Due to advances in sensors, growing large and complex medical image data have the ability to visualize the pathological change in the cellular or even the molecular level or anatomical changes in tissues and organs. As a consequence, the medical images have the potential to enhance diagnosis of disease, prediction of clinical outcomes, characterization of disease progression, management of health care and development of treatments, but also pose great methodological and computational challenges for representation and selection of features in image cluster analysis. To address these challenges, we first extend one dimensional functional principal component analysis to the two dimensional functional principle component analyses (2DFPCA) to fully capture space variation of image signals. Image signals contain a large number of redundant and irrelevant features which provide no additional or no useful information for cluster analysis. Widely used methods for removing redundant and irrelevant features are sparse clustering algorithms using a lasso-type penalty to select the features. However, the accuracy of clustering using a lasso-type penalty depends on how to select penalty parameters and a threshold for selecting features. In practice, they are difficult to determine. Recently, randomized algorithms have received a great deal of attention in big data analysis. This paper presents a randomized algorithm for accurate feature selection in image cluster analysis. The proposed method is applied to ovarian and kidney cancer histology image data from the TCGA database. The results demonstrate that the randomized feature selection method coupled with functional principal component analysis substantially outperforms the current sparse clustering algorithms in image cluster analysis.