Gaussian Blur

The problem, from first principles

Blurring averages each pixel with its neighbours, which suppresses noise and fine detail. A box blur weights all neighbours equally, which leaves visible blocky artifacts. A Gaussian blur weights neighbours by a bell curve centred on the pixel, so near pixels count more than far ones, and the result looks smooth and natural. The width of the bell is the standard deviation sigma, and the kernel's size should cover about three standard deviations on each side. The kernel must sum to 1 so that blurring preserves overall brightness. Because the 2-D Gaussian is the product of two 1-D Gaussians, it is separable: blurring rows then columns with a 1-D kernel gives the same result as one 2-D convolution at a fraction of the cost.

From theory to code

Implement gaussian_kernel and gaussian_blur.

Constraints

• gaussian_kernel(size, sigma) returns the 1-D kernel of odd length size: w[k] = exp(-(k - r)**2 / (2 sigma**2)) for k = 0..size-1 with r = size // 2, normalized to sum to 1.
• gaussian_blur(img, size, sigma) blurs a 2-D image by applying the 1-D kernel along rows and then along columns. Pad by size // 2 pixels with edge replication (np.pad(mode='edge')) so the output has the same shape as the input.
• Use cross-correlation; the kernel is symmetric so flipping makes no difference.

Hints