Deep convolutional neural networks have proven to be well suited for image classification applications. However, if there is distortion in the image, the classification accuracy can be significantly degraded, even with state-of-the-art neural networks. The accuracy cannot be significantly improved by simply training with distorted images. Instead, this paper proposes a multiple neural network topology referred to as a selective deep convolutional neural network. By modifying existing state-of-the-art neural networks in the proposed manner, it is shown that a similar level of classification accuracy can be achieved, but at a significantly lower cost. The cost reduction is obtained primarily through the use of fewer weight parameters. Using fewer weights reduces the number of multiply-accumulate operations and also reduces the energy required for data accesses. Finally, it is shown that the effectiveness of the proposed selective deep convolutional neural network can be further improved by combining it with previously proposed network cost reduction methods.