For vertical edge detection, we usually use the following kernels:
- **Sobel Filter**: gives more weight to the pixels in the central rows, which may make it more robust.
- **Scharr Filter**

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To illustrate vertical edge detection, consider a $$6 \times 8$$ pixel image where the middle columns are black ($$0$$) and the outer columns are white ($$1$$). Applying a $$1 \times 2$$ kernel of $$[1, -1]$$ via cross-correlation outputs $$0$$ when horizontally adjacent pixels are the same. At the boundaries, it detects the pixel change, outputting $$1$$ for a white-to-black edge and $$-1$$ for a black-to-white edge. If this same kernel is applied to the transposed image—where the edges are now horizontal—the output vanishes, confirming that this specific kernel exclusively detects vertical edges.

Example of Vertical Edge Detection Using Convolution

Convolution Edge Detection in Positive vs. Negative OR Light to Dark vs. Dark to Light Edge Transitions

Vertical edge detection can be implemented programmatically by applying a two-dimensional cross-correlation operation to an image tensor. First, we construct a $$ 6 \times 8 $$ image tensor `X` where the middle columns are black ($$ 0 $$) and the outer columns are white ($$ 1 $$). Next, we define a $$ 1 \times 2 $$ kernel `K` with values $$ [1.0, -1.0] $$. We then perform the cross-correlation operation `corr2d(X, K)` to detect the vertical edges. Finally, applying the same kernel to the transposed image `X.t()` results in an output of zeros, verifying that the kernel only detects vertical edges.

```python
X = torch.ones((6, 8))
X[:, 2:6] = 0
K = torch.tensor([[1.0, -1.0]])
Y = corr2d(X, K)

# Apply the kernel to the transposed image (horizontal edges)
corr2d(X.t(), K)
```

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