Counting immunopositive cells on biological tissues generally requires either manual annotation or (when available) automatic rough systems, for scanning signal surface and intensity in whole slide imaging. In this work, we tackle the problem of counting microglial cells in lumbar spinal cord cross-sections of rats by omitting cell detection and focusing only on the counting task. Manual cell counting is, however, a time-consuming task and additionally entails extensive personnel training. The classic automatic color-based methods roughly inform about the total labeled area and intensity (protein quantification) but do not specifically provide information on cell number. Since the images to be analyzed have a high resolution but a huge amount of pixels contain just noise or artifacts, we first perform a pre-processing generating several filtered images {(providing a tailored, efficient feature extraction)}. Then, we design an automatic kernel counter that is a non-parametric and non-linear method. The proposed scheme can be easily trained in small datasets since, in its basic version, it relies only on one hyper-parameter. However, being non-parametric and non-linear, the proposed algorithm is flexible enough to express all the information contained in rich and heterogeneous datasets as well (providing the maximum overfit if required). Furthermore, the proposed kernel counter also provides uncertainty estimation of the given prediction, and can directly tackle the case of receiving several expert opinions over the same image. Different numerical experiments with artificial and real datasets show very promising results. Related Matlab code is also provided.

翻译：生物组织上免疫阳性细胞的计数通常需要手动标注，或在可用时采用自动粗略系统来扫描整个切片成像中的信号表面和强度。在本研究中，我们通过省略细胞检测并仅聚焦于计数任务，来解决大鼠腰椎脊髓横切面中微胶质细胞的计数问题。然而，手动细胞计数是一项耗时的工作，并且还需要大量的人员培训。经典的基于颜色的自动方法粗略地提供了总标记面积和强度（蛋白质定量）的信息，但并未专门提供细胞数量的信息。由于待分析的图像具有高分辨率，但大量像素仅包含噪声或伪影，我们首先执行预处理，生成多张滤波图像（提供定制化、高效的特征提取）。随后，我们设计了一种自动核计数器，这是一种非参数且非线性的方法。所提出的方案可以在小型数据集中轻松训练，因为其基本版本仅依赖于一个超参数。然而，由于是非参数和非线性的，所提出的算法也足够灵活，能够表达丰富且异构数据集中包含的所有信息（在需要时可提供最大程度的过拟合）。此外，所提出的核计数器还能提供给定预测的不确定性估计，并可直接处理接收同一图像多个专家意见的情况。使用人工和真实数据集进行的多项数值实验均显示出非常有前景的结果。相关Matlab代码也已提供。