The performance and the efficiency of recent computing platforms have been deeply influenced by the widespread adoption of hardware accelerators, such as Graphics Processing Units (GPUs) or Field Programmable Gate Arrays (FPGAs), which are often employed to support the tasks of General Purpose Processors (GPP). One of the main advantages of these accelerators over their sequential counterparts (GPPs) is their ability of performing massive parallel computation. However, in order to exploit this competitive edge, it is necessary to extract the parallelism from the target algorithm to be executed, which is in general a very challenging task. This concept is demonstrated, for instance, by the poor performance achieved on relevant multimedia algorithms, such as Chambolle, which is a well-known algorithm employed for the optical flow estimation. The implementations of this algorithm that can be found in the state of the art are generally based on GPUs, but barely improve the performance that can be obtained with a powerful GPP. In this paper, we propose a novel approach to extract the parallelism from computation-intensive multimedia algorithms, which includes an analysis of their dependency schema and an assessment of their data reuse. We then perform a thorough analysis of the Chambolle algorithm, providing a formal proof of its inner data dependencies and locality properties. Then, we exploit the considerations drawn from this analysis by proposing an architectural template that takes advantage of the fine-grained parallelism of FPGA devices. Moreover, since the proposed template can be instantiated with different parameters, we also propose a design metric, the expansion rate, to help the designer in the estimation of the efficiency and performance of the different instances, making it possible to select the right one before the implementation phase. We finally show, by means of experimental results, how the proposed analysis and parallelization approach leads to the design of efficient and high performance FPGA-based implementations that are orders of magnitude faster than the state-of-the-art ones.