Quantization noise in the transform-based codec, e.g., JEPG and JEPG2000, has been extensively studied from various perspectives. However, in the lossy predictive codec (also called `∞ codec), it has rarely been addressed. In this work, we study the quantization noise in the lossy predictive codec from the following two aspects: coding performance and security. We first propose a sparsity-driven restoration technique to improve the coding performance of the `∞-decoded images. This is achieved by incorporating a `1 minimization term and `∞ constraints into a `2 optimization framework, where the initial image is estimated in an adaptive manner based on context modeling, and the weighting vectors balancing the relative contribution of each term are appropriately determined. Experimental results show that our proposed scheme significantly improves the `2 performance of the `∞-decoded images, while still preserving a tight error bound on every single pixel. When compared with the existing techniques, the PSNR gain can be up to 1 dB. In addition, we propose a novel anti-forensic framework to hide the compression footprints of `∞-decoded images. Due to the high sensitivity of prediction error sequence against disturbances, the task of erasing the evidence of being predictively compressed while still ensuring high quality of the processed image is more challenging than the counterparts for transform-based codecs. The anti-forensically modified image is obtained through adding dither over the prediction error domain, where the dither is generated based on piecewise Laplacian assumption. We further prove that the corresponding parameter selection can be cast as a convex optimization problem. To suppress the catastrophic effect of error propagation inherent to predictive coding, we employ a prediction-direction preserving strategy during the process of adding the dither. Experimental results demonstrate that our proposed method can effectively hide the fingerprints of lossy predictive compression, while still ensuring high quality of the resulting images.