In neural style transfer, the learned synthesized image often accumulates high-frequency noise, manifesting as particularly bright or dark pixels. To mitigate this, a noise reduction method known as total variation denoising is applied. Denoting $x_{i, j}$ as the pixel value at coordinate $(i, j)$, the total variation loss is defined as:

$\sum_{i, j} \left|x_{i, j} - x_{i+1, j}\right| + \left|x_{i, j} - x_{i, j+1}\right|$

Reducing this total variation loss encourages the values of neighboring pixels on the synthesized image to become closer, thereby promoting a smoother and visually cleaner final output.

# PyTorch
def tv_loss(Y_hat):
    return 0.5 * (torch.abs(Y_hat[:, :, 1:, :] - Y_hat[:, :, :-1, :]).mean() +
                  torch.abs(Y_hat[:, :, :, 1:] - Y_hat[:, :, :, :-1]).mean())