Normal Mesh Compression

Andrei KhodakovskyCaltech

Igor GuskovCaltech

original18111B (e:1.77, p:75dB)3037B (e:8.1, p:62dB)1225B (e:20, p:54dB)549B (e:54, p:45dB)

Figure 1: Partial reconstructions from a progressive encoding of the dinosaur model. File sizes are given in bytes, errors in multiples of 10−4

and PSNR in dB. The right most model is the original which has 14K vertices (model courtesy of Cyberware).

Abstract

Normal meshes were recently introduced as a new way to repre-sent geometry. A normal mesh is a multiresolution representationwhich has the property that all details lie in a known normal direc-tion and hence the mesh depends only on a single scalar per vertex.Such meshes are ideally suited for progressive compression. Wedemonstrate such a compression algorithm for normal meshes rep-resenting complex, arbitrary topology surfaces as they appear in 3Dscanning and scientific visualization. The resulting coder is shownto exhibit gains of and additional 2-5dB over the previous state ofthe art.

1 Introduction

The unrelenting growth of computing power of personal comput-ers and recent progress in shape acquisition technology facilitatethe wide use of highly detailed meshes in industry and entertain-ment. Similarly, scientific visualization applications tend to pro-duce ever finer meshes, such as iso-surfaces. In their raw, ir-regular form, acquired meshes are complex and often unmanage-able due to their sheer size and irregularity. It is therefore im-portant to find more efficient and compact representations. Al-gorithms for efficient encoding of such meshes have been de-scribed in both single rate [29, 9, 27, 22, 21] and progressive set-tings [19, 20, 16, 12, 3, 26, 1]. For an overview of 3D geometrycompression see [28].

One should recognize the fact that compression is always a trade-off between size and accuracy. That is especially true for meshesthat come from shape acquisition or iso-surface extraction, whichalways carry sampling error and acquisition noise. Hence the com-pression can be lossy as long as the approximation error is com-parible with the sampling error. Recently, a number of efficient“remeshing” techniques have appeared that replace the originalmesh with a mesh consisting of a number of “regular” pieces, such

as B-spline [8], NURBS [14], or subdivision connectivity [15, 7]patches. Naturally, one should also expect that the remeshed modelshould behave much better with regards to compression algorithms.This expectation was confirmed in [13], where the MAPS algo-rithm [15] was included as part of a progressive geometry coder. Inparticular, the paper makes it clear that any mesh representation canbe considered as having three components: geometry, connectivity,and parameterization; moreover, the last two components are notrelevant for the representation of the geometry. For semi-regularmesh hierarchies, one can make the (reasonable) assumption thatthe normal component of the detail coefficients stores the geometricinformation, whereas the tangential components carry the paramet-ric information (see also [10, 5]).

It is clear that many existing remeshing algorithms remove al-most all of the connectivity information from the mesh, and alsoreduce the parametric information. One may wonder: is it possi-ble to get rid of the parametric information entirely? The answeris yes as demonstrated in [11]. In particular, fixing the transformalgorithm to use unlifted Butterfly wavelets it is possible to builda semi-regular mesh whose details lie in the local normal direc-tion. Consequently, the geometry of “normal” meshes is fully rep-resented by one scalar per vertex, instead of the usual three. There-fore it is natural to use normal meshes for compression as we do inthis paper.

Contribution The goal of this paper is to demonstrate the addi-tional coding gains possible when employing normal semiregularmeshes rather than standard semiregular remeshes (such as thoseproduced by MAPS [15]).

2 Compression Algorithms

The progressive geometry coding described in [13] requires threenecessary components: a remeshing algorithm, a wavelet trans-form, and a zerotree coder. In [13] the MAPS remesher [15] was

used, followed by the Loop or Butterfly wavelet transform. In thispaper, we are using the normal remesher of [11] which was specif-ically designed to produce detail coefficients with no tangentialcomponents when an unlifted Butterfly wavelet transform is appliedto the produced semi-regular mesh. We use the same zerotree coderas in [13]. The following sections will briefly overview the algo-rithms used for compression. For a more detailed exposition, thereader is referred to [13] and [11].

2.1 Normal Meshes

The normal remesher starts with an arbitrary irregular mesh, andproceeds to build a semi-regular mesh hierarchy approximating theoriginal model. The algorithm is described in [11] and consists oftwo stages. Using mesh simplification, a base mesh is chosen thatis topologically equivalent to the original mesh. The connectivityof this base mesh will eventually become the connectivity of thecoarsest level in the semi-regular hierarchy. Also at this stage anet of surface curves is initialized that splits the irregular mesh intoa number of non-intersecting regions (these regions are in a one-to-one correspondence with the coarsest level faces of the semi-regular mesh we are constructing). Next, the net of surface curvesis propagated to the finest level of the original irregular mesh, and arelaxation of global vertex positions within the surface is performedto even out their distribution and improve aspect ratios of the basemesh triangles.

In the second stage of the algorithm, a “piercing procedure” isapplied recursively to obtain positions of finer level points of thesemi-regular mesh. Thus, the semi-regular mesh is refined and, tomaintain the status quo, the corresponding regions of the mesh aresplit into smaller subregions. A global adaptive parameterization ismaintained on the original mesh in order to keep the piercing pro-cess under control and to enable fast construction of surface curves.The described two-stage process produces a semi-regular mesh thathas mostly normal detail coefficients except for a small number oflocations where the piercing did not find any “valid” intersectionpoint with the original surface. The percentage of non-normal co-efficients varies depending on the geometric properties of a givenmesh and the corresponding coarse level points chosen in the firststage of the algorithm. Typically, the number of non-normal coeffi-cients is below 10% for adaptive meshes that have the same numberof vertices as the original irregular mesh. For a detailed descriptionof the algorithm refer to [11].

2.2 Wavelet Transform

The wavelet transform replaces a fine semi-regular mesh with somecoarsest level base mesh and a sequence of wavelet coefficients ex-pressing the difference between successive levels of the mesh hier-archy. In [13] a novel Loop [17] wavelet transform was described.It has the advantage that the inverse transform uses Loop subdi-vision. Several other methods for building wavelet transforms onsemi-regular meshes exist [18, 24]. In this work we use the unliftedversion of Butterfly wavelets [6, 30] because exactly the same trans-form is used to produce normal meshes. A detailed description ofthe construction of Butterfly wavelets can be found in [24].

Note that the Butterfly wavelet transform uses finite filters forboth analysis and reconstruction, and is therefore faster than theforward Loop transform which requires solving a sparse linear sys-tem. The Loop reconstruction filter has support of the same sizeas the Butterfly filter. On the other hand, we found that the Loopwavelet transform typically yields better visual appearance than theButterfly transform, though comparable error measures.

Typically, the x, y, and z wavelet components are corre-lated [13]. We exploit this correlation by expressing wavelet coeffi-cents in local frames induced by the coarser level. This is especially

true for normal meshes. Almost all wavelet coefficients computedfor normal meshes have only a normal component. Figure 2 showshistograms of the latitude angles θ (the angle from the normal axis)of the Butterfly and Loop wavelet coefficients in a local coordinateframe. Since the normal mesh is built using butterfly subdivisionalmost all Butterfly coefficients have only normal components.

0 90 180 0 90 180

Figure 2: Histograms of the Loop (left) and Butterfly (right) waveletcoefficients latitude θ angle for the Venus head model in a localframe.

2.3 Zerotree Coding

We encode components of wavelet coefficients separately, that is,our coder essentially consists of three independent zerotree coders.The bits from the three coders are interleaved to maintain progres-sivity.

A general principle of wavelet coefficient encoding is to send thehighest order bits of the largest magnitude coefficients first. Theywill make the most significant contributions towards reducing error.The zerotree algorithm [25, 23, 4] groups all coefficients in hierar-chical zerotree sets such that with high probability all coefficientsin a given set are simultaneously below threshold.

Figure 3: A coarse edge (left) is parent to four finer edges of thesame orientation (right).

The main distinction of our setting from the image case is theconstruction of the zerotrees. For images, one associates the coef-ficients with a quadrilateral face and the trees follow immediatelyfrom the face quadtree. For semi-regular mesh hierarchies, the maininsight is that while scale coefficients are associated with vertices,wavelet coefficients have a one to one association with edges of thecoarser mesh. Vertices do not have a tree structure, but edges do.Each edge is the parent of four edges of the same orientation in thefiner mesh as indicated in Figure 3. Hence, each edge of the basedomain forms the root of a zerotree; it groups all the wavelet coeffi-cients of a fixed wavelet subband from its two incident base domaintriangles.

The scale coefficients from the coarsest level are quantized uni-formly and also progressively encoded. Each bitplane is sent asthe zerotree descends another bit plane. Finally, the output of thezerotree algorithm is encoded using an arithmetic coder.

3 Adaptive Reconstruction

There is a trade-off between remeshing error and the size of the re-sulting semi-regular mesh. With adaptive remeshing this trade-offis local. We refine the mesh where the error is maximal and leaveit coarse where the remeshing error is small (Figure 5). Note, thatthe coarse remesh and its refinement give the same compressionperformance at low bit-rates, since refinement usually introduces

venus

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Figure 4: Rate distortion for our coder with Butterfly and Loop wavelets using MAPS and normal meshes, TG, and CPM coders for the Venusmodel. On the left relative L2 error in multiples of 10−4 as a function of bits/vertex. On the right the PSNR in dB.

small details at finer levels. These details are ignored by the ze-rotree coder at low bit-rates.

Figure 5: Adaptive normal mesh for the skull (19138 vertices) withrelative L2 error of 0.02% and relative L∞ error of 0.09%. Thebase mesh is a tetrahedron (4 vertices) while the original mesh has20002 vertices (model courtesy of Headus).

The compression algorithm is also adaptive. All regions whichare not subdivided to the finest level define zero wavelet coeffi-cients. These coefficients “virtually” exist as an extension of non-uniform zerotrees. The zerotree coder is naturally adaptive: Thereis no difference between non-uniform and uniform zerotrees withzeros attached to extra nodes of the latter. During the reconstructionstep we subdivide faces only when we decode a wavelet coefficientswhich belongs to those faces. After we decode all coefficients wesubdivide all faces until we meet an adaptive flatness threshold [31].This step is important since we want to produce smooth surfaceseven at low bit-rates. The flatness threshold is controlled by theuser and can be chosen depending on the reconstruction bit-rate.

4 Results and Discussion

We measured the performance of our coder [13] with normalmeshes using both Butterfly and Loop wavelet transforms. Forcomparison we use the CPM coder of Pajarola and Rossignac [19]and the single-rate coder of Touma and Gotsman [29]. We plottedrate-distortion curves for Touma-Gotsman coder by changing thecoordinate quantization between 8 and 12 bits. Also, we compareour results with the performance of the coder of [13] for the Venus,horse and rabbit models.

Error was measured using the publicly available METROtool [2]. All graphs show error between the reconstruction andthe irregular original model as a function of the number of bitsper vertex of the original mesh. All errors are given as PSNR(PSNR= 20 log10 BBoxDiag/L2-error).

Venus rabbit horse dinosaur skull molecule

Vor 50K 67K 48K 14K 20K 10KVb 42 71 112 128 4 53Er, 10

−4 0.47 0.47 0.51 1.7 2.3 6.3rn 1580 759 754 2973 1861 794

Table 1: Number of vertices in original models (Vor), base domainof remeshes (Vb), relative L2 error (Er) in units of 10−4, and thenumber of non-normal coefficients (rn).

We found that the coding with normal meshes has better perfor-mance compared to MAPS meshes. Even using a Loop wavelettransform we observe an improvement. For example, for the horsemodel Loop (Figure 6) wavelets on the normal mesh allow 1.5 times(3.5dB) smaller distortion than on the MAPS mesh. Note, that animprovement in the high bit-rate limit is due to the smaller remesh-ing error. More important is that we have better distortion for allbit-rates which happens because of the normality of the mesh. Theadditional improvement of about a factor of 1.2 (1.5dB) comes fromusing Butterfly wavelets summing up to a total of 5dB improvementcompared to [13]. For the rabbit and Venus models MAPS meshesare closer to normal so the coders using MAPS and normal mesheshave closer performance. However we still observe an improve-ment of at least 2dB.

The fact that encoding with Loop wavelets benefits from But-terfly normal meshes is remarkable. Experimentally we found thatif we treat wavelet components completely separately, Loop nor-mals compress better than Butterfly normals. Given this evidencewe should expect further improvement with Loop based normalmeshes.

Figure 7 shows rate-distortion curves for the dinosaur, skull, andmolecule models. These have coarser original meshes, thereforewe allow a larger remeshing error (see the discussion on the natu-ral choice of remeshing error in [13]). Note, that the remesh of theskull model has less vertices than the original, and has a reasonablysmall remeshing error. The base domain for the skull is a tetrahe-dron. For all the meshes mentioned in this paper, the remeshing stepof our algorithm takes less than 10 minutes to complete. The for-ward Loop transform of the coder requires the solution of a sparselinear system, which takes about 30 seconds. Loop reconstructionand both Butterfly forward and backward transforms take 2-3 sec-

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Figure 6: Rate-distortion for rabbit and horse models showingPSNR as a function of bits/vertex.

onds. Zerotree encoding and decoding take about 1 second.

5 Conclusions

In this paper we show that normal meshes improve performanceof progressive geometry compression. We observe improvementof 2-5 dB depending on the model. Also we described adaptivecompression which allows finer control on the number of verticesin the reconstructed model.

One of the directions for the future work is the design of normalmeshes using Loop wavelets.

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