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diff --git a/articles/2021-09-26_noise_and_ray_marching.org b/articles/2021-09-26_noise_and_ray_marching.org new file mode 100644 index 0000000..d3b231d --- /dev/null +++ b/articles/2021-09-26_noise_and_ray_marching.org @@ -0,0 +1,276 @@ +* Noise and Ray Marching +[[https://literatelb.org][LiterateLB's]] volumetric visualization functionality relies on a simple ray marching implementation +to sample both the 3D textures produced by the simulation side of things and the signed distance +functions that describe the obstacle geometry. While this produces surprisingly [[https://www.youtube.com/watch?v=n86GfhhL7sA][nice looking]] +results in many cases, some artifacts of the visualization algorithm are visible depending on the +viewport and sample values. Extending the ray marching code to utilize a noise function is +one possibility of mitigating such issues that I want to explore in this article. + +While my [[https://www.youtube.com/watch?v=J2al5tV14M8][original foray]] into just in time visualization of Lattice Boltzmann based simulations +was only an aftertought to [[https://tree.kummerlaender.eu/projects/symlbm_playground/][playing around]] with [[https://sympy.org][SymPy]] based code generation approaches I have +since put some work into a more fully fledged code. The resulting [[https://literatelb.org][LiterateLB]] code combines +symbolic generation of optimized CUDA kernels and functionality for just in time fluid flow +visualization into a single /literate/ [[http://code.kummerlaender.eu/LiterateLB/tree/lbm.org][document]]. + +For all fortunate users of the [[https://nixos.org][Nix]] package manager, tangling and building this from the [[https://orgmode.org][Org]] +document is as easy as executing the following commands on a CUDA-enabled NixOS host. + +#+BEGIN_SRC sh +git clone https://code.kummerlaender.eu/LiterateLB +nix-build +./result/bin/nozzle +#+END_SRC + +** Image Synthesis +The basic ingredient for producing volumetric images from CFD simulation data is to compute +some scalar field of samples \(s : \mathbb{R}^3 \to \mathbb{R}_0^+\). Each sample \(s(x)\) can be assigned a color +\(c(x)\) by some convenient color palette mapping scalar values to a tuple of red, green and blue +components. + +[[https://literatelb.org/tangle/asset/palette/4wave_ROTB.png]] + +The task of producing an image then consists to sampling the color field along a ray assigned +to a pixel by e.g. a simple pinhole camera projection. For this purpose a simple discrete +approximation of the volume rendering equation with constant step size \(\Delta x \in \mathbb{R}^+\) already +produces suprisingly good pictures. Specifically +$$C(r) = \sum_{i=0}^N c(i \Delta x) \mu (i \Delta x) \prod_{j=0}^{i-1} \left(1 - \mu(j\Delta x)\right)$$ +is the color along ray \(r\) of length \(N\Delta x\) with local absorption values \(\mu(x)\). This +local absorption value may be chosen seperately of the sampling function adding an +additional tweaking point. + +#+BEGIN_EXPORT html +<video style="width:100%" src="https://literatelb.org/media/nozzle.webm" controls="controls"> +</video> +#+END_EXPORT + +The basic approach may also be extended arbitrarily, e.g. it is only the inclusion of a couple +of phase functions away from being able [[https://tree.kummerlaender.eu/projects/firmament/][recover the color produced by light travelling through the participating media that is our atmosphere]]. + +** The Problem +There are many different possibilities for the choice of sampling function \(s(x)\) given the results of a +fluid flow simulation. E.g. velocity and curl norms, the scalar product of ray direction and shear layer +normals or vortex identifiers such as the Q criterion +\[ Q = \|\Omega\|^2 - \|S\|^2 > 0 \text{ commonly thresholded to recover isosurfaces} \] +that contrasts the local vorticity and strain rate norms. The strain rate tensor \(S\) is easily +recovered from the non-equilibrium populations \(f^\text{neq}\) of the simulation lattice — and is in +fact already used for the turbulence model. Similarly, the vorticity \(\Omega = \nabla \times u\) can be +computed from the velocity field using a finite difference stencil. + +The problem w.r.t. rendering when thresholding sampling values to highlight structures in the flow +becomes apparent in the following picture: + +#+BEGIN_EXPORT html +<div class="flexcolumns"> +<div> +<span>Q Criterion</span> +<img src="https://static.kummerlaender.eu/media/q_criterion_default.png"/> +</div> +<div> +<span>Curl Norm</span> +<img src="https://static.kummerlaender.eu/media/curl_default.png"/> +</div> +</div> +#+END_EXPORT + +While the exact same volume discretization was used for both visualizations, the slices are much +less apparent for the curl norm samples due to the more gradual changes. In general the issue is +most prominent for scalar fields with large gradients (specifically the sudden jumps that occur +when restricting sampling to certain value ranges as is the case for the Q criterion). + +** Colors of Noise +The reason for these artifacts is primarily choice of start offsets w.r.t. the traversed volume +in addition the the step width. While this tends to become less noticable when decreasing said +steps, this is not desirable from a performance perspective. + +What I settled on for LiterateLB's renderer are view-aligned slicing and random jittering to remove +most visible artifacts. The choice of /randomness/ for jittering the ray origin is critical here as plain +random numbers tend to produce a distracting static-like pattern. A common choice in practice is +to use so called /blue noise/ instead. While both kinds of noise eliminate most slicing artifacts, the +remaining patterns tend to be less noticeable for blue noise. Noise is called /blue/ if it contains only +higher frequency components which makes it harder for the pattern recognizer that we call brain to +find patterns where there should be none. + +The [[https://www.spiedigitallibrary.org/conference-proceedings-of-spie/1913/0000/Void-and-cluster-method-for-dither-array-generation/10.1117/12.152707.short?SSO=1][void-and-cluster algorithm]][fn:vac] provides a straight forward method for +pre-computing tileable blue noise textures that can be reused during the actual visualization. +Tileability is a desirable property for this as we otherwise would either need a noise texture +large enough to cover the entire image or instead observe jumps at the boundary between +the tiled texture. + +The first ingredient for /void-and-cluster/ is a =filteredPattern= function that applies a +plain Gaussian filter with given $\sigma$ to a cyclic 2d array. Using cyclic wrapping during the +application of this filter is what renders the generated texture tileable. + +#+BEGIN_SRC python +def filteredPattern(pattern, sigma): + return gaussian_filter(pattern.astype(float), sigma=sigma, mode='wrap', truncate=np.max(pattern.shape)) +#+END_SRC + +This function will be used to compute the locations of the largest void and tightest +cluster in a binary pattern (i.e. a 2D array of 0s and 1s). In this context a /void/ describes +an area with only zeros and a /cluster/ describes an area with only ones. + +#+BEGIN_SRC python +def largestVoidIndex(pattern, sigma): + return np.argmin(masked_array(filteredPattern(pattern, sigma), mask=pattern)) +#+END_SRC + +These two functions work by considering the given binary pattern as a float array that is blurred by +the Gaussian filter. The blurred pattern gives an implicit ordering of the /voidness/ of each pixel, the +minimum of which we can determine by a simple search. It is important to exclude the initial binary +pattern here as void-and-cluster depends on finding the largest areas where no pixel is set. + +#+BEGIN_SRC python +def tightestClusterIndex(pattern, sigma): + return np.argmax(masked_array(filteredPattern(pattern, sigma), mask=np.logical_not(pattern))) +#+END_SRC + +Computing the tightest cluster works in the same way with the exception of searching the largest array +element and masking by the inverted pattern. + +#+BEGIN_SRC python +def initialPattern(shape, n_start, sigma): + initial_pattern = np.zeros(shape, dtype=np.bool) + initial_pattern.flat[0:n_start] = True + initial_pattern.flat = np.random.permutation(initial_pattern.flat) + cluster_idx, void_idx = -2, -1 + while cluster_idx != void_idx: + cluster_idx = tightestClusterIndex(initial_pattern, sigma) + initial_pattern.flat[cluster_idx] = False + void_idx = largestVoidIndex(initial_pattern, sigma) + initial_pattern.flat[void_idx] = True + return initial_pattern +#+END_SRC + +For the initial binary pattern we set =n_start= random locations to one and then repeatedly +break up the largest void by setting its center to one. This is also done for the tightest cluster +by setting its center to zero. We do this until the locations of the tightest cluster and largest +void overlap. + +#+BEGIN_SRC python +def blueNoise(shape, sigma): +#+END_SRC + +The actual algorithm utilizes these three helper functions in four steps: +1. Initial pattern generation + #+BEGIN_SRC python + n = np.prod(shape) + n_start = int(n / 10) + + initial_pattern = initialPattern(shape, n_start, sigma) + noise = np.zeros(shape) + #+END_SRC +3. Eliminiation of =n_start= tightest clusters + #+BEGIN_SRC python + pattern = np.copy(initial_pattern) + for rank in range(n_start,-1,-1): + cluster_idx = tightestClusterIndex(pattern, sigma) + pattern.flat[cluster_idx] = False + noise.flat[cluster_idx] = rank + #+END_SRC +4. Elimination of =n/2-n_start= largest voids + #+BEGIN_SRC python + pattern = np.copy(initial_pattern) + for rank in range(n_start,int((n+1)/2)): + void_idx = largestVoidIndex(pattern, sigma) + pattern.flat[void_idx] = True + noise.flat[void_idx] = rank + #+END_SRC +5. Elimination of =n-n/2= tightest clusters of the inverted pattern + #+BEGIN_SRC python + for rank in range(int((n+1)/2),n): + cluster_idx = tightestClusterIndex(np.logical_not(pattern), sigma) + pattern.flat[cluster_idx] = True + noise.flat[cluster_idx] = rank + #+END_SRC + +For each elimination the current =rank= is stored in the noise texture +producing a 2D arrangement of the integers from 0 to =n=. As the last +step the array is divided by =n-1= to yield a grayscale texture with values +in $[0,1]$. + +#+BEGIN_SRC python +return noise / (n-1) +#+END_SRC + +In order to check whether this actually generated blue noise, we can take a +look at the Fourier transformation for an exemplary \(100 \times 100\) texture: + +#+BEGIN_EXPORT html +<div class="flexcolumns"> +<div> +<span>Blue noise texture</span> +<img src="https://static.kummerlaender.eu/media/blue_noise.png"/> +</div> +<div> +<span>Fourier transformation</span> +<img src="https://static.kummerlaender.eu/media/blue_noise_fourier.png"/> +</div> +</div> +#+END_EXPORT + +One can see qualitatively that higher frequency components are significantly more +prominent than lower ones. Contrasting this to white noise generated using uniformly +distributed random numbers, no preference for any range of frequencies can be +observed: + +#+BEGIN_EXPORT html +<div class="flexcolumns"> +<div> +<span>White noise texture</span> +<img src="https://static.kummerlaender.eu/media/white_noise.png"/> +</div> +<div> +<span>Fourier transformation</span> +<img src="https://static.kummerlaender.eu/media/white_noise_fourier.png"/> +</div> +</div> +#+END_EXPORT + +** Comparison +Contasting the original Q criterion visualization with one produced using blue noise jittering +followed by a soft blurring shader, we can see that the slicing artifacts largely vanish. +While the jittering is still visible to closer inspection, the result is significantly more pleasing +to the eye and arguably more faithful to the underlying scalar field. + +#+BEGIN_EXPORT html +<div class="flexcolumns"> +<div> +<span>Simple ray marching</span> +<img src="https://static.kummerlaender.eu/media/q_criterion_default.png"/> +</div> +<div> +<span>Ray marching with blue noise jittering</span> +<img src="https://static.kummerlaender.eu/media/q_criterion_blue_noise.png"/> +</div> +</div> +#+END_EXPORT + +While white noise also obcures the slices, its lower frequency components +produce more obvious static in the resulting image compared to blue noise. +As both kinds of noise are precomputed we can freely choose the kind of +noise that will produce the best results for our sampling data. + +#+BEGIN_EXPORT html +<div class="flexcolumns"> +<div> +<span>Blue noise</span> +<img src="https://static.kummerlaender.eu/media/q_criterion_blue_noise_close.png"/> +</div> +<div> +<span>White noise</span> +<img src="https://static.kummerlaender.eu/media/q_criterion_white_noise_close.png"/> +</div> +</div> +#+END_EXPORT + +In practice where the noise is applied just-in-time during the visualization of +a CFD simulation, all remaining artifacts tend to become invisible. This can +be seen in the following video of the Q criterion evaluated for a simulated +nozzle flow in LiterateLB: + +#+BEGIN_EXPORT html +<video style="width:100%" src="https://static.kummerlaender.eu/media/nozzle_q_criterion.webm" controls="controls"> +</video> +#+END_EXPORT + +[fn:vac] Ulichney, R. Void-and-cluster method for dither array generation. In Electronic Imaging (1993). DOI: [[https://www.spiedigitallibrary.org/conference-proceedings-of-spie/1913/0000/Void-and-cluster-method-for-dither-array-generation/10.1117/12.152707.short?SSO=1][10.1117/12.152707]]. |