## 1 Introduction  
Digital pathology underwent a significant transformation during the last years, to become a key tool in modern clinical practice and research. This was facilitated by the developments both in hardware, with cheaper and faster computers and storage, the availability of more powerful graphics processing units (GPUs) with larger memory, faster slide scanners, etc., and computational methods, with deep learning approaches being the most prominent example. Having faster and more performant pathology slide scanners, enables almost mass-production of digital slide archives and puts high pressure on the computational infrastructure. For example, a single whole slide image (WSI) of a tissue sample scanned at 20\(\times\) magnification easily contains 10\({}^{10}\) multi-channel pixels, describing highly complex multi-scale structures. Clearly, there is a need for highly performant image analysis methods able to detect, extract, measure and summarize these structures. For a detailed review of the current challenges and trends in digital pathology, see [1, 2]. Immunohistochemistry (IHC) is a method for visualizing at light microscopy level the spatial distribution of targets (antigens) of interest by exploiting the specificity of antibody binding with its antigen. Routine IHC is limited in the number of antibodies (colours) that can be used in a single tissue section meaning that for studying the (co-)distribution of several targets one needs several consecutive sections. More recently, a number of protocols (e.g., stain multiplexing) allow the use of several antibodies for visualizing multiple targets at the same time. Here, we are interested in multi-stain experiments that employ the routine IHC approach which rely on analysing consecutive sections stained with different antibodies. The central question is how one could jointly investigate the resulting images in order to identify correlations or co-occurrences (or lack thereof) of various types of cells. In order to correlate the observations across different slides/images, they need either to be registered together such that at least the major structures overlap, or similar features need to be extracted from all images. In fact, registration usually requires itself extraction of such features for the identification of landmark points [3]. One way of extracting cross-stain features would require a first step in which the colour/stain information is separated from the texture/structure information. Such stain deconvolution can be performed, for example, by matrix operations if the stain mixing matrix is known (usually derived experimentally). The results of [PERSON] and [PERSON] [4] represent a canonical application of this principle. However, in some cases the estimation is problematic since the stain may not even obey the Beer-Lambert law (e.g., the Diaminobenzidine (DAB)staining [5]). Alternatively, the stain mixing matrix may be estimated directly from image (e.g., [6; 7]). The extraction of relevant features from the images is a key step of any image analysis method. In the case of WSIs this is even more challenging, not only due to the large image size and the need of integrating multiple resolutions in the features, but also because the very nature of these features is, in many cases, difficult to define. The traditional approaches were based on standard image descriptors (like Gabor filters, local binary patterns, textural descriptors, etc.) and tried to match the pathologists' expertise. They were generally hand-crafted and relied on expert tuning for better performance. See [8] for a discussion and [9] for a comparison of such features in the case of segmenting colorectal cancer tissue sections, as an example. With the advent of deep learning approaches, the features extracted by the convolutional neural networks (CNNs) gained more traction and proved their utility in identifying key image characteristics for various classification tasks.