Means no padding.
$$n \times n * f \times f = (n - f + 1) \times (n - f + 1)$$

Valid Convolution in DNN

Means padding so that output size is the same as the input size.
$$(n + 2p) \times (n + 2p) * f \times f = (n + 2p - f + 1) \times (n + 2p - f + 1)$$
In order to make the output size the same as the input size:
$$n + 2p - f + 1 = n \Rightarrow p = \frac{f - 1}{2}$$

Same Convolution in DNN

University of Michigan - Ann Arbor

Since kernels generally have width and height greater than 1, after convolving an image by the kernel, the image size will be reduced. By adding padding to the input image, we can have the output with the same size as input.

If we start with a 240×240-pixel image, 10 layers of 5×5 convolutions reduce the image to 200×200 pixels, slicing off 30% of the image and with it obliterating any interesting information on the boundaries of the original image. Padding is the most popular tool for handling this issue.

In the below example, we pad a 3×3 input, increasing its size to 5×5. The corresponding output then increases to a 4×4 matrix. The shaded portions are the first output element as well as the input and kernel tensor elements used for the output computation:  0×0+0×1+0×2+0×3=0.

Padding Convolution

In general, if we add a total of  $p_h$  rows of padding (roughly half on top and half on bottom) and a total of  $p_w$ columns of padding (roughly half on the left and half on the right), the output shape will be:
$(n_h−k_h+ 2p_h+1)×(n_w−k_w+2p_w+1)$

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