Anti-aliased Convolutional Neural Networks (CNNs) have been proposed to overcome the shift variant nature of the CNNs. The fundamental building block of the anti-aliased CNN has been the application of Gaussian or wavelet-based smoothing before the pooling operation. However, in all these approaches, the feature maps' edges are also smoothed while suppressing high-frequency components. In this work, two novel pooling approaches are presented, namely the Laplacian-Gaussian Concatenation with Attention (LGCA) pooling and Wavelet-based Approximate-Detailed Coefficient concatenation with Attention (WADCA) that can preserve the edges in the feature maps. The results suggest that the proposed pooling approaches outperform conventional as well as blur pooling for classification, segmentation and auto-encoders. In terms of average binary classification accuracy (cats vs dogs), the proposed LGCA approach outperforms the conventional pooling and blur pooling by 4\% and 2\%, 3\% and 4\%, 3\% and 0.5\% for MobileNetv2, DenseNet121 and ResNet50 respectively. On the other hand, the proposed WADCA approach outperforms the normal pooling and blur pooling by 5\% and 3\%, 2\% and 3\%, 2\% and 0.17\% for MobileNetv2, DenseNet121 and ResNet50 respectively. It is also observed from the results that edge-preserving pooling does not have any significance in segmentation tasks possibly due to high to low-resolution translation. Meanwhile, high-resolution reconstruction has been observed for the LGCA pooling in the case of convolutional auto-encoders.