Compressive Sensing And Its Applications In Digital Image Forensics

Traditional method of signal sampling obeys the Shannon/Nyquist sampling theorem which specifies that one must sample at least two times faster than the signal bandwidth when capturing a signal. However, in recent years, a method called compressive sensing breaks through the restriction of this theorem. We can use the measurement matrix to integrate the sampling and compression steps. Then the signal can be reconstructed using fewer samples via some optimization algorithms. Compressive sensing is based on the sparseness of signals and the theory of sparse representation has also achieved significant progress in the past decade. The method of sparse representation has developed from using orthogonal transform such as Fourier transform, DCT and wavelet transform to redundant dictionary. Thus we can present the signals over the redundant dictionary adaptively and sparsely.Compressive sensing theory changes the method of accessing the data and aiming to acquire more information from less data. Consequently, variety of tasks have got remarkable achievements including image compression, image processing, information security, computer vision, pattern recognition, face recognition, communication and so on. Some of these works such as image denoising and face recognition both have state-of-the-art performance. In this paper, we apply the compressive sensing and sparse representation to signal processing and image forensics. The main research results are summarized as following:1) Secret image sharing based on compressive sensing. We use the sparse of the image and the limitation of existing algorithm to transform image sharing problem into compressive signal recovery problem. This scheme is simple and safe.2) Image compression and reconstruction based on sparse representation and partial differential equation. This algorithm firstly extracts the corner points to get the sparse representation in the spatial domain. Then we can compress the image by using the measurement matrix. Conversely, we can reconstruct the corner points via optimization algorithms and recover the original image using diffusion equations based on PDE.3) An image fragile watermarking algorithm based on compressive sensing. The core idea of this algorithm is that the sparse recovery technology in compressive sensing can recover signal from the noised signal measurement, so we can regard the sparse signal of the image DCT frequency domain as noise and the watermarking as original signal, then we can measure the watermarking and add it to the frequency domain, but the watermark can’t be detected correctly if the image has been attacked, thus we design the fragile watermarking successfully.4) Dual-channel noise reduction via sparse representations. Firstly, we use the overlapping patches sampled from two channels together to train the dictionary. Then reconstruction algorithm is applied to obtain the sparse coefficients of patches using the dictionary. After that, we can get the denoising speech by the updated coefficients. Experiment’s results show that this algorithm can perform better under the Gaussian noise.