In this paper, we propose an algorithm that enhances the number of pixels for high-speed imaging. High-speed cameras have a principle problem that the number of pixels reduces when the number of frames per second (fps) increases. To enhance the number of pixels, we suppose an optical structure that block-randomly selects some percent of pixels in an image. Then, we need to reconstruct the entire image. For this, a state-of-the-art method takes three-dimensional reconstruction strategy, which requires a heavy computational cost in terms of time. To reduce the cost, the proposed method reconstructs the entire image frame-by-frame using a new cost function exploiting two types of sparsity. One is within each frame and the other is induced from the similarity between adjacent frames. The latter further means not only in the image domain, but also in a sparsifying transformed domain. Since the cost function we define is convex, we can find the optimal solution using a convex optimization technique with small computational cost. We conducted simulations using grayscale image sequences. The results show that the proposed method produces a sequence, mostly the same quality as the state-of-the-art method, with dramatically less computational time.