The two-dimensional cross-correlation operation can be implemented programmatically by iterating over the valid spatial positions of the output tensor. For an input tensor `X` and a kernel tensor `K`, the output dimensions are first initialized based on the formula $$(n_\textrm{h}-k_\textrm{h}+1) \times (n_\textrm{w}-k_\textrm{w}+1)$$. A nested loop then systematically slides the kernel across the input, performing an elementwise multiplication and summation for each local window to yield the final output tensor.

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To execute a two-dimensional cross-correlation, the convolution window starts at the upper-left of the input tensor and systematically translates across it, moving rightwards and downwards. At each valid position, the kernel tensor undergoes an elementwise multiplication with the currently overlaid input subtensor. The sum of these individual products produces a single scalar, which becomes the corresponding element in the resulting output tensor.

Two-Dimensional Convolution Operation Procedure

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In a convolutional layer, the convolution window (also known as the kernel window) represents the local spatial region of the input tensor that is currently being processed. The dimensions of this window are defined exactly by the spatial height and width of the kernel tensor.

Convolution Window

To illustrate a two-dimensional cross-correlation operation, consider an input tensor with a shape of $$3 \times 3$$ and a kernel tensor with a shape of $$2 \times 2$$. The convolution window begins at the upper-left corner of the input. The input subtensor within this window and the kernel are multiplied elementwise, and the products are summed to yield a single scalar value: $$0\times0+1\times1+3\times2+4\times3=19$$. As the window slides across the input tensor from left to right and top to bottom, this process is repeated. The resulting output tensor has a shape of $$2 \times 2$$ and contains the four derived scalar values: $$19$$, $$25$$, $$37$$, and $$43$$.

Example of Two-Dimensional Cross-Correlation

Two-Dimensional Cross-Correlation Code Implementation

When applying a two-dimensional cross-correlation without padding, the resulting output tensor is dimensionally smaller than the input. Since the kernel must fit completely within the boundaries of the input tensor, an input of size $$n_\textrm{h} \times n_\textrm{w}$$ convolved with a kernel of size $$k_\textrm{h} \times k_\textrm{w}$$ will yield an output shape calculated by the formula: $$(n_\textrm{h}-k_\textrm{h}+1) \times (n_\textrm{w}-k_\textrm{w}+1)$$.

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