Filtering in PRMan

Brent Burley, Feb 2007

The following are some very interesting insights on filtering in PRMan, by Brent. Keep in mind that Brent is not representing or speaking for Disney or Pixar. These are purely his insights as he tried to understand how filtering works in PRMan. And I thought it would be something really cool to share with the RenderMan community. I do want to thank Disney for letting us share this.-- Tal Jul 24, 2008

Here are some typical default filtering settings for PRMan:


 ShadingRate
 PixelSamples
 ShadingInterpolation PixelFilter
PRMan
 1.0
 2 2
 "constant" "gaussian" 2.0 2.0
MTOR
 1.0
 2 2
 "smooth" "box" 1.0 1.0
"TOR.ini"
 5.0
 3 3
 "smooth" "separable-catmull-rom" 2.0 2.0
Lumiere
 1.0
 5 5
 "smooth" "catmull-rom" 2.0 2.0
Shot Finaling CHIC 0.5 8 8 "smooth" "catmull-rom" 2.0 2.0
Shot Finaling WILB
 0.5
 8 8
 "smooth" "mitchell" 5.0 5.0
 
How do these settings affect the filtering, and what are the optimal settings?


PRMan filtering occurs in several stages:
  1. Dicing of geometry into micropolygons whose size is based on the ShadingRate
  2. Shading, which includes texture and/or procedural filtering
  3. Stochastic sampling, with a resolution based on PixelSamples, and micropolygon sample values determined by ShadingInterpolation
  4. Final image reconstruction using the PixelFilter
Each step attenuates high-frequencies to some extent and also adds some aliasing.   First, the FFT is reviewed as a tool for frequency analysis, and then the frequency response of each of the filter settings is analyzed in the follow sections.  Finally, filtered sample images are compared and recommendations are summarized.
 

Frequency Analysis with FFT

The frequency content, or spectrum, of a signal can be measured using the Fast Fourier Transform (FFT) in 1 or 2 dimensions.  To measure the frequency response of a filter, a random (white noise) signal is generated, processed by the filter in question, and then analyzed with the FFT to measure the spectrum after filtering.

Sample 1d FFT:

Top left: 1d kernel plot; top right: 2d kernel image.  Bottom: FFT frequency response.

The FFT response curve shows the attenuation based on frequency.  The attenuation is in decibels (dB) and is a logarithmic scale.  Every 6 dB down from zero corresponds to a reduction in intensity by 1/2, or 1 bit of resolution.  For example, in an 8-bit image, after 48 dB the intensity is no longer measurable; 16 bit resolution gives 96 dB dynamic range.  The filter shown above may be sufficient for 8-bit images but will produce significant aliasing in a 16-bit image.

The frequency for an image filter is measured in cycles per pixel.  The frequency of 0.5 is the Nyquist limit and is the highest representable frequency corresponding to a signal with a 2-pixel period (e.g. alternating bright/dark pixels).  Everything above that frequency that gets through the filter is by definition aliasing.  Any attenuation below the Nyquist limit is lost detail and results in a blurry image.  The gaussian 2.0 filter shown above has an attenuation of about 10dB (or 70%) at the Nyquist limit but also loses about 2.5dB of detail (or 25%) in the 0.25 frequency range.

For separable filters (like mitchell and separable-catmull-rom), the 1d frequency response also applies to 2d filtering.  But for non-separable filters (like gaussian and catmull-rom), and other filtering effects like stochastic sampling and shading interpolation, the frequency response must be examined using a 2d FFT.

       
Left to Right: 2d FFTs of gaussian 2.0, sinc 7.0, catmull-rom 4.0, and separable-catmull-rom 4.0 filters.

The 2d FFT shows the frequency response of a filtered rand image.  The lower left corner of the image represents frequency zero in x and y, and the upper right corner is the Nyquist limit (the lower right and upper left corners are the 1d Nyquist limits in x and y).  Low pass filters attenuate the higher frequencies and thus the image is typically darker on the top and right edges.  The sinc response is much flatter than the gaussian with negligible attenuation below the nyquist limit.  For the gaussian, the intensity in the lower right corner is approximately 70% lower than at the lower left corner (which is consistent with the 1d FFT); for a separable filter, the attenuation in the upper right corner will be the product of the x and y attenuations which for the gaussian is about 90% (.3 * .3 = .09 = 91% attenuation).  Notice how the catmull-rom and separable-catmull-rom have significantly different 2d responses even though their 1d response is identical.

Various 2d images and their FFTs (inset over the images) are found here.



ShadingRate

The ShadingRate determines the size of the micropolygons and represents the desired area of the micropolygons in pixel units.  The length of a micropolygon on each side is equal to sqrt(ShadingRate).  For example, a ShadingRate of 0.25 generates micropolygons that are 1/2 a pixel wide on each side.

Using a ShadingRate of less than 1.0 results in subpixel shading and gives a more detailed image, but at significant cost: the shading time (which is often most of the rendering time) is roughly proportional to 1.0/ShadingRate.  For example, a shading rate of 0.25 incurs a shading time 4 times longer than a ShadingRate of 1.0; the effect on overall render time depends on other factors but will likely be somewhat less than 4 times.


Left to right:  2d FFT for shading rates of 4, 2, 1, 0.5, 0.25, .125, and 0.0625 (with smooth shading interpolation).


1d FFT for various shading rates.

With a shading rate of 0.5, there is still significant attenuation (about -3dB, or 50%) at the Nyquist limit.  It is recommended to use a shading rate of 0.25 or smaller to minimize detail loss.


Texture Filtering

There are a number of texture filters available:

box
 aliases too much
disk
 supposed to avoid mipmap artifacts, but expensive and also aliases too much
gaussian
 best filter for color, not smooth enough for displacements
radial-bspline
 best filter for displacements, too blurry for color
lagrangian
 supposed to be 4x faster, but has strange artifacts

Prman texture filtering is anisotropic, but only in a rectangular way.  That is, you can scale the filter rectangle separately in s and t, but the filter area must remain a rectangle.  When the texture coordinates don't form a rectangle (i.e. when the shading grid and texture axes are not aligned) the filter rectangle is formed from the bounding box of the coordinates which which results in overly soft filtering.  Fully anisotropic filtering (using "elliptical weighted averaging") is reported to be in development but is not yet available.  But even when it is, it will still likely be somewhat soft as elliptical filtering is typcially based on the gaussian filter which is a soft filter.

Our [Disney's] proprietary texture system, Ptex, uses the Mitchell filter with controllable sharpness. Smooth filtering (sharpness=0) is equivalent to bspline filtering and is good for displacements, and sharp filtering (sharpness=1) is good for color. Values in-between are also possible (0.4 is similar to gaussian). Because the per-face UVs are always grid-aligned, the filter area will always be a rectangle and proper anisotropic filtering will be achieved. A paper on Ptex was published in the 2008 Eurographics. Go here to find out more about this texture format and to request a copy of the paper.

Procedural Filtering

Anti-aliasing of a procedural-noise-based shader generally involves attenuating noise octaves based on the filter size.  An optimally attenuated noise function is now available with the "wnoise" shadeop.  However, one problem that remains is that the filtering is isotropic.  On silhouettes where one dimension should be filtered more than the other, both dimensions must be filtered equally and thus too much detail is lost.  Baking a procedural noise pattern into a texture (at a slightly higher res than needed) will then enable anisotropic filtering and generally produce a more detailed result.


PixelSamples

The PixelSamples setting controls the number of times the micropolygons are sampled for each pixel.  Prman generates a "subpixel" image that is PixelSamples times larger and then uses a 2d "pixel filter" to reduce the image to the final resolution.

    E.g.
       PixelSamples 5 5
       Format 2048 1280

    Prman renders a 10240 by 6400 image and then reduces it to 2048 x 1280 using the PixelFilter

A special mode can be used to turn off the pixel filtering (Hider "hidden" "subpixel" [1.0]) and render the subpixel image.  This can be useful for debugging or for filtering in another package like Shake.

  
Left: subpixel (PixelSamples 5 5) render, Right: final render after filtered reduction

Notice the noisy edges in the above image, particularly on geometric silhouettes.  This is due to stochastic sampling which jitters the micropolygon sample locations to help anti-alias sharp edges.  The amount of noise is reduced as the number of pixel samples is increased. 

   
An 8x8 checker pattern with its 2d FFT inset.  All of the energy is concentrated in the fundamental frequency of 1/16 and the odd harmonics (3/16, 5/16, 7/16).  On the right, the checker pattern was stochastically sampled.  The even harmonics are slightly emphasized, but otherwise the main effect is the addition of white noise with a strength of about 20%.  Doubling the pixel samples appears to cut the strength of the noise in half.  Note that the 20% noise level is particular to the frequency content of this image, but is fairly typical.

In addition to scaling the resolution of the subpixel image, PixelSamples also determines the number of filter kernel samples as shown in the figures below.



Notes:

At a bare minimum, PixelSamples should be at least high enough to sample each micropolygon twice in each direction (sampling only once per polygon defeats the purpose of smooth shading and adds a lot of noise to the image).  The minimum reasonable value is thus 2 / sqrt(ShadingRate) and is shown in the table below.  Increasing PixelSamples further doesn't affect shading cost, but does become expensive at very high levels (20 or more).  More PixelSamples will generally look better (with less noise), but have negligible effect on beyond a certain point (given by the Recommended PixelSamples values below).  Note that more samples may be needed for other reasons such as motion blur.

Recommended PixelSample Values
Shading Rate
 Min PixelSamples for Shading Rate
 Recommended PixelSamples for Shading Rate
1.0
 2 2
 4 4
0.5
 4 4*
 6 6
0.25
 4 4
 8 8
0.1
 6 6
 10 10
* 3 3 would normally be sufficient, but using an odd number of samples is not recommended for some filters as previously discussed

A comparison of frequency response vs pixel samples (using mitchell 4.0 filter) is shown here.  For each shading rate, increasing the number of pixel samples beyond a certain point has no visible effect.


ShadingInterpolation

ShadingInterpolation has two options:
"constant" - micropolygons are flat shaded
"smooth" - micropolygons are smooth shaded (aka "Gouraud" shading or bi-linear interpolation)

8x8 pixel samples with randomly colored, pixel-sized micropolygons rendered as subpixel image:
      
Constant Shading       Smooth Shading


Frequency response vs ShadingInterpolation, using a ShadingRate of 1.0.

Constant shading is like "box" filtering and is sharper than smooth shading, but has significant aliasing.  Smooth shading is like "triangle" filtering.  Note that smooth shading is required to avoid certain artifacts), particularly with displacements, so it's not really a choice (see "centered derivatives" discussion in prman docs); but the shading interpolation affects the overall frequency response and thus must be taken into account. 


PixelFilter

There is a wide choice of pixel filters, each with a characteristic kernel shape and a settable kernel width.  Unfortunately, there seems to be no good published advice for choosing a filter kernel or setting the appropriate width.  What's more, you can't compare filters by setting their kernel widths to be equal; for a fair comparison, you must know the optimal kernel width for each filter and that information is not easily obtainable (and is certainly not in the prman docs)!  For the filters shown below, the "canonical" widths were gathered from a number of sources including original research papers, by comparing with Shake (where possible), and by examining the FFTs.  The blackman-harris filter's 3.3 width in particular was determined by matching the frequency response to the gaussian.

Here are all the PRMan filter kernel profiles (with canonical widths):


The top row of filters approximate the infinite gaussian function, exp(-w * x^2).  The box filter just averages all values within the box.  If you apply a 1 unit wide box filter twice, you get a 2 unit triangle, also known as a linear b-spline; three times, a 3 unit quadratic b-spline; four times, a 4 unit cubic b-spline; and so on; at the limit you'll have the infinite gaussian (which is infinitely smooth).  But in a practical implementation, the gaussian function must be windowed.  PRMan's "gaussian" filter uses a box window which adds aliasing similar to a box filter (but attenuated).  A wider window would reduce the aliasing, but the window width is not settable in PRMan.  The blackman-harris filter has a gaussian-like kernel that goes smoothly to zero at the edges and has much less aliasing; it is built from cosines, a0 + a1 * cos(pi*x) + a2 * cos(2*pi*x) + a3 * cos(3*pi*x), where a0..a3 are carefully chosen constants.  The lower order gaussian-like filters (like box and triangle) are much sharper, but have strong aliasing.

The bottom row of filters approximate the infinite sinc function, sin(x)/x, with alternating (and progressively smaller) positive and negative lobes.  The filters are shown left to right in order of increasing sharpness (which is relative to the size of the negative lobes).  Mitchell and catmull-rom are cubic filters with single, negative side lobes.  The lanczos filter (named after a Hungarian mathmetician and pronounced "LAHN-tsosh") is a smoothly windowed sinc with very good performance.  The bessel filter is another variant of sinc that is softer than lanczos due to the wider lobes and lower middle peak.  Beware of a bug in bessel with odd filter widths - the sample at 0.0 is too low (at least in the PRMan 13.0).  The sinc filter is just the straight sinc function windowed with a box and thus will have significant aliasing for smaller widths.


Frequency response of the recommended filters, in order of increasing sharpness.


Negative lobes and ringing.  The infinite sinc is theoretically the ideal filter, perfectly reconstructing sampled periodic signals below the Nyquist limit, and perfectly suppressing higher frequencies and the sinc-like filters do preserve the most detail.  But for non-periodic signals (like the step that occurs at the edge of an object) this results in ringing due to the oscillating lobes.  With one negative lobe such as with mitchell or catmull-rom, the ringing is confined to a single band that merely increases the contrast at the edge.  With an additional positive lobe (as shown in the bessel, lanczos, and sinc filters above) there will be two rings - one dark and one light - creating a halo effect.  Filters will more lobes will have more rings that look like pond ripples.  For this reason, only filters with a single negative side lobe and no additional lobes are recommended.  This means confining lanczos and sinc to widths of 4.

Radial vs. separable filters.  The disk, gaussian, and catmull-rom filters are radial which means that the 1d filter kernel is revolved to produce a radially-symmetric 2d kernel.  The rest of the filters are separable which means that the 1d kernels are computed separately in x and y and then multipled together to produce the 2d kernel.  The "separable-catmull-rom" filter has a 1d kernel that is identical to the radial "catmull-rom", but is extended to 2d rectangularly as shown below.  Note how the negative lobes cancel out on the diagonal for the separable-catmull-rom but form a symmetric "moat" around the radial catmull-rom.



Radial filters are theoretically better at suppressing sampling grid structure.  But, the disk filter has too much aliasing to be useful; the gaussian filter is windowed with a box and thus behaves more like a separable filter than a radial one; and the radial catmull-rom filter is much sharper than the 1d and separable forms due to the enlarged negative region and the higher-than-one central peak resulting from normalization.  Ironically, blackman-harris, the apparently best-performing radial filter isn't actually radial!  This doesn't prove that radial filters are bad, just that they don't behave as expected from the 1d analysis.

Separable 2d filters on the other hand perform equivalently to their 1d counterparts.  Note that all of the Shake filters are separable and this apparently has not been an issue. 

Filter width.  All filters have a width parameter (actually there are separate "xwidth" and "ywidth" parameters though the 2 values are generally the same).  But the behavior of the width parameters varies from filter to filter.  For the box, disk, triangle, gaussian, blackman-harris, and mitchell filters, the width parameter controls the kernel width.  The kernel is simply scaled spatially, and the frequency response is similarly scaled.  Put another way, this is like controlling the blur width in Shake.  For these filters, there are canonical widths that put the cutoff frequency near the Nyquist limit.  These are the widths shown in the profiles above.  The Blackman-Harris filter doesn't have a published canonical width that I could find, but I found that the 3.3 value closely matches the frequency response of PRMan's Gaussian 2.0 filter.  The canonical widths should be considered minimum widths.  Larger widths can be used to reduce aliasing.  But to avoid a result that is too blurry, the width should probably not be scaled much beyond about 25% larger than the canonical width.

For the remaining filters (catmull-rom, separable-catmull-rom, bessel, lanczos, and sinc), the width parameter controls the kernel window width and thus controls the number (and in some cases the strength) of the side lobes.  The frequency response however is fixed, and wider kernels just increase the sharpness of the image.  Also note that both forms of the catmull-rom filter have a kernel that goes to zero beyond a width of 4 and thus values larger than 4 have no effect (other than increasing the cpu time).  Also note that the kernel window is applied as a rectangle, so even though a "catmull-rom 2.0" kernel will have no negative lobes in 1d, it will have partial negative lobes on the diagonal in 2d.  The "separable-catmull-rom 2.0" filter will have no negative lobes in 1d or 2d (but clipping the kernel like this will add significant aliasing and is not recommended).  With these filters, going below 4 is not recommended due to the increased aliasing, and going above 6 is not recommended due to the excessive ringing as previously discussed.

Kernel plots and 1d FFT images for all of the prman filters over a range of widths are shown
here)

2d FFT images of the filters with three different shading rates (1.0, 0.5, and 0.25) and using 8x8 pixel samples and smooth shading are shown in:
fft2d_1.0
fft2d)_0.5
fft2d_0.25

Based on these tests, I would recommended the following filters and filter widths:
Minimum Width
 Maximum Recommended Width
softest blackman-harris
 3.3
 4.0
.
 mitchell 4.0 5.0
.
 separable-catmull-rom 4.0 4.0
sharpest lanczos
 4.0 6.0 (be careful of ringing!)
For the first three, larger widths will be softer, but will produce less aliasing.
For lanczos, larger widths will be sharper, but with increased ringing.


Shake filters.  The primary Shake filters all have PRMan equivalents:

 Shake Filter PRMan Filter
softest gauss gaussian 2.0, blackman-harris 3.3
.
 mitchell (default enlargement filter) mitchell 4.0
.
 lanczos lanczos 6.0
sharpest sinc (default reduction filter) sinc 4.0

Shake also has box and triangle filters, but because of the way shake pre-filters the image, these are not equivalent to the prman versions.


Adjustable Mitchell filter.  The Mitchell filter is really a family of filters that includes Catmull-Rom, the cubic b-spline, and everything in between.  Mitchell can even go sharper than Catmull-Rom.  The sharpness parameter is not controllable in PRMan, but it can be easily implemented as a Rif plugin.

Here are are kernel plots and frequency responses for the mitchell 4.0 filter with a sharpness values ranging from 0.0 to 1.8:

The response is flattest at a sharpness of 1.2 and is still pretty flat up to about 1.5.  Values above 1.5 should probably be avoided.

The mitchell filter can provide sharpness that is comparable to most of the builtin filters:
PRMan filter
 comparable sharpness
for mitchell 4.0 filter
 comments
(b-spline)
 0.0
 exact match
gaussian 2.0
blackman-harris 3.3
 0.4
separable-catmull-rom 4.0
 1.0
 exact match
lanczos 4.0
 1.0
lanczos 5.0
 1.25
 larger lanczos widths have a second positive lobe
and will exhibit halo artifacts
sinc 4.0
 1.4
 mitchell has much flatter response!
sinc 5.0
 1.5
 sinc 5.0 is much flatter than sinc 4.0 (comparable to mitchell),
but has second positive lobe and will exhibit halo artifacts


Notice in the above images that as the sharpness increases, the attenuation at the Nyquist frequency decreases (from about 10db at sharpness 0.4 to only about 3db at sharpness 1.5).  As an alternative, a constant Nyquist attenuation can be achieved by matching the filter width to the sharpness level as shown below.  This should give consistent anti-aliasing for a range sharpness values, though the sharpness of the image will be slightly reduced.


Mitchell filters with matched sharpness level and filter width.   Sharpness 0.7, 4.0 is close to the "standard" Mitchell filter.  All of these values have about an 8dB attenuation at the Nyquist frequency.


Image comparison

   
Mitchell 4.0 filter w/ a range of ShadingRate settings (w/ standard sharpness of 0.667).

   
Mitchell 4.0 filter w/ a range of sharpness settings (ShadingRate 0.25).

Additional filters are shown in diner.  All images used 5x5 PixelSamples.


Summary of Recommendations