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The unprecedented growth of computer vision applications, such as surveillance systems and social media, raises security and visual privacy concerns, especially when data is stored on cloud servers. Image obfuscation offers a way to preserve visual privacy while maintaining an adequate level of usability; thus, it has been a topic of great interest in recent years. However, prior obfuscation schemes are either vulnerable to malicious attacks, such as model inversion to reconstruct original images from obfuscated images, or generate non-trainable obfuscated images, making them unusable for achieving reasonable accuracy. This paper proposes a novel bit-plane-based image obfuscation scheme, {\em Bit-ViP}, to preserve visual privacy for image-based recognition tasks. The Bit-ViP scheme produces secure, usable images by incorporating an innovative end-to-end obfuscation function. While doing so, the obfuscated image would contain non-invertible noise (generated by Lorenz's chaotic system and differential privacy), making it hard for an adversary to reconstruct the original image. We conduct extensive experiments on two popular activity recognition datasets, namely UCF101 and HMDB51, to validate the effectiveness of Bit-ViP. In the face of attacks on reconstruction, pixel frequency, information entropy, and pixel inter-correlation, we present a rigorous security analysis demonstrating tangible improvements over existing schemes.