This paper presents an algorithm for rotating a subimage in place without using any extra working array. Due to this constraint, we have to overwrite pixel values by interpolated values. Key ideas are local reliability test which determines whether interpolation at a pixel is carried out correctly without using interpolated values, and lazy interpolation which stores interpolated values in a region which is never used for output images and then fills in interpolated values after safety is guaranteed. It is shown that linear interpolation is always safely implemented. An extension to cubic interpolation is also discussed.

The maximum a posteriori (MAP) algorithm is the optimum solution for decoding concatenated codes, such as turbo codes. Since the MAP algorithm is computationally complex, more efficient algorithms, such as the Max-Log-MAP algorithm and the soft-output Viterbi algorithm (SOVA), can be used as suboptimum solutions. Especially, the Max-Log-MAP algorithm is widely used, due to its near-optimum performance and lower complexity compared with the MAP algorithm. In this paper, we propose an efficient algorithm for decoding concatenated codes by modifying the Max-Log-MAP algorithm. The efficient implementation of the backward recursion and the log-likelihood ratio (LLR) update in the proposed algorithm improves its computational efficiency. Memory is utilized more efficiently if the sliding window algorithm is adopted. Computer simulations and analysis show that the proposed algorithm requires a considerably lower number of computations compared with the Max-Log-MAP algorithm, while providing the same overall performance.