Why Does Convolution with a Custom Gaussian Kernel Produce Different Noise Patterns than a Standard Gaussian Blur?

Below is an explanation of what’s happening and why the two outputs look so different.

What the Code Does

1. Generating a Kernel:
   The code creates a 2D “Gaussian‐like” kernel via

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   correlation_scale = 150
   x = np.arange(-correlation_scale, correlation_scale)
   y = np.arange(-correlation_scale, correlation_scale)
   X, Y = np.meshgrid(x, y)
   dist = np.sqrt(X**2 + Y**2)
   filter_kernel = np.exp(-dist**2/(2*correlation_scale))

   Note: In a standard Gaussian, the exponent is written as
   exp(–r²/(2σ²)). Here, the denominator is 2·correlation_scale instead of 2·(correlation_scale²).
   As a result, the kernel’s decay is controlled differently than with a typical σ value—in effect, the “width” of the kernel is much broader than if you had set σ = 1. This kernel is used to “spread out” or mix nearby noise values.

2. Generating Noise:
   A grid of random noise is created with

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   n = 512
   noise = np.random.randn(n, n)

3. Convolving the Noise with the Kernel:
   The code then convolves the random noise with this kernel

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   noise1 = scipy.signal.fftconvolve(noise, filter_kernel, mode='same')

   This convolution indeed is a Gaussian blur, but the nature of the blur is determined by the kernel. Because of the kernel’s construction (specifically its size and “effective σ”), the convolution results in noise whose spatial features vary on a much larger scale than typical “local” blur.

Comparing with scipy.ndimage.gaussian_filter

When you use:

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from scipy.ndimage import gaussian_filter
noise_filtered = gaussian_filter(noise, sigma=1, radius=10)

you are applying a Gaussian filter with a small standard deviation (σ = 1). This only slightly smooths the noise over a very local neighborhood. The resulting image still has lots of high frequency detail, as it isn’t averaged over a wide area.

Why the Images Look Different

• Effective Scale Difference:
  The convolution in step 3 uses a kernel that is constructed with a much larger spatial scale (controlled by correlation_scale = 150 and the way it’s used in the exponent). In effect, even though both operations are forms of Gaussian blur, the first one “spreads” or correlates the noise over a much larger area. This creates smooth, large-scale variations (or patterns) in the output image—what you see as slowly varying patches.

• Blur Strength:
  The gaussian_filter example with sigma=1 barely blurs the noise; it only slightly mixes adjacent pixels. This yields an image that still appears speckled (almost white noise). In contrast, the convolution with a large kernel “averages” over a much larger region, resulting in a distinctly different appearance; you get a noise field with spatially varying intensities that are correlated over large distances.

In Summary

Both approaches are applying a Gaussian convolution. However, the convolution via fftconvolve uses a deliberately broad kernel (with parameters that do not match the typical σ used in gaussian_filter) to produce noise with large-scale but spatially varying patterns, while the gaussian_filter example with sigma=1 produces a much less blurred (and hence different-looking) noise image.

Hopefully, this clears up the confusion between the two methods and why their outputs appear so different.