radiation) of its basic elements @cite_12 @cite_15 . Recently, this field has attracted intensive research interests which are mostly centered on analysis of minimum volume simplex, c.f. @cite_5 @cite_3 @cite_12 @cite_1 . Another tightly related application w.r.t. learning of simplices is the recognition of common patterns in cancerous tumors @cite_2 . Tumors usually have a high cell-type heterogeneity, and a collective knowledge from the characterstics of each cell-type is vital for an effective treatment. However, biological datasets are mostly in bulk format and therefore each sample is an aggregation of cells from all the cell-types. The idea of finding the smallest inclusive simplex for capturing the hidden profiles of cell-types is exploited in several recent researches in this area @cite_11 @cite_2 @cite_21 @cite_20 .
@cite_21 @cite_1 @cite_3 @cite_2 @cite_5 @cite_15 @cite_20 @cite_13 @cite_12 @cite_11
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Linear spectral unmixing aims at estimating the number of pure spectral substances, also called endmembers ,t heir spectral signatures, and their abundance fractions in remotely sensed hyperspectral images. This paper describes a method for unsupervised hyperspectral unmixing called minimum volume simplex analysis (MVSA) and introduces a new computationally efficient implementation. MVSA approaches hyperspectral un- mixing by fitting a minimum volume simplex to the hyperspec- tral data, constraining the abundance fractions to belong to the probability simplex. The resulting optimization problem, which is computationally complex, is solved in this paper by implementing a sequence of quadratically constrained subproblems using the interior point method, which is particularly effective from the computational viewpoint. The proposed implementation (available online: www.lx.it.pt 7ejun DemoMVSA.zip) is shown to exhibit state-of-the-art performance not only in terms of unmixing accu- racy, particularly in nonpure pixel scenarios, but also in terms of computational performance. Our experiments have been con- ducted using both synthetic and real data sets. An important assumption of MVSA is that pure pixels may not be present in the hyperspectral data, thus addressing a common situation in real scenarios which are often dominated by highly mixed pixels. In our experiments, we observe that MVSA yields competitive perfor- mance when compared with other available algorithms that work under the nonpure pixel regime. Our results also demonstrate that MVSA is well suited to problems involving a high number of endmembers (i.e., complex scenes) and also for problems involving a high number of pixels (i.e., large scenes). Introduction: In the framework of the Cognitive Microscope (MICO) project, we have set up a contest about mitosis detection in images of H and E stained slides of breast cancer for the conference ICPR 2012. Mitotic count is an important parameter for the prognosis of breast cancer. However, mitosis detection in digital histopathology is a challenging problem that needs a deeper study. Indeed, mitosis detection is difficult because mitosis are small objects with a large variety of shapes, and they can thus be easily confused with some other objects or artefacts present in the image. We added a further dimension to the contest by using two different slide scanners having different resolutions and producing red-green-blue (RGB) images, and a multi-spectral microscope producing images in 10 different spectral bands and 17 layers Z-stack. 17 teams participated in the study and the best team achieved a recall rate of 0.7 and precision of 0.89. Context: Several studies on automatic tools to process digitized slides have been reported focusing mainly on nuclei or tubule detection. Mitosis detection is a challenging problem that has not yet been addressed well in the literature. Aims: Mitotic count is an important parameter in breast cancer grading as it gives an evaluation of the aggressiveness of the tumor. However, consistency, reproducibility and agreement on mitotic count for the same slide can vary largely among pathologists. An automatic tool for this task may help for reaching a better consistency, and at the same time reducing the burden of this demanding task for the pathologists. Subjects and Methods: Professor Frederique Capron team of the pathology department at Pitie-Salpetriere Hospital in Paris, France, has selected a set of five slides of breast cancer. The slides are stained with H and E. They have been scanned by three different equipments: Aperio ScanScope XT slide scanner, Hamamatsu NanoZoomer 2.0‑HT slide scanner and 10 bands multispectral microscope. The data set is made up of 50 high power fields (HPF) coming from 5 different slides scanned at ×40 magnification. There are 10 HPFs slide. The pathologist has annotated all the mitotic cells manually. A HPF has a size of 512 µm × 512 µm (that is an area of 0.262 mm 2 , which is a surface equivalent to that of a microscope field diameter of 0.58 mm. These 50 HPFs contain a total of 326 mitotic cells on images of both scanners, and A number of recent attempts have been made to describe early sensory coding in terms of a general information processing strategy. In this paper, two strategies are contrasted. Both strategies take advantage of the redundancy in the environment to produce more effective representations. The first is described as a "compact" coding scheme. A compact code performs a transform that allows the input to be represented with a reduced number of vectors (cells) with minimal RMS error. This approach has recently become popular in the neural network literature and is related to a process called Principal Components Analysis (PCA). A number of recent papers have suggested that the optimal compact code for representing natural scenes will have units with receptive field profiles much like those found in the retina and primary visual cortex. However, in this paper, it is proposed that compact coding schemes are insufficient to account for the receptive field properties of cells in the mammalian visual pathway. In contrast, it is proposed that the visual system is near to optimal in representing natural scenes only if optimality is defined in terms of "sparse distributed" coding. In a sparse distributed code, all cells in the code have an equal response probability across the class of images but have a low response probability for any single image. In such a code, the dimensionality is not reduced. Rather, the redundancy of the input is transformed into the redundancy of the firing pattern of cells. It is proposed that the signature for a sparse code is found in the fourth moment of the response distribution (i.e., the kurtosis). In measurements with 55 calibrated natural scenes, the kurtosis was found to peak when the bandwidths of the visual code matched those of cells in the mammalian visual cortex. Codes resembling "wavelet transforms" are proposed to be effective because the response histograms of such codes are sparse (i.e., show high kurtosis) when presented with natural scenes. It is proposed that the structure of the image that allows sparse coding is found in the phase spectrum of the image. It is suggested that natural scenes, to a first approximation, can be considered as a sum of self-similar local functions (the inverse of a wavelet). Possible reasons for why sensory systems would evolve toward sparse coding are presented. For the instrument limitation and imperfect imaging optics, it is difficult to acquire high spatial resolution hyperspectral imagery. Low spatial resolution will result in a lot of mixed pixels and greatly degrade the detection and recognition performance, affect the related application in civil and military fields. As a powerful statistical image modeling technique, sparse representation can be utilized to analyze the hyperspectral image efficiently. Hyperspectral imagery is intrinsically sparse in spatial and spectral domains, and image super-resolution quality largely depends on whether the prior knowledge is utilized properly. In this article, we propose a novel hyperspectral imagery super-resolution method by utilizing the sparse representation and spectral mixing model. Based on the sparse representation model and hyperspectral image acquisition process model, small patches of hyperspectral observations from different wavelengths can be represented as weighted linear combinations of a small number of atoms in pre-trained dictionary. Then super-resolution is treated as a least squares problem with sparse constraints. To maintain the spectral consistency, we further introduce an adaptive regularization terms into the sparse representation framework by combining the linear spectrum mixing model. Extensive experiments validate that the proposed method achieves much better results.