Helical gearboxes play a critical role in power transmission of industrial applications. They are vulnerable to various faults due to long-term and heavy-duty operating conditions. To improve the safety and reliability of helical gearboxes, it is necessary to monitor their health conditions and diagnose various types of faults. The conventional measurements for gearbox fault diagnosis mainly include lubricant analysis, vibration, airborne acoustics, thermal images, electrical signals, etc. However, a single domain measurement may lead to unreliable fault diagnosis and the contact installation of transducers is not always accessible, especially in harsh and dangerous environments. In this article, a Compressive Sensing (CS)-based Dual-Channel Convolutional Neural Network (CNN) method was proposed to accurately and intelligently diagnose common gearbox faults based on two complementary non-contact measurements (thermal images and acoustic signals) from a mobile phone. The raw acoustic signals were analysed by the Modulation Signal Bispectrum (MSB) to highlight the coupled modulation components relating to gear faults and suppress the irrelevant components and random noise, which generates a series of two-dimensional matrices as sparse MSB magnitude images. Then, CS was used to reduce the image redundancy but retain key information owing to the high sparsity of thermal images and acoustic MSB images, which significantly accelerates the CNN training speed. The experimental results convincingly demonstrate that the proposed CS-based Dual-Channel CNN method significantly improves the diagnostic accuracy (99.39% on average) of industrial helical gearbox faults compared to the single-channel ones.