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We present highly efficient algorithms for performing forward and backwardpropagation of Convolutional Neural Network (CNN) for pixelwise classificationon images. For pixelwise classification tasks, such as image segmentation andobject detection, surrounding image patches are fed into CNN for predicting theclasses of centered pixels via forward propagation and for updating CNNparameters via backward propagation. However, forward and backward propagationwas originally designed for whole-image classification. Directly applying it topixelwise classification in a patch-by-patch scanning manner is extremelyinefficient, because surrounding patches of pixels have large overlaps, whichlead to a lot of redundant computation.  The proposed algorithms eliminate all the redundant computation inconvolution and pooling on images by introducing novel d-regularly sparsekernels. It generates exactly the same results as those by patch-by-patchscanning. Convolution and pooling operations with such kernels are able tocontinuously access memory and can run efficiently on GPUs. A fraction ofpatches of interest can be chosen from each training image for backwardpropagation by applying a mask to the error map at the last CNN layer. Itscomputation complexity is constant with respect to the number of patchessampled from the image. Experiments have shown that our proposed algorithmsspeed up commonly used patch-by-patch scanning over 1500 times in both forwardand backward propagation. The speedup increases with the sizes of images andpatches.

Highly Efficient Forward and Backward Propagation of Convolutional Neural Networks for Pixelwise Classification