Rendering — turning particles into pixels

MPM computes a continuum sampled by particles. Nothing about that says what it should look like, and the gap between “the physics is right” and “the frame is right” is where much of the open work in real-time MPM sits. This document covers the rendering subject: which reconstruction methods exist and what each one structurally can and cannot deliver, the optical stack a liquid needs, the per-material look problems that remain unsolved, and the secondary particles — spray, splash, foam — that a splash needs.

1. The reconstruction method space

The choice is not a quality dial, it is a choice of what kind of error you accept. One rule binds every density-field method below: the splat kernel’s shape must match the solver’s particle-to-grid kernel. A Gaussian, a bilinear splat, or a differently-supported B-spline all look approximately right in a still frame and reintroduce grid-locked shimmer in motion, because the surface then samples the material on a different footprint from the one the solver moved it with. Matching the shape does not mean matching the width: the transfer kernel is tuned for transfer accuracy, and used unwidened as a reconstruction kernel it produces the lumpy surface §1.1 explains. Widening the same kernel is the standard move.

1.1 Why particle surfaces are lumpy — and what the literature does about it

A density field splatted from particles is a sum of narrow kernels over a finite random sample, and that makes its noise predictable. For a kernel $W$ with unit integral over a sample of $n$ particles per unit volume, the field’s relative standard deviation at the kernel scale is shot noise:

$\frac{\sigma}{\rho} \approx \sqrt{\frac{D \int W^{2}}{n}}$

(3D quadratic B-spline at solver width: $\int W^{2} = 0.55^{3} \approx 0.166$ )

where $D$ is the index of dispersion of the sample — variance over mean of particle counts per cell — which is 1 for uniform-random (Poisson) seeding and roughly 0.45 for a lattice jittered by 80 % of its spacing. At 8 particles per cell and solver width that is 7–14 %; at 25 it is still 8 %, and the noise lives at a correlation length of about one cell. An iso-surface of such a field is displaced by $\delta\rho / |\nabla\rho|$ — of order 0.05–0.15 cells — at a one-cell wavelength, which is a 5–12° wobble of the normal, and a sharp specular lobe renders a 5° wobble as orange peel. Everything else follows from that one relation.

What is not the lever. Field resolution: a finer field samples the same noisy function more finely (rms noise identical at 1×, 2×, 4× the solver grid). Raising the iso threshold: it fixes inflation (below) but increases roughness, because the kernel fringe a low threshold sits in averages more particles than the boundary at half density does. Render-side Laplacian smoothing of particle positions with the transfer kernel: with a kernel only three or four spacings wide the step is a mean-shift iteration and converges particles onto density modes — one pass nearly doubles the dispersion — so it manufactures exactly the clumps it was meant to hide. The pre-pass in the anisotropic-kernel paper works at SPH neighborhood scale with a mild blend; it is a surface smoother that also shrinks volume, not a denoiser.

What is. In order of cost:

Kernel width relative to particle spacing. Doubling the smoothing length (the same B-spline, support scaled) halves the normal noise, since the noise falls as $h^{-1.5}$ and the gradient only as $h^{-1}$ . Every production particle surfacer does this: Houdini documents its plain spherical method as “a very lumpy surface that will usually require smoothing”, and splashsurf defaults the smoothing length to twice the particle radius before adding 25 Laplacian passes. A separable Gaussian blur of the composited field with $\sigma \approx 1$ cell reaches the same noise for three cheap passes instead of $h^{3}$ more scatter atomics per particle.
Threshold normalization, physically. An absolute threshold on a field whose interior reads “particles per cell” sits in the kernel fringe — at a few percent of the interior — which pushes every face about a cell outside its particles and bridges any gap under ~1.5 cells: fat sheets, fat limbs, joined lobes, merged droplets. The SPH color field fixes the magnitude rather than the dial: weight each particle by its volume ( $m/\rho$ in SPH; $V_0 \det(\mathbf{F}) J_p$ in MPM, carried on every particle already) so the interior reads ≈ 1 regardless of particle count or compaction, and a threshold of 0.5 means “halfway”. Widen first, then normalize — in that order.
Seeding. Uniform-random seeding is Poisson; a jittered lattice at the same count has a third of the surface noise, a blue-noise set less. A 25-particle-per-cell uniform-random ball reconstructs rougher than an 8-per-cell lattice-seeded one.
Tricubic filtering. Trilinear sampling of the voxel field is C⁰, so once the shot noise is gone a texel-scale faceting remains; cubic B-spline filtering with eight trilinear fetches (Sigg & Hadwiger 2005) gives a C² field and analytic gradients from the same voxels.
Anisotropic kernels (Yu & Turk 2013). A per-particle covariance of the neighborhood stretches each kernel along the local particle distribution, so flat pools and thin sheets reconstruct flat and thin; this is the default in Houdini’s “average position” surfacing and in splashsurf. In MPM the neighborhood moments are one extra grid transfer on the solver’s own lattice.
Level-set post-smoothing. Dilate → smooth → erode (the morphological closing OpenVDB and Houdini ship as a filter chain), mean-curvature or Laplacian iterations, or the constrained thin-plate smoothing of Bhattacharya et al. 2015, which minimizes bending energy while staying inside the band between the union of particle spheres and its erosion — the form that does not round corners. The averaged-center distance of Zhu & Bridson 2005, with the Solenthaler et al. 2007 correction for concave regions, is the older route to the same end and needs a smooth, wide weight; on a compact B-spline at twice the grid’s resolution it staircases.
Screen-space surfaces (van der Laan, Green & Sainz 2009; Truong & Yuksel 2018; Oliveira et al. 2022). Splat particle depth, smooth the depth image, shade from its normals; no volume texture, so no resolution ceiling and droplets at any size. The mainstream real-time route, and the one with a live patent — see the sources below.

What the particle set itself contributes. After the reconstruction noise is gone a residue can remain that no kernel width hides, because it is correlated over several cells: the density structure of plastically compacted material, or the cell-scale clumping that particle-in-cell transfers accumulate. Its signature is noise that does not fall as $h^{-1.5}$ when the kernel widens. Volume weighting (item 2) reads real compaction correctly; clumping is a solver question — position correction, resampling, or a different transfer — and not a rendering one.

The reference frames are a different regime. The published MLS-MPM demos were simulated at one to five million particles on grids of 250–500 cells per unit at minutes per frame and visualized offline in Houdini (particles exported as bgeo, solids as obj); the sand in those frames is drawn as individual particles rather than a surface, with an offline renderer’s shadowing and occlusion — the exact renderer settings are not published — and the snow reference of the original MPM snow paper was rendered as a scattering volume straight from the rasterized grid (Henyey–Greenstein, $g = 0.5$ , $\sigma_t = 724\ \mathrm{m}^{-1}$ , albedo 0.9–1.0), never as a surface. The paper that introduced MLS-MPM also concedes that “it is hard to reconstruct sharp cutting surfaces from particles.” A real-time frame at a tenth of the particle count on a grid four times coarser is not going to match their feature scale; it can match their smoothness, which is the part that is free.

Sources. Every paper behind this section — the anisotropic-kernel, level-set and screen-space reconstruction work, the filtering and VDB references, and the MPM papers the numbers come from — is listed with venue, year and DOI in References, under Surface reconstruction & rendering.

What may be read. Code that may be read for commercial work: splashsurf (MIT; anisotropic-kernel marching cubes with Laplacian and normal smoothing — its defaults are a calibrated recipe), fluid-engine-dev (MIT; spherical, SPH blobby, Zhu–Bridson and anisotropic points-to-implicit side by side), OpenVDB (Apache-2.0; ParticlesToLevelSet and the dilate/smooth/erode filters). Not to be read: the narrow-range filter’s reference implementation (repository) carries no license; the paper describes the filter completely. Patent: NVIDIA’s US8947430B1 (filed 2010, reported active to 2032-09-15) claims rendering particles as screen-oriented quads into a depth image, smoothing it, deriving positions and normals, and shading — in conjunction with a specific per-pixel thickness accumulation — so any screen-space design is a counsel question first (Patents).

Binary iso-surface. Density decides in/out only, never opacity: fading opacity by density exposes the coarse grid as blocky alpha and turns density noise into spikes that never settle, so it is a dead end rather than a preference — the binary threshold is what hides the density-magnitude noise. Produces a gap-free opaque body with a controllable silhouette. Structurally cannot show internal density variation, which is exactly what the eye reads as clumping in sand and snow.
Additive/soft accumulation. Shows density variation. Structurally reads as spheres, because that is what the accumulation kernel is.
Hybrid iso + interior texture. Iso for the gap-free body, plus albedo modulation from either the density one inward sample below the hit or the slope between two inward samples. The shader leaves the opaque hit and written depth unchanged, so the silhouette stays the iso-surface’s; this is emphatically not density-as-opacity. The mechanism is cheap and deterministic, but only a per-material human look comparison can choose density, slope, another texture axis, or points.
Faceted (tetrahedral) reconstruction. Replaces trilinear density sampling with a continuous piecewise-linear tetrahedral field and uses that field’s analytic, cell-flat gradient — real planar geometry, not screen-space normal noise. Measured on one GPU density raymarch at 720p, it made close and grazing patches read more planar, left the broad hero silhouette of a liquid recognizably rounded, and raised the surface pass from 3.51 to 5.98 ms — so it was inconclusive for water. For rock it is the right axis, and should be judged against a rock target rather than a water one.
Screen-space surface (depth splat + narrow-range filter, Truong & Yuksel 2018). Decouples droplet size from a volume texture entirely, which is the standing answer to a volume texture’s resolution ceiling (at 256³, the smallest representable droplet in a 1 m box is roughly 4 mm: fine for a pour, marginal for spray). The open alternative is a narrow-band/sparse field. Patent-encumbered in its canonical form — §1.1, Sources.
Volume rendering of the density itself. For high-albedo scattering media — snow above all — the offline reference rasterizes particles to the grid and path-traces the grid as a participating medium; no surface is extracted. A real-time stand-in is a smooth iso plus wrapped diffuse and a thickness term (§3), but the primitive to aim at is the volume.

Grains as grains stays on the table. Point rendering is a legitimate granular path, and a close view of actual grains is a legitimate target. Decide per scene which path ships — points, hybrid iso + texture, or both at different distances — rather than promoting one globally.

2. The optical stack

What separates a convincing liquid from a blue solid, roughly in order of how much each changes a frame.

Accumulated thickness. Integrate the above-threshold interval along the ray. The alternative — using the domain’s far wall as a thickness proxy — makes a thin sheet absorb like a full tank and makes spray indistinguishable from depth. The cheap implementation linearly resolves and sums every disjoint above-threshold interval on the existing density raymarch.
Scene-color refraction. A raymarch without it substitutes something like a two-color vertical gradient for the world behind the surface. The mechanism: copy the current scene color into one transient non-MSAA texture before the water pass, then sample it with a thickness-scaled view-normal offset, applied as a delta to the water color.
Environment reflection, a GGX lobe, and a sun disc, replacing a fixed pow(dot(n,h), k) highlight. Most of what reads as water in a good reference frame is sky and sun glint sitting in the surface — not the water itself.
Beer–Lambert absorption keyed to real thickness, which is what makes depth read as depth rather than as tint.
Foam, spray, and bubbles. Usually the largest gap: breaking and splashing are what a height field structurally cannot do, and real splash reads white. The choice is between secondary diffuse particles on a trapped-air/curvature criterion (Ihmsen et al. 2012) and a cheaper shading term derived from surface curvature and speed; weigh both against any one-way tracer machinery the solver already has. See §4 below for the mechanism — and first: presentation particles are physics unless their coupling is explicitly one-way.
Caustics. Optional, last; a lit floor sells depth cheaply once the rest is right.

Half the realism in a good reference frame is not the water. Those frames also carry global illumination, sky and atmosphere, aerial perspective, a tonemapped exposure, and surroundings that give the eye scale. Correct optics in a gray box will lose to mediocre water in a good environment, so budget sky, ground, lighting and tonemap as part of the work rather than as dressing after it.

3. The per-material look problems

Each material has a different kind of look problem, and conflating them wastes effort.

Water — given a gap-free reconstructed surface, what remains is optics. This is purely §2’s list.
Granular (sand, snow) — an untextured binary iso does not read as granular, and the failure is structural rather than a tuning miss: the eye reads clumping from internal density variation, which a binary iso erases (§1). Two cautions from §1.1: the “clay” bumpiness a noisy reconstruction produces is not a sand texture — it disappears with the field noise and the granular read has to be put back deliberately — and what makes the offline references read as grains is per-grain shadowing and occlusion on individual particles (Meng et al. 2015 is the physics of why), which a sprite without depth, shadow, or occlusion cannot supply at any size. The hybrid iso + interior texture of §1 makes density and slope available as surface candidates without changing the solver or the silhouette. Then the two materials diverge sharply: sand is opaque, high-albedo, diffuse to the point of flatness, so its realism lives off the surface — contact darkening where grains pack, a crisp silhouette at the angle of repose under motion, dust lofting off a collapse front, and the wet/dry split. Snow is the hardest optically, because most of what identifies it is subsurface: multiple scattering (why snow is bright but not shiny), thickness-dependent blue in deep pack, sparkle from specular facets, and a packed-versus-powder distinction visible in both silhouette and shading. Cheapest credible route is wrapped diffuse plus a thickness term rather than a real subsurface solve.
Rock — the problem is geometry, not optics. Rock is opaque and rough, so none of §2 applies. What makes a reconstructed body read as stone is the inverse of what makes water read as water: hard silhouettes rather than smooth ones, planar faces, surface detail finer than the grid. A smooth iso-surface is right for elastic bodies, where a rounded iso is correct; for rock it is precisely wrong. Fresh fracture faces are the highest-value single item — a broken rock reads as broken because the new face is flat, sharp-edged, and a different color and roughness from the weathered outside. The cheapest reusable foundation density-weights a per-particle damage variable into the reconstructed field and maps it to an albedo/roughness response. That is cheap and deterministic, but it does not make damage into exposure age: an old failed region stays visually fresh. A human comparison has to decide whether the proxy is sufficient before adding a timestamped lane.
Ice — reuses water’s stack with different constants plus one addition: internal structure (cracks, trapped bubbles, cloudiness) scattering inside the body. The physical axis is well established — clear young ice is low-roughness with sharp refraction and little scattering, while old ice carrying trapped air goes rougher, denser, and blue. A recipe worth reusing: accumulate a per-particle cloudiness proportional to impact intensity, diffuse it, and derive scratch direction from its gradient.

4. Secondary particles — spray, splash, foam

The published option with actual constants comes from real-time height-field water, where secondary particles exist because the 2.5D representation fails; MPM does not have that failure, so what transfers is the classification, the spawn rule, the lifetime treatment, and the rendering — not the detection criteria.

Foam is spawned from splash impact, not detected. When a splash particle hits the surface, create a foam particle with a probability that depends on impact speed. Cheaper than any curvature or trapped-air criterion, because MPM already knows a particle’s impact speed. Foam then advects with the horizontal velocity field, projects onto the surface, and takes a user lifetime modulated with noise; it is what conveys horizontal swirl.
Spray and splash are separate species with different drag, and spray renders as an ellipse elongated along velocity to fake motion blur. That impostor change is a few lines and is the cheapest thing that makes fast droplets read as spray rather than as balls — directly against the sphere impostor’s ceiling, where sharp reads as balls and soft reads as blur with no setting between.
Reabsorption is the part with no other candidate. A generated particle takes mass and momentum from the field and deposits them back where it lands; for effective volume $V_\mathrm{eff}$ the height change is $V_\mathrm{eff}/(\Delta x)^{2}$ and the receiving velocity is a mass-weighted blend. Without this, secondary particles are a mass leak.
Surface tension is the mechanism that forms what this section draws. A per-material node-side density gradient, gathered back through the same B-spline weights and applied as a body force, is the cheapest continuum-surface-force in MPM: the gradient points along the interface normal and vanishes in the bulk, so the force is automatically confined to the surface. It also produces immiscibility between materials for free. Caution: a gradient-derived force is noisy at low particles-per-cell, and grid-side density estimates read spuriously expanded at a free surface, so any grid-derived volume correction must act on the compression side only.

The binding rule: presentation particles are physics unless their coupling is explicitly one-way. If foam and spray are real MPM particles they change the simulation; if they are not, the reabsorption bookkeeping above is what keeps mass honest.