dhruvdh / convolutions

Performance Modeling: 2D Convolution

Assignment 4: Performance Modeling: 2D Convolution

Modelling

• How many Flops needs to be done to compute a convolution of dimension k on a image of size n × m?

Assuming the edge pixels are ignored, the dimensions of the output image would be (n - k + 1) x (m - k + 1). Meaning that we'd have to calculate the values for (n - k + 1) * (m - k + 1) pixels; which implies that we'd need to perform (n - k + 1) * (m - k + 1) convolutions.

The amount of flops needed for one convolution is k * k multiplications and k * k - 1 additions, coming to (2 * k * k) - 1 flops.

Thusly, the total amount of floating point operations needed are ((2 * k * k) - 1) * (n - k + 1) * (m - k + 1).

• How much memory needs to be moved to compute a convolution of dimension k on a image of size n × m?

As calculated above, we need to perform (n - k + 1) * (m - k + 1) convolutions. For each, convolution we'll need to move k * k floating point numbers for the kernel, and k * k pixels from the image.

Total memory moved would be (2 * k * k) * (n - k + 1) * (m - k + 1) * 32 bits.

• Assuming the performance numbers you measured in assignment 1 and 2, how long should computing a convolution of dimension 3 on an image of 1024 × 768 take?

According to what we found in question 1 it would take 13,308,484 flops or 0.013308484 GFlops to get to the final output image. According to my previous estimations a compute node on mamba should have a peak performance of 1638.4 GFlops/sec.

It should thusly take 0.013308484 ÷ 1638.4 seconds which comes to 8.12285400390625e-6 seconds or 8.12285 microseconds.

• What about a convolution of dimension 11 ? On this kind of problem, performance is usually reported in pixel processed per second.

For a 1024 x 768 image, a kernel of dimension 11 would take 185,235,492‬‬ flops. Which would take 1.130587719726563e-4‬ seconds.

Since in all we processed and produced 1014 x 758 output pixels,we produced 6,798,340,248.962656‬ pixels per second, or 6.7983 gigapixels/sec.

• Plot maximum expected performance as a function of k.

Y-axis is pixels processed in gigapixels per second, and X-axis is the kernel size.