Convolution Neural Networks

$(f * g)(t) = \int_{-\infty}^{\infty} f(\tau)g(t - \tau)d\tau$

1. What does the above equation represent?
2. What does $g(t)$ represent?

1. A convolution operation is both linear and shift invariant.
2. A convolution operation is just like correlation, except that we flip over the filter before applying the correlation operator.
3. The convolution operation reaches a maximum, only in cases where the filter is mostly similar to a specific section of the input signal. Is he right in assuming so? Explain in detail the meaning of these statements.

1. The cross-correlation operator is used to find the location where two different signals are most similar.
2. The autocorrelation operator is used to find when a signal is similar to a delayed version of itself.

$f(x,y) * h(x,y) = \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} f(m,n)h(x - m, y - n)$ Using above, write the equivalent 2D correlation operation.

1. Define SAME
2. Define VALID

\begin{equation} A = \begin{bmatrix} 0 & 0 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & 0 \\ \end{bmatrix} \tag{8.4} \end{equation}

arr001 = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}

\begin{equation} f(x,y) * h(x,y) = \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} f(m,n)h(x - m, y - n) \tag{8.6} \end{equation} $1. Given the following discrete kernel in the X direction, what would be the equivalent Y direction?$ K = \frac{1}{2} \begin{bmatrix} -1 & 1 \\ -1 & 1 \end{bmatrix}

\begin{equation} G(x) = \sqrt\frac{1}{2\pi\sigma^2}e^{-\frac{x^2}{2\sigma^2}} \tag{8.8} \end{equation}

sim(u, v) = \frac{u.v}{|u||v|} = \frac{\sum_{i=1}^Nu_iv_i}{\sqrt{(\sum_{i=1}^Nu_i^2)(\sum_{i=1}^Nv_i^2)}}