A convolution slides a small grid of numbers over the image.
In the orange frame: the window times the kernel, then add up
- Pick a kernel: which edges does it light up?
- Click a pixel of the output: check the sum by hand.
- Change padding and stride: how big is the output?
Kernel
Image
Padding
Stride
Each pixel of the feature map: yi,j = Σa,b wa,b xi+a, j+b, with a and b from −1 to 1.
One convolution learns the stencil of the heat equation from data.
Loss and largest difference to the stencil (log scale)
- Train: watch the nine numbers approach the heat step.
- Switch to smooth fields: why does it crawl?
- Add noise: how close does it get?
Random fields
One heat step: unew = u + r (sum of the four neighbours − 4u). The stencil: 0.2 at the four neighbours, 1 − 4r = 0.2 in the middle.
Stacking and pooling let a pixel see far.
Layer by layer: what the output pixel depends on
- Point at the image: what does one output pixel see?
- Stack one, two, three convolutions: 3, 5, 7 pixels.
- Pool in between: how far does it see now?
3 × 3 convolutions, stacked
2 × 2 pooling
A small network reads the conductivity from the image; it knows shifts, not rotations.
- Pick a tile: compare the network with the solver.
- Shift it, then rotate it: which numbers change?
- Choose the network that saw only horizontal structures: rotate.
The network was trained on
30 test tiles, never used in training
Rows: long in x, round, long in y.
More options
Table: the 30 test tiles
| Tile | Share | Elongation | Solver kxx | Solver kyy | All images | Horizontal only | + rotated copies |
|---|
The networks' columns: their kxx and kyy for the tile as stored (lecture, Python).
How it is computed
Micrograph of step 1: Samson00, ferrite (light) and pearlite (dark) in a C45 steel with 0.45 % carbon, micrograph at 500× (2012, cropped), CC BY-SA 3.0, via Wikimedia Commons; here downsampled to 96 × 96 grey values, shared under the same licence. Microstructures, conductivities and the three networks: computed for the lecture (Python, PyTorch, CPU); the networks' numbers are stored in this page.