INFOMAGR – Advanced Graphics Jacco Bikker - November 2019 - February 2020 Lecture 15 - “Bits & Pieces” Welcome! 𝑱 𝒚, 𝒚 ′ = 𝒉(𝒚, 𝒚 ′ ) 𝝑 𝒚, 𝒚 ′ + න 𝝇 𝒚, 𝒚 ′ , 𝒚 ′′ 𝑱 𝒚 ′ , 𝒚 ′′ 𝒆𝒚′′ 𝑻
Today’s Agenda: ▪ On Offsetting Shadows and Reflections ▪ Consistent Normal Interpolation ▪ Packing Normals ▪ Spatial Splits in BVHs ▪ Research Directions
Advanced Graphics – Bits & Pieces 3 Offsets You have been told… …To offset your shadow rays by ‘epsilon times R or L’. ▪ How should be chose epsilon? ▪ Is epsilon the same everywhere? ▪ Is this always sufficient? 𝜕 𝑗 𝑂 𝜕 𝑝 p
Advanced Graphics – Bits & Pieces 4 Offsets You have been told… Geek solution: use fixed point numbers. This time: epsilon = 1 . 𝜕 𝑗 𝑂 𝜕 𝑝 p
Advanced Graphics – Bits & Pieces 5 Offsets SafeOrigin When R (or L) is almost parallel to the surface, the epsilon offset fails. Alternative: 𝑃 += cos 𝜄 3 𝑓 𝑆 + 1 − cos 𝜄 3 𝑓 𝑂 𝜕 𝑗 𝑂 𝜕 𝑝 p
Advanced Graphics – Bits & Pieces 6 Offsets SafeOrigin When R (or L) is almost parallel to the surface, the epsilon offset fails. Alternative: 𝑃 += cos 𝜄 3 ∙ 𝑆 + 1 − cos 𝜄 3 ∙ 𝑂 𝑂 𝜕 𝑝 𝜕 𝑗 p
Advanced Graphics – Bits & Pieces 7 Offsets Second Opinion Ray Tracing Gems, chapter 6: A Fast and Robust Method for Avoiding Self-Intersection. Carsten Wächter and Nikolaus Binder, 2019.
Today’s Agenda: ▪ On Offsetting Shadows and Reflections ▪ Consistent Normal Interpolation ▪ Packing Normals ▪ Spatial Splits in BVHs ▪ Research Directions
Advanced Graphics – Bits & Pieces 9 Normals The Problem Vertex normal: the average of the normals of all polygons connected to the vertex. Generating / updating vertex normals in O(N): ▪ set each vertex normal to (0,0,0); ▪ loop over polygons; ▪ add normal of each polygon to each polygon vertex; ▪ normalize all vertex normals.
Advanced Graphics – Bits & Pieces 10 Normals The Problem Vertex normal: the average of the normals of all polygons connected to the vertex. Using vertex normals: ▪ Möller-Trumbore yields ‘ u,v ’ coordinates: these are barycentric coordinates; ▪ calculate 𝑥 = 1 – (𝑣 + 𝑤) ; ▪ now 𝑂 𝑗 = 𝑥 ∗ 𝑂 0 + 𝑣 ∗ 𝑂 1 + 𝑤 ∗ 𝑂 2 . Mind the side!
Advanced Graphics – Bits & Pieces 11 Normals The Problem Vertex normal: the average of the normals of all polygons connected to the vertex. For grazing directions, the reflection may go into the surface. Now what? ▪ Use geometric normal instead of interpolated normal? ▪ Clamp dot between R and N to 0? ▪ Just return black?
Advanced Graphics – Bits & Pieces 12 Normals
Advanced Graphics – Bits & Pieces 13 Normals
Advanced Graphics – Bits & Pieces 14 Normals Consistent Normal Interpolation* Can we come up with an interpolated normal 𝑜 𝑑 that behaves properly? - Requirements: ▪ It is smooth ▪ Reflections in 𝑜 𝑑 point away from the surface ▪ If all vertex normals are equal to 𝑜 , 𝑜 𝑑 = 𝑜 ▪ If 𝑒𝑝𝑢 𝑗, 𝑜 = 1 , 𝑜 𝑑 = 𝑜 𝑞 It turns out that this is indeed possible. The paper outlines how to calculate, for a given 𝑗, a vector 𝑠 that satisfies the constraints. 𝑜 𝑑 now is simply 𝑜𝑝𝑠𝑛𝑏𝑚𝑗𝑨𝑓 𝑗 + 𝑠 . *: Reshetov et al., Consistent Normal Interpolation. ACM Transactions on Graphics, 2010.
Advanced Graphics – Bits & Pieces 15 Normals
Advanced Graphics – Bits & Pieces 16 Normals
Advanced Graphics – Bits & Pieces 17 Normals
Advanced Graphics – Bits & Pieces 18 Normals
Advanced Graphics – Bits & Pieces 19 Normals Normal Mapping A normal map can be used to modify normals.
Advanced Graphics – Bits & Pieces 20 Normals Normal Mapping A normal map can be used to modify normals. This is typically done in tangent space. Problem: the orientation of tangent space matters. We need to align 𝑈 and 𝐶 with the texture, in other words: it needs to be based on the U and V vectors over the surface. This is non- trivial. For a derivation of the calculation of T and B, see: learnopengl.com/Advanced-Lighting/Normal-Mapping
Advanced Graphics – Bits & Pieces 21 Normals What if the normal sends you into the surface
Advanced Graphics – Bits & Pieces 22 Normals What if the normal sends you into the surface
Advanced Graphics – Bits & Pieces 23 Normals What if the normal sends you into the surface Microfacet-based Normal Mapping for Robust Monte Carlo Path Tracing. Schüssler et al., 2017.
Today’s Agenda: ▪ On Offsetting Shadows and Reflections ▪ Consistent Normal Interpolation ▪ Packing Normals ▪ Spatial Splits in BVHs ▪ Research Directions
Advanced Graphics – Bits & Pieces 25 Packing Efficiently Storing Normals A float color pixel contains r, g and b. Wouldn’t it be convenient if the fourth component could store the normal? ➔ Can we store a normal accurately in 32 bits? Observation: For a normal, we need two components (plus a sign). ➔ Can we store the two components in 16 bits each?
Advanced Graphics – Bits & Pieces 26 Packing Efficiently Storing Normals For an extensive study on this topic, see: aras-p.info/texts/CompactNormalStorage.html Bottom line: We can encode a normal in 32-bit, and the precision will be excellent: expect approximately four digits of precision. Pack: uint PackNormal( float3 N ) { float f = 65535.0f / sqrtf( 8.0f * N.z + 8.0f ); return (uint)(N.x * f + 32767.0f) + ((uint)(N.y * f + 32767.0f) << 16); }
Advanced Graphics – Bits & Pieces 27 Packing Efficiently Storing Normals For an extensive study on this topic, see: aras-p.info/texts/CompactNormalStorage.html Bottom line: We can encode a normal in 32-bit, and the precision will be excellent: expect approximately four digits of precision. Unpack: float3 UnpackNormal( uint p ) { float4 nn = make_float4( (float)(p & 65535) * (2.f / 65535.f), (float)(p >> 16) * (2.f / 65535.f), 0, 0 ); nn += make_float4( -1, -1, 1, -1 ); float l = dot( make_float3( nn.x, nn.y, nn.z ), make_float3( -nn.x, -nn.y, -nn.w ) ); nn.z = l, l = sqrtf( l ), nn.x *= l, nn.y *= l; return make_float3( nn ) * 2.0f + make_float3( 0, 0, -1 ); }
Today’s Agenda: ▪ On Offsetting Shadows and Reflections ▪ Consistent Normal Interpolation ▪ Packing Normals ▪ Spatial Splits in BVHs ▪ Research Directions
Advanced Graphics – Bits & Pieces 29 Splitting Problematic Large Polygons Large polygons lead to poor BVHs. (far more common than you’d think)
Advanced Graphics – Bits & Pieces 30 Splitting Problematic Large Polygons Large polygons lead to poor BVHs. Using the spatial splits in kD-trees, this is far less of an issue: The triangle will simply be assigned to each subspace.
Advanced Graphics – Bits & Pieces 31 Splitting Problematic Large Polygons Large polygons lead to poor BVHs. Using the spatial splits in kD-trees, this is far less of an issue: The triangle will simply be assigned to each subspace. Solution 1: split large polygons*. Observations: 1. A polygon can safely reside in multiple leafs; 2. The bounds of a leaf do not have to include the entire polygon. *: Early Split Clipping for Bounding Volume Hierarchies, Ernst & Greiner, 2007
Advanced Graphics – Bits & Pieces 32 Splitting Early Split Clipping Issues: ▪ Observations: Threshold parameter ▪ Individual polygons are split, 1. A polygon can safely reside in multiple leafs; regardless of surrounding geometry 2. The bounds of a leaf do not have to include the ▪ Primitives may end up multiple times entire polygon. in the same leaf 3. BVH construction only uses primitive bounding boxes. Algorithm: (some of these issues are resolved in: The Edge Prior to BVH construction, we recursively subdivide Volume Heuristic - Robust Triangle Subdivision for any polygon with a surface area that exceeds a Improved BVH Performance, Dammertz & Keller, certain threshold. 2008)
Advanced Graphics – Bits & Pieces 33 Splitting 𝐷 𝑡𝑞𝑚𝑗𝑢 = 𝐵 𝑚𝑓𝑔𝑢 ∗ 𝑂 𝑚𝑓𝑔𝑢 + 𝐵 𝑠𝑗ℎ𝑢 ∗ 𝑂 𝑠𝑗ℎ𝑢 < 𝐵 ∗ 𝑂 Spatial Splits for BVHs Observation: spatial splits are not limited to kD-trees. But: spatial splits tend to increase the cost of a split. Idea: 1. Determine cost of optimal object partition; 2. Determine cost of optimal spatial split; 3. Apply spatial split if cost is lower than object partition*. *: Spatial Splits in Bounding Volume Hierarchies, Stich et al., 2009
Advanced Graphics – Bits & Pieces 34 Splitting State of the Art: SBVH Summary: high quality bounding volume hierarchies can be obtained by combining the surface area heuristic and spatial splits. Compared to a regular SAH BVH, spatial splits improve the BVH by ~25% (see paper for scenes and figures).
Advanced Graphics – Bits & Pieces 35 Better BVHs
Advanced Graphics – Bits & Pieces 36 Better BVHs
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