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Image smoothing is a fundamental task in signal processing. For such task, box filter is well-known. However, box filter can not keep some features of the signal, such as edges, corners and the jump in the step function. In this paper, we present a one-sided box filter that can smooth the signal but keep the discontinuous features in the signal. More specifically, we perform box filter on eight one-sided windows, leading to a one-sided box filter that can preserve corners and edges. Our filter inherits the constant $O(1)$ computational complexity of the original box filter with respect to the window size and also the linear $O(N)$ computational complexity with respect to the total number of samples. We performance several experiments to show the efficiency and effectiveness of this filter. We further compare our filter with other the-state-of-the-art edge preserving methods. Our filter can be deployed in a large range of applications where the classical box filter is adopted.

Weighted Gaussian Curvature is an important measurement for images. However, its conventional computation scheme has low performance, low accuracy and requires that the input image must be second order differentiable. To tackle these three issues, we propose a novel discrete computation scheme for the weighted Gaussian curvature. Our scheme does not require the second order differentiability. Moreover, our scheme is more accurate, has smaller support region and computationally more efficient than the conventional schemes. Therefore, our scheme holds promise for a large range of applications where the weighted Gaussian curvature is needed, for example, image smoothing, cartoon texture decomposition, optical flow estimation, etc.

One of the methods for stratifying different molecular classes of breast cancer is the Nottingham Prognostic Index Plus (NPI+) which uses breast cancer relevant biomarkers to stain tumour tissues prepared on tissue microarray (TMA). To determine the molecular class of the tumour, pathologists will have to manually mark the nuclei activity biomarkers through a microscope and use a semi-quantitative assessment method to assign a histochemical score (H-Score) to each TMA core. Manually marking positively stained nuclei is a time consuming, imprecise and subjective process which will lead to inter-observer and intra-observer discrepancies. In this paper, we present an end-to-end deep learning system which directly predicts the H-Score automatically. Our system imitates the pathologists' decision process and uses one fully convolutional network (FCN) to extract all nuclei region (tumour and non-tumour), a second FCN to extract tumour nuclei region, and a multi-column convolutional neural network which takes the outputs of the first two FCNs and the stain intensity description image as input and acts as the high-level decision making mechanism to directly output the H-Score of the input TMA image. To the best of our knowledge, this is the first end-to-end system that takes a TMA image as input and directly outputs a clinical score. We will present experimental results which demonstrate that the H-Scores predicted by our model have very high and statistically significant correlation with experienced pathologists' scores and that the H-Score discrepancy between our algorithm and the pathologists is on par with the inter-subject discrepancy between the pathologists.