We hide grayscale secret images into a grayscale cover image, which is considered to be a challenging steganography problem. Our goal is to develop a steganography scheme with enhanced embedding capacity while preserving the visual quality of the stegoimage as well as the extracted secret image, and ensuring that the stego-image is resistant to steganographic attacks. The novel embedding rule of our scheme helps to hide secret image sparse coefficients into the oversampled cover image sparse coefficients in a staggered manner. The stego-image is constructed by using the Alternating Direction Method of Multipliers (ADMM) to solve the Least Absolute Shrinkage and Selection Operator (LASSO) formulation of the underlying minimization problem. Finally, the secret images are extracted from the constructed stego-image using the reverse of our embedding rule. Using these components together, to achieve the above mentioned competing goals, forms our most novel contribution. We term our scheme SABMIS (Sparse Approximation Blind Multi-Image Steganography). We perform extensive experiments on several standard images. By choosing the size of the length and the width of the secret images to be half of the length and the width of cover image, respectively, we obtain embedding capacities of 2 bpp (bits per pixel), 4 bpp, 6 bpp, and 8 bpp while embedding one, two, three, and four secret images, respectively. Our focus is on hiding multiple secret images. For the case of hiding two and three secret images, our embedding capacities are higher than all the embedding capacities obtained in the literature until now (3 times and 6 times than the existing best, respectively). For the case of hiding four secret images, although our capacity is slightly lower than one work (about 2/3rd), we do better on the other two goals (quality of stego-image & extracted secret image as well as resistance to steganographic attacks). For our experiments, there is very little deterioration in the quality of the stego-images as compared to their corresponding cover images. Like all other competing works, this is supported visually as well as over 30 dB of Peak Signal-to-Noise Ratio (PSNR) values. The good quality of the stego-images is further validated by multiple numerical measures. None of the existing works perform this exhaustive validation. When using SABMIS, the quality of the extracted secret images is almost same as that of the corresponding original secret images. This aspect is also not demonstrated in all competing literature. SABMIS further improves the security of the inherently steganographic attack resistant transform based schemes. Thus, it is one of the most secure schemes among the existing ones. Additionally, we demonstrate that SABMIS executes in few minutes, and show its application on the real-life problems of securely transmitting medical images over the internet
