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Image quality assessment (IQA) in medical imaging can be used to ensure that downstream clinical tasks can be reliably performed. Quantifying the impact of an image on the specific target tasks, also named as task amenability, is needed. A task-specific IQA has recently been proposed to learn an image-amenability-predicting controller simultaneously with a target task predictor. This allows for the trained IQA controller to measure the impact an image has on the target task performance, when this task is performed using the predictor, e.g. segmentation and classification neural networks in modern clinical applications. In this work, we propose an extension to this task-specific IQA approach, by adding a task-agnostic IQA based on auto-encoding as the target task. Analysing the intersection between low-quality images, deemed by both the task-specific and task-agnostic IQA, may help to differentiate the underpinning factors that caused the poor target task performance. For example, common imaging artefacts may not adversely affect the target task, which would lead to a low task-agnostic quality and a high task-specific quality, whilst individual cases considered clinically challenging, which can not be improved by better imaging equipment or protocols, is likely to result in a high task-agnostic quality but a low task-specific quality. We first describe a flexible reward shaping strategy which allows for the adjustment of weighting between task-agnostic and task-specific quality scoring. Furthermore, we evaluate the proposed algorithm using a clinically challenging target task of prostate tumour segmentation on multiparametric magnetic resonance …