Non-overlapping patch-wise convolution is the default image tokenizer for all state-of-the-art vision Transformer (ViT) models. Even though many ViT variants have been proposed to improve its efficiency and accuracy, little research on improving the image tokenizer itself has been reported in the literature. In this paper, we propose a new image tokenizer based on wavelet transformation. We show that ViT models with the new tokenizer achieve both higher training throughput and better top-1 precision for the ImageNet validation set. We present a theoretical analysis on why the proposed tokenizer improves the training throughput without any change to ViT model architecture. Our analysis suggests that the new tokenizer can effectively handle high-resolution images and is naturally resistant to adversarial attack. Furthermore, the proposed image tokenizer offers a fresh perspective on important new research directions for ViT-based model design, such as image tokens on a non-uniform grid for image understanding.