Sheared Polygonal Texture Filtering

Guowei Lu, Jerry Jinfeng Guo, Petr Kellnhofer, Elmar Eisemann

Letter Of Changes:

We thank the reviewers for their valuable feedback. We reflected on it to improve our paper. This document details
how we have addressed individual issues raised in the reviews.

### Meta Review

- *The EWA and SPTF results in Figure 1 appear pixel-for-pixel
identical.*

We verified that the images presented in the paper are correct.
Therefore, we adjusted the **caption** to clarify that we consider
EWA is a high-quality but costly method. Hence, the hardly noticeable
differences are favorable for our computationally cheaper method.
Additionally, we increased the contrast in the **FLIP error maps**
to better guide the reader toward the differences.

- *Eq. 2 looks like an application of the Calculus and not like an
application of Green's theorem.*

Indeed, Eq. 2 does involve an application of the Calculus for the
texture function $M$ along each vertical line with respect to a
horizontal baseline ($\mathrm{d}u$). The function $M$ can be expressed as
$M(u,v) = \int_0^{v}{\mathcal{P} (u,v^\prime)\mathrm{d}v^\prime}$, where $(u,v)$ is the end point of this vertical line and $\mathcal{P}(u,v^\prime)$
is the texel value at $(u,v^\prime)$. Green's theorem relates this line
integral $M$ around a boundary $C$ to an area integral over the
region $Q$ bounded by $C$ as
$$\iint_Q{\mathop {\underbrace{\mathcal{P} (u,v)}} \limits_{\frac{\partial M}{\partial v}}\mathrm{d}u \mathrm{d}v} = -\oint_C M(u,v) \mathrm{d}u.$$

We added a clarification to justify our use of Green's Theorem when
deriving **Eq. 2** in Section 3.1.

- *The object in the video is commonly known as a 'Snow Globe'.*

We renamed the object in the **video** to a "Snow Globe".

### Official Review 1

- *The coordinate transformations in Section 3.3 and Figure.*

We clarify that **above Eq.4** in Section 3.3 and **Figure 4**, the
SSAT texture is one row larger than the original texture and we
added a new **Eq. 4** that describes the associated coordinate
transformation. Furthermore, we integrated this transformation into
**Eq. 7**.

- *Motivation for the additional vertical integration.*

We added a new **Figure 5** to illustrate how SSAT works for a slope
$\lambda > 1$ and why an additional SAT sample is needed to complete
the intended integral.

- *The step size equation in the text is confusing in Section 3.3.*

In Section 3.3 (**Step Size**), we corrected a typo in the step size
equation $s = 4/k$ and we adjusted the derived values in Section 4.1
(**Memory Analysis**) accordingly.

- *A typo in Fig. 6.*

We rectified the typo "EWI" to "EWA" in **Fig. 6**.

- *In Table 1, the relative memory overhead does not take into account
the additional pixels of the SSAT textures.*

In the caption of **Table 1**, we clarify that the additional row of
each SSAT table adds an asymptotically negligible overhead depending
on the texture resolution. Furthermore, **below Eq. 7** we comment
that the SAT does not require an additional computation nor memory
since it is already available as a special case of the SSAT for zero
slope. 123

- *Source code.*

We will consider releasing the source code at a later point.

### Official Review 2

- *Temporal stability*

We added a new **temporal analysis** to Section 4.1. We measure the
filtering error with respect to the ground truth during a smooth
camera motion and observe both a lower absolute error and a lower
variation when compared to ANISO16 which itself is widely used and
accepted as temporally stable for many applications including a new
**Figure 9(a)**. Furthermore, we added a preview of this experiment
to our supplemental **video**.

Keywords: Image sampling, signal reconstruction, texture filtering, rendering

TL;DR: A new efficient method for accurate real-time texture filtering.

Abstract: Efficient and precise texture filtering is essential in various applications. However, there is often a trade-off between coarse real-time approximations and accurate computationally expensive supersampling. We introduce a novel efficient texture-filtering method over arbitrary quadrilateral footprints, achieving high accuracy at a low computational cost. We achieve this by pre-computing integration tables that sparsely sample the space of possible footprints. Finally, we compare the qualitative and computational performance of our method to commonly used techniques and demonstrate various applications for high-quality real-time image synthesis, including normal filtering, soft shadow mapping, and glint rendering.

Submission Number: 10