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target="_blank" rel="nofollow" href="#ulink_5ae11b65-34da-534a-9f75-3b061c70ebd7">Figure 3.23, we have

(3.81)

      where the first matrix is denoted as A, and * is the convolution operator.

      The MATLAB command B = im2col(A, [2 2]) gives the B matrix, which is an expanded version of A:

(3.82)

      Note that the first column of B corresponds to the first 2 × 2 region in A, in a column‐first order, corresponding to (i^{l + 1}, j^{l + 1}) = (0, 0). Similarly, the second to last column in B correspond to regions in A with (i^{l + 1}, j^{l + 1}) being (1, 0), (0, 1), (1, 1), (0, 2) and (1, 2), respectively. That is, the MATLAB im2col function explicitly expands the required elements for performing each individual convolution to create a column in the matrix B. The transpose, B^T, is called the im2row expansion of A. If we vectorize the convolution kernel itself into a vector (in the same column‐first order) (1, 1, 1, 1)^T, we find that

(3.83)

If we reshape this resulting vector properly, we get the exact convolution result matrix in Eq. (3.81).

      If D^l > 1 (x^l has more than one channel, e.g., in Figure 3.24 of RGB image/three channels), the expansion operator could first expand the first channel of x^l, then the second, … , until all D^l channels are expanded. The expanded channels will be stacked together; that is, one row in the im2row expansion will have H × W × D^l elements, rather than H × W.

      Suppose x^l is a third‐order tensor in , with one element in x^l represented by (i^l, j^l, d^l), and f is a set of convolution kernels whose spatial extent are all H × W. Then, the expansion operator (im2row) converts x^l into a matrix φ(x^l) with elements indexed as (p, q). The expansion operator copies the element at (i^l, j^l, d^l) in x^l to the (p, q)‐th entry in φ(x^l). From the description of the expansion process, given a fixed (p, q), we can calculate its corresponding (i^l, j^l, d^l) triplet from the relation

(3.84)

      As an example, dividing q by HW and take the integer part of the quotient, we can determine which channel (d^l) belongs to.

We can use the standard vec operator to convert the set of convolution kernels f (an order‐4 tensor) into a matrix. Starting from one kernel, which can be vectorized into a vector in , all convolution kernels can be reshaped into a matrix with HWD^l rows and D columns with D^l+1 = D referred to as matrix F. With these notations, we have an expression to calculate convolution results (in Eq. (3.83), φ(x^l) is B^T):

      (3.85)

with , φ(x^l) , and . The matrix multiplication φ(x^l)F results in a matrix of size (H^{l + 1} W^{l + 1}) × D. The vectorization of this resultant matrix generates a vector in , which matches the dimensionality of vec(y).

      The Kronecker product: Given two matrices A ∈ ℝ^{m × n} and B ∈ ℝ^{p × q}, the Kronecker product A ⊗ B is a mp × nq matrix, defined as a block matrix

      (3.86)

      The Kronecker product has the following properties that will be useful for us:

(3.87)

(3.88) Скачать книгу