Many real-world applications in image processing and computer vision require splitting an input image into a cartoon component and a texture component. We propose a nonconvex variational image decomposition model for simultaneously recovering cartoon and texture images. To induce the sparsity of gradient norms of the cartoon image more strongly than the classical total variation regularization, we applied the nonconvex firm penalty function as a regularizer for the cartoon image. The nonconvex firm penalty regularizer function has a better ability to separate the piecewise constant component with neat edges. The G-norm was used as an oscillating prior for the texture image. Converting the proposed optimization model to a constrained problem by variable splitting, we addressed it with the alternating direction method of multipliers. Experimental results and comparisons were given to verify the superiority of existing state-of-the-art methods in terms of correlation, peak signal-to-noise ratio, structural similarity, and visual quality. Finally, we demonstrated the effectiveness of the proposed model by several applications such as image abstraction and pencil sketching, artifact removal, image denoising, image composition, and detail enhancement.