Numeric Jungle: Navigating the Road to Reality. Notes from the Frontier

CV Lab 1 - Image Filters and Convolution

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Every pixel in a convolutional layer’s output is a weighted sum of a small neighbourhood of input pixels. The weights form the kernel — a tiny matrix that slides across the image, leaving a transformed copy in its wake. This lab makes that operation concrete: choose a kernel, watch the result appear, and read what each weight is actually asking the image to do.

What is a convolution?

Given an input image $I$ and a kernel $K$ of size $k \times k$, the convolution at position $(x, y)$ is

\[(I * K)(x, y) = \sum_{i=-\lfloor k/2 \rfloor}^{\lfloor k/2 \rfloor} \sum_{j=-\lfloor k/2 \rfloor}^{\lfloor k/2 \rfloor} I(x+i,\; y+j) \cdot K(i, j)\]

The result at each location depends only on a $k \times k$ neighbourhood — this is called local connectivity, and it is what makes CNNs dramatically more parameter-efficient than fully-connected layers applied to images.

The kernels

Kernel Effect
Identity Pass-through — output equals input.
Sharpen Amplify centre, subtract blurred neighbours.
Gaussian blur Weighted average biased toward the centre.
Sobel X/Y First-order derivative along each axis — detects edges perpendicular to that axis.
Edge detect Subtracts the average of all 8 neighbours — responds to any edge direction.
Emboss Directional derivative at 45° — creates a bas-relief illusion.
Laplacian Second-order derivative — responds to high curvature in any direction.

From hand-crafted to learned

Every kernel above was designed by a human. The revolutionary insight behind CNNs is that these weights can instead be learned from data by backpropagation. A conv layer with 16 filters learns 16 different kernels, each specialising in a different image feature. In CV Lab 2 you can watch exactly that happen on real handwritten-digit data.

Live demo

Things to notice

  • Identity output is pixel-perfect identical to the input.
  • Sobel X produces bright vertical stripes and dark horizontal ones — it measures the left-to-right gradient.
  • Edge detect blacks out smooth regions entirely, leaving only contours.
  • Gaussian softens the checkerboard patch more than the circles — there is more high-frequency energy to suppress.
  • Emboss adds a fixed 128 bias (the bias term in the formula above) so zero-response pixels appear mid-grey rather than black.

Key takeaways

  • Convolution is just a dot product between the kernel and each local patch of the input.
  • Positive kernel weights amplify; negative weights subtract; a mix creates a derivative.
  • Adding a bias shifts the output range — useful when you want signed responses to appear visible.
  • The same operation, run over many channels with learned weights, is what a CNN layer computes.