Question about FPS measurement

[RuixiangXue]

Hi, thanks for sharing this great work!

I noticed that FPS is computed differently between your repo and the OctreeGS implementation.
In your repo, the FPS is measured as follows:
times = []
for _ in range(10):
__ = render(views[0], gaussians, pipeline, background, use_trained_exp=train_test_exp, separate_sh=separate_sh)
for idx, view in enumerate(tqdm(views, desc="Rendering progress")):
rendering_pack = render(view, gaussians, pipeline, background, use_trained_exp=train_test_exp, separate_sh=separate_sh)
rendering = rendering_pack["render"]
render_time = rendering_pack["render_time"]
fpss.append(1.0 / render_time)
print("fps=", torch.tensor(fpss).mean())

while in other implementations, it is sometimes measured like this:

t_list = []
for idx, view in enumerate(tqdm(views, desc="Rendering progress")):
torch.cuda.synchronize(); t0 = time.time()
rendering = render(view, gaussians, pipeline, background)["render"]
torch.cuda.synchronize(); t1 = time.time()
t_list.append(t1-t0)
t = np.array(t_list)
fps = 1.0 / t.mean()
print(f'Test FPS: {fps:.5f}')

Could you please clarify the main difference between these two FPS measurement methods?
In my experiments, the results differ quite a lot — the FPS computed by your method is significantly higher than latter.(using same 3DGS model, same rasterization from your work)

[zmliao]

If the rendering time of the $$i$$-th frame is $$t_i$$, the former FPS is the arithmetic mean of FPS per frame:

$$ \text{FPS}_{TCGS}=\frac{1}{n}\sum_{i=1}^n\frac{1}{t_i}. $$

and the latter FPS is the harmonic mean of FPS per frame:

$$ \text{FPS}_{other}=\frac{n}{\sum_{i=1}^n{t_i}}=\frac{n}{\sum_{i=1}^n\left(\frac{1}{t_i}\right)^{-1}}. $$

The inequality holds:

$$ \text{FPS}_{TCGS}=\frac{1}{n}\sum_{i=1}^n\frac{1}{t_i}\geq \frac{1}{\left(\prod_{i=1}^n{t_i}\right)^\frac{1}{n}}\geq\frac{1}{\frac{1}{n}\left(\sum_{i=1}^nt_i\right)}=\text{FPS}_{other}. $$

Therefore, the former FPS is higher.

In the paper TC-GS, all FPSs are measured in the former form.