Signed Distance Functions (SDF) are a mathematical representation of 3D geometry as distance to the nearest surface — defining shapes by storing the signed distance from any point in space to the closest surface point, with negative values inside and positive outside, enabling powerful geometric operations and high-quality surface representation.

What Is a Signed Distance Function?

• Definition: Function SDF(x, y, z) → distance to nearest surface.
• Sign Convention: Negative inside object, positive outside, zero on surface.
• Properties: Continuous, differentiable, metric information.
• Surface: Defined by zero level set (SDF = 0).

SDF Formula:

SDF(p) = {
  -d  if p inside object
   0  if p on surface
  +d  if p outside object
}
where d = distance to nearest surface point

Why Signed Distance Functions?

• Geometric Operations: Easy boolean operations (union, intersection, difference).
• Collision Detection: Fast distance queries for physics.
• Ray Marching: Efficient rendering via sphere tracing.
• Surface Reconstruction: Robust to noise, handles topology changes.
• Shape Representation: Continuous, resolution-independent.
• Gradient: Surface normal = gradient of SDF.

SDF Properties

Metric Information:

• Distance: Exact distance to surface at any point.
• Use: Collision detection, proximity queries.

Surface Normal:

• Formula: n = ∇SDF / |∇SDF|
• Benefit: Normals available everywhere via gradient.

Lipschitz Continuity:

• Property: |SDF(p₁) - SDF(p₂)| ≤ |p₁ - p₂|
• Benefit: Bounded rate of change, stable numerics.

Zero Level Set:

• Surface: Points where SDF(p) = 0.
• Extraction: Marching Cubes, dual contouring.

SDF Representations

Analytic SDF:

• Method: Mathematical formulas for primitive shapes.
• Examples: Sphere, box, cylinder, torus.
• Benefit: Exact, efficient evaluation.
• Use: Procedural modeling, CSG.

Discrete SDF:

• Method: Store SDF values on grid (voxels).
• Benefit: Represent arbitrary shapes.
• Use: 3D reconstruction, simulation.

Neural SDF:

• Method: Neural network learns SDF as implicit function.
• Examples: DeepSDF, Neural SDF.
• Benefit: Compact, continuous, learnable.

Truncated SDF (TSDF):

• Method: Truncate SDF to narrow band around surface.
• Benefit: Focus computation on surface region.
• Use: Real-time 3D reconstruction (KinectFusion).

Analytic SDF Examples

Sphere:

SDF(p) = |p - center| - radius

Box:

SDF(p) = max(|p.x| - size.x, |p.y| - size.y, |p.z| - size.z)

Plane:

SDF(p) = dot(p - point_on_plane, normal)

Torus:

q = (|p.xz| - major_radius, p.y)
SDF(p) = |q| - minor_radius

SDF Operations

Boolean Operations:

• Union: min(SDF₁, SDF₂)
• Intersection: max(SDF₁, SDF₂)
• Difference: max(SDF₁, -SDF₂)
• Benefit: Combine shapes easily.

Smooth Boolean:

• Smooth Union: Blend SDFs smoothly.
• Formula: -log(exp(-k·SDF₁) + exp(-k·SDF₂)) / k
• Benefit: Organic blending between shapes.

Transformations:

• Translation: SDF(p - offset)
• Rotation: SDF(R⁻¹ · p)
• Scale: SDF(p / scale) · scale
• Benefit: Transform shapes via coordinate transformation.

Deformations:

• Twist, Bend, Taper: Apply deformation to coordinates before SDF evaluation.
• Benefit: Complex shape variations.

Applications

3D Reconstruction:

• Use: Fuse depth maps into TSDF volume.
• Method: KinectFusion, Voxblox, BundleFusion.
• Benefit: Robust to noise, incremental updates.

Ray Marching / Sphere Tracing:

• Use: Render SDFs efficiently.
• Method: March along ray by SDF distance (safe step size).
• Benefit: Accurate rendering without discretization.

Collision Detection:

• Use: Fast distance queries for physics simulation.
• Benefit: Exact distance, penetration depth.

Shape Generation:

• Use: Neural networks learn SDF for shape synthesis.
• Method: DeepSDF — latent code → SDF.
• Benefit: Continuous, high-quality shapes.

Procedural Modeling:

• Use: Combine primitive SDFs for complex shapes.
• Benefit: Compact, parametric, editable.

SDF-Based Rendering

Sphere Tracing:

• Method: Ray marching using SDF as step size.
• Algorithm:

1. Start at ray origin. 2. Evaluate SDF at current position. 3. Step forward by SDF distance (safe, won't overshoot surface). 4. Repeat until close to surface (SDF ≈ 0) or max steps.

• Benefit: Efficient, accurate, no discretization.

Soft Shadows:

• Method: Accumulate minimum SDF along shadow ray.
• Benefit: Soft, realistic shadows.

Ambient Occlusion:

• Method: Sample SDF in hemisphere around point.
• Benefit: Approximate global illumination.

Neural SDF

DeepSDF:

• Architecture: MLP maps (x, y, z, latent) → SDF value.
• Training: Learn to predict SDF from ground truth.
• Use: Shape representation, generation, completion.
• Benefit: Compact (KB), continuous, interpolatable.

Neural Implicit Surfaces:

• Method: Neural network represents SDF.
• Benefit: Smooth, high-quality surfaces.
• Examples: DeepSDF, IGR, SAL.

Conditional Neural SDF:

• Method: Condition on observations (point cloud, image).
• Use: 3D reconstruction from partial data.

Challenges

Computation:

• Problem: Evaluating SDF at many points is expensive.
• Solution: Hierarchical evaluation, octrees, hash encoding.

Learning:

• Problem: Neural networks struggle with exact SDF (Lipschitz constraint).
• Solution: Eikonal loss, geometric regularization.

Topology Changes:

• Problem: Discrete SDFs struggle with topology changes.
• Solution: Adaptive grids, implicit representations.

Thin Structures:

• Problem: Thin features require high resolution.
• Solution: Adaptive resolution, multi-scale.

TSDF (Truncated SDF)

Definition:

• Truncation: Clamp SDF to [-τ, +τ] near surface.
• Benefit: Focus computation on surface region.

KinectFusion:

• Method: Fuse depth maps into TSDF volume.
• Process:

1. Align depth map to volume. 2. Integrate depth into TSDF (weighted average). 3. Extract mesh via Marching Cubes.

• Benefit: Real-time 3D reconstruction.

Voxblox:

• Method: Efficient TSDF for robotics.
• Benefit: Fast, memory-efficient, incremental.

Quality Metrics

• Accuracy: Distance to ground truth surface.
• Completeness: Coverage of object surface.
• Lipschitz Property: Verify |∇SDF| ≈ 1.
• Surface Quality: Smoothness, detail preservation.
• Rendering Quality: Visual realism of rendered SDF.

SDF Tools

Analytic SDF:

• Shadertoy: Online SDF ray marching demos.
• Inigo Quilez: Comprehensive SDF resource.

Discrete SDF:

• Open3D: TSDF integration and mesh extraction.
• PCL: Point cloud to SDF conversion.

Neural SDF:

• DeepSDF: Official implementation.
• PyTorch3D: Implicit function support.
• Kaolin: 3D deep learning with SDF.

Rendering:

• ShaderToy: Real-time SDF rendering.
• Blender: SDF-based procedural modeling.

SDF vs. Other Representations

SDF vs. Occupancy:

• SDF: Metric distance information.
• Occupancy: Binary inside/outside.
• Benefit: SDF provides more information.

SDF vs. Meshes:

• SDF: Implicit, continuous, easy boolean ops.
• Meshes: Explicit, efficient rendering.
• Use: SDF for modeling, mesh for rendering.

SDF vs. Voxels:

• SDF: Continuous, resolution-independent.
• Voxels: Discrete, fixed resolution.
• Benefit: SDF smoother, more accurate.

Future of SDF

• Real-Time: Fast neural SDF evaluation.
• High-Resolution: Capture fine geometric details.
• Generalization: Single model for all shapes.
• Hybrid: Combine with explicit representations.
• Dynamic: Represent deforming shapes.
• Semantic: Integrate semantic information with geometry.

Signed Distance Functions are a powerful geometric representation — they encode 3D shapes as continuous distance fields, enabling efficient rendering, robust reconstruction, and intuitive geometric operations, making them fundamental to modern computer graphics, vision, and robotics.