Title: Measuring Image Complexity as a Discrete Hierarchy using MDL Clustering. Abstract: Being able to quantify the complexity of data is an important question in machine learning, computer science, and data science. In the case of image data, a number of methods have been proposed. However, existing methods are based only on the degree of variation across the image, and cannot distinguish meaningful content from noise. In particular, existing methods assign a very high complexity to white noise images, despite such images containing no meaningful information. In this paper, we present a method to measure the complexity of images by analyzing them has a discrete hierarchy of patches, using MDL clustering. Beginning with individual pixels, each level of the hierarchy is formed using the cluster labels from the level below. The complexity is the sum, across all levels, of the entropy of cluster labels inside all patches on that level. Clustering is performed using the minimum description length principle (MDL), which we leverage in a novel way to distinguish signal from noise. We test against existing methods on seven different sets of images, four from public image datasets and three synthetic, and show that ours is the only method that can assign an accurate measure of complexity to all images considered. Every other method measures white noise as highly complex, while our method gives it zero complexity. We then present ablation studies showing the contribution of the components of our method, and further experiments showing robustness to image quality.