While bilinear interpolation may be applied in various applications, an image processing example is used here. In this case, function values correspond to pixel values in the range [0, 255]. Single precision, floating-point numerical format is assumed for interpolated pixel values and interpolation coordinates $(x_q,y_q)$.
A reference image is used to generate a lookup table which provides input to the AI Engine. An input image with resolution $x_{res} \times y_{res}$ is assumed to have pixels defined on a grid with unit spacing. The $x$ and $y$ pixel coordinates may be combined using the equation $I = x \times y_{res} + y$ to derive a LUT index $I$, as shown in Figure 4.
Figure 4 - Image as a Lookup Table
For any query point, $(x_q,y_q)$, the floating point coordinates can be separated into integer and fractional parts, where $x_q = x_{int}.x_{frac}$ and $y_q = y_{int}.y_{frac}$. The integer parts are used to extract pixel values used in the interpolation equation. The four values required for interpolation are obtained from the LUT using the following relations:
$$ \begin{aligned} &f(x_1,y_1) = LUT(x_{int} * y_{res} + y_{int}) \ &f(x_1,y_2) = LUT(x_{int} * y_{res} + y_{int} + 1) \ &f(x_2,y_1) = LUT((x_{int} + 1) * y_{res} + y_{int}) \ &f(x_2,y_2) = LUT((x_{int} + 1) * y_{res} + y_{int} + 1). \end{aligned} $$
An example of LUT indexing is shown in Figure 4 using the pixels marked with X. Once the four pixel values required for interpolation are obtained, the integer parts of the coordinates $(x_q,y_q)$ are no longer needed and can be assumed to be zero. This simplifies the interpolation equation to
$$ f(x_q,y_q) = \begin{bmatrix} 1-x_{frac} & x_{frac} \end{bmatrix} \begin{bmatrix} f(x_1,y_1) & f(x_1,y_2) \ f(x_2,y_1) & f(x_2,y_2) \end{bmatrix} \begin{bmatrix} 1-y_{frac} \ y_{frac} \end{bmatrix} $$
or expressed as an inner product
$$ f(x_q,y_q) = \begin{bmatrix} (1-x_{frac})(1-y_{frac}) & (1-x_{frac})y_{frac} & x_{frac}(1-y_{frac}) & x_{frac}y_{frac} \end{bmatrix} \begin{bmatrix} f(x_1,y_1) \ f(x_1,y_2) \ f(x_2,y_1) \ f(x_2,y_2) \end{bmatrix}. $$