Federated learning (FL) aims at keeping client data local to preserve privacy. Instead of gathering the data itself, the server only collects aggregated gradient updates from clients. Following the popularity of FL, there has been considerable amount of work, revealing the vulnerability of FL approaches by reconstructing the input data from gradient updates. Yet, most existing works assume an FL setting with unrealistically small batch size, and have poor image quality when the batch size is large. Other works modify the neural network architectures or parameters to the point of being suspicious, and thus, can be detected by clients. Moreover, most of them can only reconstruct one sample input from a large batch. To address these limitations, we propose a novel and completely analytical approach, referred to as the maximum knowledge orthogonality reconstruction (MKOR), to reconstruct clients' input data. Our proposed method reconstructs a mathematically proven high quality image from large batches. MKOR only requires the server to send secretly modified parameters to clients and can efficiently and inconspicuously reconstruct the input images from clients' gradient updates. We evaluate MKOR's performance on the MNIST, CIFAR-100, and ImageNet dataset and compare it with the state-of-the-art works. The results show that MKOR outperforms the existing approaches, and draws attention to a pressing need for further research on the privacy protection of FL so that comprehensive defense approaches can be developed.