TensoRF: Tensorial Radiance Fields
Anpei Chen1⋆

Zexiang Xu2⋆

Andreas Geiger3

Jingyi Yu1

Hao Su4

1

3

ShanghaiTech University 2 Adobe Research
University of Tübingen and MPI-IS, Tübingen 4 UC San Diego
https://apchenstu.github.io/TensoRF/

Abstract. We present TensoRF, a novel approach to model and reconstruct radiance fields. Unlike NeRF that purely uses MLPs, we model theradiance field of a scene as a 4D tensor, which represents a 3D voxel gridwith per-voxel multi-channel features. Our central idea is to factorize the4D scene tensor into multiple compact low-rank tensor components. We
demonstrate that applying traditional CANDECOMP/PARAFAC (CP)decomposition – that factorizes tensors into rank-one components withcompact vectors – in our framework leads to improvements over vanilla
NeRF. To further boost performance, we introduce a novel vector-matrix(VM) decomposition that relaxes the low-rank constraints for two modesof a tensor and factorizes tensors into compact vector and matrix factors.
Beyond superior rendering quality, our models with CP and VM decompositions lead to a significantly lower memory footprint in comparison toprevious and concurrent works that directly optimize per-voxel features.
Experimentally, we demonstrate that TensoRF with CP decomposition
achieves fast reconstruction (< 30 min) with better rendering quality
and even a smaller model size (< 4 MB) compared to NeRF. Moreover,TensoRF with VM decomposition further boosts rendering quality andoutperforms previous state-of-the-art methods, while reducing the reconstruction time (< 10 min) and retaining a compact model size (< 75MB).

1Introduction

Modeling and reconstructing 3D scenes as representations that support highquality image synthesis is crucial for computer vision and graphics with various
applications in visual effects, e-commerce, virtual and augmented reality, androbotics. Recently, NeRF [31] and its many follow-up works [61,26] have shown
success on modeling a scene as a radiance field and enabled photo-realistic
rendering of scenes with highly complex geometry and view-dependent appearance
effects. Despite the fact that (purely MLP-based) NeRF models require small
memory, they take a long time (hours or days) to train. In this work, we pursue⋆

Equal contribution.
Research done when Anpei Chen was in a remote internship with UCSD.

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A. Chen, Z. Xu et al.
Scene

TensoRF-VM

Quantitative Results on the Synthetic NeRF Dataset
PlenOctrees
SNeRG

Plenoxels
DVGO

Ours

NeRF

PSNR Method
31.01 NeRF
30.38 SNeRG
31.71 PlenOctrees
31.71 Plenoxels
31.95 DVGO
31.56 Ours-CP-384-30k
32.52 Ours-VM-192-15k
33.14 Ours-VM-192-30k(Point sizes correspond to PNSRs)

Fig. 1: Left: We model a scene as a tensorial radiance field using a set of vectors(v) and matrices (M) that describe scene appearance and geometry along theircorresponding axes. These vector/matrix factors are used to compute volume
density σ and view-dependent RGB color via vector-matrix outer products forrealistic volume rendering. Right: In comparison with previous and concurrent
methods, our TensoRF models can achieve the best rendering quality and are
the only methods that can simultaneously achieve fast reconstruction and highcompactness. (Our models are denoted with their decomposition techniques,
number of components, and training steps.)a novel approach that is both efficient in training time and compact in memory
footprint, and at the same time achieves state-of-the-art rendering quality.
To do so, we propose TensoRF, a novel radiance field representation that is
highly compact and also fast to reconstruct, enabling efficient scene reconstruction
and modeling. Unlike coordinate-based MLPs used in NeRF, we represent radiancefields as an explicit voxel grid of features. Note that it is unclear whether voxel
grid representation can benefit the efficiency of reconstruction: While previous
work has used feature grids [26,59,17], they require large GPU memory to store
the voxels whose size grows cubically with resolution, and some even requirepre-computing a NeRF for distillation, leading to very long reconstruction time.
Our work addresses the inefficiency of voxel grid representations in a principledframework, leading to a family of simple yet effective methods. We leverage thefact that a feature grid can naturally be seen as a 4D tensor, where three of its
modes correspond to the XYZ axes of the grid and the fourth mode represents thefeature channel dimension. This opens the possibility of exploiting classical tensor
decomposition techniques – which have been widely applied to high-dimensional
data analysis and compression in various fields [23] – for radiance field modeling.
We, therefore, propose to factorize the tensor of radiance fields into multiple lowrank tensor components, leading to an accurate and compact scene representation.
Note that our central idea of tensorizing radiance fields is general and can bepotentially adopted to any tensor decomposition technique.
In this work, we first attempt the classic CANDECOMP/PARAFAC (CP)decomposition [6]. We show that TensoRF with CP decomposition can alreadyachieve photo-realistic rendering and lead to a more compact model than NeRF
that is purely MLP based (see Fig. 1 and Tab. 1). However, experimentally, tofurther push reconstruction quality for complex scenes, we have to use morecomponent factors, which undesirably increases training time.

TensoRF: Tensorial Radiance Fields

3

Therefore, we present a novel vector-matrix (VM) decomposition techniquethat effectively reduces the number of components required for the same expression
capacity, leading to faster reconstruction and better rendering. In particular,
inspired by the CP and block term decomposition [12], we propose to factorize thefull tensor of a radiance field into multiple vector and matrix factors per tensor
component. Unlike the sum of outer products of pure vectors in CP decomposition,
we consider the sum of vector-matrix outer products (see Fig. 2). In essence,
we relax the ranks of two modes of each component by jointly modeling twomodes in a matrix factor. While this increases the model size compared to pure
vector-based factorization in CP, we enable each component to express more
complex tensor data of higher ranks, thus significantly reducing the required
number of components in radiance field modeling.
With CP/VM decomposition, our approach compactly encodes spatiallyvarying features in the voxel grid. Volume density and view-dependent color can be
decoded from the features, supporting volumetric radiance field rendering. Becausea tensor expresses discrete data, we also enable efficient trilinear interpolationfor our representation to model a continuous field. Our representation supports
various types of per-voxel features with different decoding functions, including
neural features – depending on an MLP to regress view-dependent colors fromthe features – and spherical harmonics (SH) features (coefficients) – allowing
for simple color computation from the fixed SH functions and leading to arepresentation without neural networks.
Our tensorial radiance fields can be effectively reconstructed from multi-view
images and enable realistic novel view synthesis. In contrast to previous works that
directly reconstruct voxels, our tensor factorization reduces space complexity fromO(n3 ) to O(n) (with CP) or O(n2 ) (with VM), significantly lowering memory
footprint. Note that, although we leverage tensor decomposition, we are notaddressing a decomposition/compression problem, but a reconstruction problem
based on gradient decent, since the feature grid/tensor is unknown. In essence,
our CP/VM decomposition offers low-rank regularization in the optimization,
leading to high rendering quality. We present extensive evaluation of our approachwith various settings, covering both CP and VM models, different numbers ofcomponents and grid resolutions. We demonstrate that all models are able toachieve realistic novel view synthesis results that are on par or better thanprevious state-of-the-art methods (see Fig. 1 and Tab. 1). More importantly, ourapproach is of high computation and memory efficiency. All TensoRF models
can reconstruct high-quality radiance fields in 30 min; our fastest model with
VM decomposition takes less than 10 min, which is significantly faster (about
100x) than NeRF and many other methods, while requiring substantially lessmemory than previous and concurrent voxel-based methods. Note that, unlike
concurrent works [43,32] that require unique data structures and customized
CUDA kernels, our model’s efficiency gains are obtained using a standard PyTorch
implementation. As far as we know, our work is the first that views radiance fieldmodeling from a tensorial perspective and pose the problem of radiance fieldreconstruction as one of low-rank tensor reconstructions.

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A. Chen, Z. Xu et al.

Related Work

Tensor decomposition. Tensor decomposition [23] has been studied for decadeswith diverse applications in vision, graphics, machine learning, and other fields[36,20,51,13,1,19]. In general, the most widely used decompositions are Tucker
decomposition [50] and CP decomposition [6,16], both of which can be seen as
generalizations of the matrix singular value decomposition (SVD). CP decomposition can also be seen as a special Tucker decomposition whose core tensor isdiagonal. By combining CP and Tucker decomposition, block term decomposition(BTD) has been proposed with its many variants [12] and used in many visionand learning tasks [2,58,57]. In this work, we leverage tensor decomposition forradiance field modeling. We directly apply CP decomposition and also introduce
a new vector-matrix decomposition, which can be seen as a special BTD.
Scene representations and radiance fields. Various scene representations,
including meshes [15,52], point clouds [40], volumes [18,41], implicit functions[29,39], have been extensively studied in recent years. Many neural representations [9,62,45,28,4] are proposed for high-quality rendering or natural signal
representation [44,48,25]. NeRF [31] introduces radiance fields to address novel
view synthesis and achieves photo-realistic quality. This representation has beenquickly extended and applied in diverse graphics and vision applications, including generative models [8,33], appearance acquisition [3,5], surface reconstruction[53,35], fast rendering[42,59,17,14], appearance editing [55,27], dynamic capture[24,37] and generative model [34,7]. While leading to realistic rendering anda compact model, NeRF with its pure MLP-based representation has known
limitations in slow reconstruction and rendering. Recent methods [59,26,17] haveleveraged a voxel grid of features in radiance field modeling, achieving fast rendering. However, these grid-based methods still require long reconstruction time andeven lead to high memory costs, sacrificing the compactness of NeRF. Based onfeature grids, we present a novel tensorial scene representation, leveraging tensor
factorization techniques, leading to fast reconstruction and compact modeling.
Other methods design generalizable network modules trained across scenes
to achieve image-dependent radiance field rendering [49,60,54,11] and fast reconstruction [10,56]. Our approach focuses on radiance field representation and only
considers per-scene optimization (like NeRF). We show that our representation
can already lead to highly efficient radiance field reconstruction without any
across-scene generalization. We leave the extensions to generalizable settings as
future work.
Concurrent work. The field of radiance field modeling is moving very fast andmany concurrent works have appeared on arXiv as preprints over the last threemonths. DVGO [47] and Plenoxels [43] also optimize voxel grids of (neural orSH) features for fast radiance field reconstruction. However, they still optimize
per-voxel features directly like previous voxel-based methods, thus requiring
large memory. Our approach instead factorizes the feature grid into compact
components and leads to significantly higher memory efficiency. Instant-NGP[32] uses multi-resolution hashing for efficient encoding and also leads to highTensoRF: Tensorial Radiance Fields

=

+…+

=

+…+

+

+…+

+

5

+…+

Fig. 2: Tensor factorization. Left: CP decomposition (Eqn. 1), which factorizes atensor as a sum of vector outer products. Right: our vector-matrix decomposition(Eqn. 3), which factorizes a tensor as a sum of vector-matrix outer products.
compactness. This technique is orthogonal to our factorization-based technique;potentially, each of our vector/matrix factor can be encoded with this hashingtechnique and we leave such combination as future work.

3

CP and VM Decomposition

We factorize radiance fields into compact components for scene modeling. To doso, we apply both the classic CP decomposition and a new vector-matrix (VM)
decomposition; both are illustrated in Fig. 2. We now discuss both decompositions
with an example of a 3D (3rd-order) tensor. We will introduce how to apply
tensor factorization in radiance field modeling (with a 4D tensor) in Sec. 4.
CP decomposition. Given a 3D tensor T ∈ RI×J×K , CP decomposition
factorizes it into a sum of outer products of vectors (shown in Fig. 2):

  \Tensor = \sum _{r=1}^R \Vector _r^{1} \OuterP \Vector _r^{2} \OuterP \Vector _r^{3} \label {eqn:cpvector} (1)where vr1 ◦ vr2 ◦ vr3 corresponds to a rank-one tensor component, and vr1 ∈ RI ,vr2 ∈ RJ , and vr3 ∈ RK are factorized vectors of the three modes for the rthcomponent. Superscripts denote the modes of each factor; ◦ represents the outer
product. Hence, each tensor element Tijk is a sum of scalar products:

  \Tensor _{ijk} = \sum _{r=1}^R \Vector _{r,i}^{1} \Vector _{r,j}^{2} \Vector _{r,k}^{3} \label {eqn:cpelement} (2)where i, j, k denote the indices of the three modes.
CP decomposition factorizes a tensor into multiple vectors, expressing multiple
compact rank-one components. CP can be directly applied in our tensorial
radiance field modeling and generate high-quality results (see Tab. 1). However,
because of too high compactness, CP decomposition can require many components
to model complex scenes, leading to high computational costs in radiance fieldreconstruction. Inspired by block term decomposition (BTD), we present a new
VM decomposition, leading to more efficient radiance field reconstruction.
Vector-Matrix (VM) decomposition. Unlike CP decomposition that utilizes
pure vector factors, VM decomposition factorizes a tensor into multiple vectors

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A. Chen, Z. Xu et al.

and matrices as shown in Fig. 2 right. This is expressed by
  \Tensor = \sum _{r=1}^{R_1} \Vector _r^{1} \OuterP \Matrix _r^{2,3} + \sum _{r=1}^{R_2}\Vector _r^{2} \OuterP \Matrix _r^{1,3} + \sum _{r=1}^{R_3}\Vector _r^{3} \OuterP \Matrix _r^{1,2} \label {eqn:vmrrr} (3)where M2,3
∈ RJ×K , M1,3
∈ RI×K , M1,2
∈ RI×J are matrix factors for tworrr(denoted by superscripts) of the three modes. For each component, we relax
its two mode ranks to be arbitrarily large, while restricting the third mode to
be rank-one; e.g., for component tensor vr1 ◦ M2,3
r , its mode-1 rank is 1, andits mode-2 and mode-3 ranks can be arbitrary, depending on the rank of thematrix M2,3
r . In general, instead of using separate vectors in CP, we combine
every two modes and represent them by matrices, allowing each mode to beadequately parametrized with a smaller number of components. R1 , R2 , R3 canbe set differently and should be chosen depending on the complexity of eachmode. Our VM decomposition can be seen as a special case of general BTD.
Note that, each of our component tensors has more parameters than acomponent in CP decomposition. While this leads to lower compactness, a VM
component tensor can express more complex high-dimensional data than a CP
component, thus reducing the required number of components when modelingthe same complex function. On the other hand, VM decomposition is still ofvery high compactness, reducing memory complexity from O(N 3 ) to O(N 2 ),compared to dense grid representations.
Tensor for scene modeling. In this work, we focus on the task of modeling andreconstructing radiance fields. In this case, the three tensor modes correspond
to XYZ axes, and we thus directly denote the modes with XYZ to make it
intuitive. Meanwhile, in the context of 3D scene representation, we consider
R1 = R2 = R3 = R for most of scenes, reflecting the fact that a scene candistribute and appear equally complex along its three axes. Therefore, Eqn. 3
can be re-written as
  \Tensor = \sum _{r=1}^{R} \Vector _r^{X} \OuterP \Matrix _r^{Y,Z} + \Vector _r^{Y} \OuterP \Matrix _r^{X,Z} + \Vector _r^{Z} \OuterP \Matrix _r^{X,Y} \label {eqn:vmr} 

(4)

In addition, to simplify notation and the following discussion in later sections,X
YZ
we also denote the three types of component tensors as AXr = v r ◦ Mr ,
Y
Y
XZ
Z
Z
XY
Ar = vr ◦ Mr , and Ar = vr ◦ Mr ; here the superscripts XYZ of A indicate
different types of components. With this, a tensor element Tijk is expressed as  \Tensor _{ijk} = & \sum _{r=1}^{R} \sum _{m} \Comp _{r,ijk}^m \label {eqn:vmrelement}

(5)

X
YZ
Y
Y
XZ
Z
Z
XY
where m ∈ XY Z, AXr,ijk = vr,i Mr,jk , Ar,ijk = vr,j Mr,ik , and Ar,ijk = vr,k Mr,ij .
γXYZSimilarly, we can also denote a CP component as A = vr ◦ vr ◦ vr , and Eqn. 5
can also express CP decomposition by considering m = γ, where the summation
over m can be removed.

TensoRF: Tensorial Radiance Fields

47

Tensorial Radiance Field Representation

We now present our Tensorial Radiance Field Representation (TensoRF). For
simplicity, we focus on presenting TensoRF with our VM decomposition. CP
decomposition is simpler and its decomposition equations can be directly appliedwith minimal modification (like Eqn. 5).
4.1

Feature grids and radiance field.

Our goal is to model a radiance field, which is essentially a function that maps any
3D location x and viewing direction d to its volume density σ and view-dependentcolor c, supporting differentiable ray marching for volume rendering. We leveragea regular 3D grid G with per-voxel multi-channel features to model such a function.
We split it (by feature channels) into a geometry grid Gσ and an appearance grid
Gc , separately modelling the volume density σ and view-dependent color c.
Our approach supports various types of appearance features in Gc , depending
on a pre-selected function S that coverts an appearance feature vector and a
viewing direction d to color c. For example, S can be a small MLP or sphericalharmonics (SH) functions, where Gc contains neural features and SH coefficients
respectively. We show that both MLP and SH functions work well with our model(see Tab.1). On the other hand, we consider a single-channel grid Gσ , whose
values represent volume density directly, without requiring an extra converting
function. The continuous grid-based radiance field can be written as
  \Dens , \Rad = \Grid _\Dens (\Pos ), \ShadFunc (\Grid _\Rad (\Pos ),\Dir ) \label {eqn:rf} (6)where Gσ (x), Gc (x) represent the trilinearly interpolated features from the two
grids at location x. We model Gσ and Gc as factorized tensors.
4.2

Factorizing radiance fields

While Gσ ∈ RI×J×K is a 3D tensor, Gc ∈ RI×J×K×P is a 4D tensor. Here I, J,
K correspond to the resolutions of the feature grid along the X, Y, Z axes, andP is the number of appearance feature channels.
We factorize these radiance field tensors to compact components. In particular,
with the VM decomposition. the 3D geometry tensor Gσ is factorized as
  \Grid _{\Dens } = \sum _{r=1}^{R_\Dens } \Vector _{\Dens , r}^{X} \OuterP \Matrix _{\Dens ,r}^{YZ} + \Vector _{\Dens ,r}^{Y} \OuterP \Matrix _{\Dens ,r}^{XZ} + \Vector _{\Dens ,r}^{Z} \OuterP \Matrix _{\Dens ,r}^{XY} = \sum _{r=1}^{R_\Dens } \sum _{m\in XYZ} \Comp ^m_{\Dens , r} \label {eqn:dense-vmr}(7)

The appearance tensor Gc has an additional mode corresponding to the featurechannel dimension. Note that, compared to the XYZ modes, this mode is often
of lower dimension, leading to a lower rank. Therefore, we do not combine thismode with other modes in matrix factors and instead only use vectors, denoted

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A. Chen, Z. Xu et al.

++− . . 22

Rendering loss
Ray DistanceFig. 3: TensoRF (VM) reconstruction and rendering. We model radiance fields as
tensors using a set of vectors (v) and matrices (M), which describe the scene along
their corresponding (XYZ) axes and are used for computing volume density σ andview-dependent color c in differentiable ray marching. For each shading locationx = (x, y, z), we use linearly/bilinearly sampled values from the vector/matrix
factors to efficiently compute the corresponding trilinearly interpolated values(A(x)) of the tensor components. The density component values ( Aσ (x)) aresummed to get the volume density (σ) directly. The appearance values (Ac (x)) areconcatenated into a vector (⊕[Amc (x)]m ) that is then multiplied by an appearance
matrix B and sent to the decoding function S for RGB color (c) regression.
by br , for this mode in the factorization. Specifically, Gc is factorized as
  \Grid _{\Rad } = \sum _{r=1}^{R_\Rad }& \Vector _{\Rad , r}^{X} \OuterP \Matrix _{\Rad ,r}^{YZ} \OuterP \AppVec _{3r-2} + \Vector _{\Rad ,r}^{Y} \OuterP \Matrix _{\Rad ,r}^{XZ} \OuterP \AppVec _{3r-1} + \Vector _{\Rad ,r}^{Z} \OuterP \Matrix _{\Rad ,r}^{XY} \OuterP \AppVec _{3r} \notag \\ =\sum _{r=1}^{R_\Rad } &\Comp _{\Rad ,r}^X \OuterP \AppVec _{3r-2} +\Comp _{\Rad ,r}^Y \OuterP \AppVec _{3r-1} +\Comp _{\Rad ,r}^Z \OuterP \AppVec _{3r} \label {eqn:rad-vmr}(8)Note that, we have 3Rc vectors br to match the total number of components.
Overall, we factorize the entire tensorial radiance field into 3Rσ +3Rc matrices
X
X(MYσ,rZ ,...,MYc,rZ ,...) and 3Rσ +6Rc vectors (vσ,r,...,vc,r,...,br ). In general, we adopt
Rσ ≪ I, J, K and Rc ≪ I, J, K, leading to a highly compact representation thatcan encode a high-resolution dense grid. In essence, the XYZ-mode vector andX
X
matrix factors, vσ,r, MYσ,rZ , vc,r, MYc,rZ , ..., describe the spatial distributions of
the scene geometry and appearance along their corresponding axes. On the other
hand, the appearance feature-mode vectors br express the global appearance
correlations. By stacking all br as columns together, we have a P × 3Rc matrix B;
this matrix B can also be seen as a global appearance dictionary that abstracts
the appearance commonalities across the entire scene.
4.3

Efficient feature evaluation.

Our factorization-based model can compute each voxel’s feature vector at low
costs, only requiring one value per XYZ-mode vector/matrix factor. We alsoenable efficient trilinear interpolation for our model, leading to a continuous field.

TensoRF: Tensorial Radiance Fields

9

Direct evaluation. With VM factorization, a density value Gσ,ijk of a single
voxel at indices ijk can be directly and efficiently evaluated by following Eqn. 5:

  \Grid _{\Dens ,ijk} = \sum _{r=1}^{R_\Dens } \sum _{m\in XYZ} \Comp ^m_{\Dens ,r,ijk} \label {eqn:dens-vmrelement}(9)Here, computing each Am
σ,r,ijk only requires indexing and multiplying two values
from its corresponding vector and matrix factors.
As for the appearance grid Gc , we always need to compute a full P -channel
feature vector, which the shading function S requires as input, corresponding to
a 1D slice of Gc at fixed XYZ indices ijk:
  \Grid _{\Rad ,ijk} = \sum _{r=1}^{R_\Rad } \Comp ^X_{\Rad ,r,ijk} \AppVec _{3r-2} + \Comp ^Y_{\Rad ,r,ijk} \AppVec _{3r-1} + \Comp ^Z_{\Rad ,r,ijk} \AppVec _{3r} \label {eqn:rad-vmrslice}(10)

Here, there’s no additional indexing for the feature mode, since we compute a full
vector. We further simplify Eqn. 10 by re-ordering the computation. For this, wemdenote ⊕[Amc,ijk ]m,r as the vector that stacks all Ac,r,ijk values for m = X, Y, Z
and r = 1, ..., Rc , which is a vector of 3Rc dimensions; ⊕ can also be considered as
the concatenation operator that concatenates all scalar values (1-channel vectors)
into a 3Rc -channel vector in practice. Using matrix B (introduced in Sec. 4.1)
that stacks all br , Eqn. 10 is equivalent to a matrix vector product:
  \Grid _{\Rad ,ijk} = \AppMat (\oplus [\Comp ^m_{\Rad ,ijk}]_{m,r}) \label {eqn:rad-vmrfeature}(11)

Note that, Eqn. 11 is not only formally simpler but also leads to a simpler
implementation in practice. Specifically, when computing a large number ofvoxels in parallel, we first compute and concatenate Am
c,r,ijk for all voxels ascolumn vectors in a matrix and then multiply the shared matrix B once.
Trilinear interpolation. We apply trilinear interpolation to model a continuous
field. Naı̈vely achieving trilinear interpolation is costly, as it requires evaluationof 8 tensor values and interpolating them, increasing computation by a factor of 8
compared to computing a single tensor element. However, we find that trilinearlyinterpolating a component tensor is naturally equivalent to interpolating its
vector/matrix factors linearly/bilinearly for the corresponding modes, thanks tothe beauty of linearity of the trilinear interpolation and the outer product.
XYZ
For example, given a component tensor AX
with its each tensorr = vr ◦ Mr
X
YZ
element Ar,ijk = vr,i Mr,jk , we can compute its interpolated values as:
  \Comp ^X_r(\Pos ) = \Vector _r^X(x)\Matrix _{r}^{YZ}(y,z) \label {eqn:xinter} (12)where AXr (x) is Ar ’s trilinearly interpolated value at location x = (x, y, z) inthe 3D space, vrX (x) is vrX ’s linearly interpolated value at x along X axis, and
MYr Z (y, z) is MYr Z ’s bilinearly interpolated value at (y, z) in the YZ plane.
Z
Z
XY
Similarly, we have AYr (x) = vrY (y)MXZr (x, z) and Ar (x) = vr (z)Mr (x, y)

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A. Chen, Z. Xu et al.

(for CP decomposition Aγr (x) = vrX (x)vrY (y)vrZ (z) is also valid). Thus, trilinearlyinterpolating the two grids is expressed as
  \Grid _{\Dens }(\Pos ) &= \sum _r \sum _m \Comp ^m_{\Dens ,r}(\Pos ) \label {eqn:dens-vmrinter} \\ \Grid _{\Rad }(\Pos ) &= \AppMat (\oplus [\Comp ^m_{\Rad ,r}(\Pos )]_{m,r}) \label {eqn:rad-vmrinter}(14)
These equations are very similar to Eqn. 9 and 11, simply replacing the tensor
elements with interpolated values. We avoid recovering 8 individual tensor elements for trilinear interpolation and instead directly recover the interpolated
value, leading to low computation and memory costs at run time.
4.4

Rendering and reconstruction.

Equations 6, 12–14 describe the core components of our model. By combining
Eqn. 6,13,14, our factorized tensorial radiance field can be expressed as  \Dens , \Rad =\sum _{r}\sum _m \Comp ^m_{\Dens ,r}(\Pos )\:,\: \ShadFunc (\AppMat (\oplus [\Comp ^m_{\Rad ,r}(\Pos )]_{m,r}), \Dir ) 

(15)

i.e., we obtain continuous volume density and view-dependent color given any
3D location and viewing direction. This allows for high-quality radiance fieldreconstruction and rendering. Note that, this equation is general and describes
TensoRF with both CP and VM decomposition. Our full pipeline of radiancefield reconstruction and rendering with VM decomposition is illustrated in Fig. 3.
Volume rendering. To render images, we use differentiable volume rendering,following NeRF [31]. Specifically, for each pixel, we march along a ray, sampling
Q shading points along the ray and computing the pixel color by
  \Color = \sum _{q=1}^Q \Trans _q (1-\exp (-\Dens _q \Step _q)) \Rad _q, \\Trans _q = \exp (-\sum _{p=1}^{q-1} \Dens _p \Step _p ) \label {eq:raymarching} 

(16)Here, σq , cq are the corresponding density and color computed by our model attheir sampled locations xq ; ∆q is the ray step size and τq represents transmittance.
Reconstruction. Given a set of multi-view input images with known camera
poses, our tensorial radiance field is optimized per scene via gradient descent,minimizing an L2 rendering loss, using only the ground truth pixel colors as
supervision. Our radiance field is explained by tensor factorization and modeledby a set of global vectors and matrices as basis factors that correlate andregularize the entire field in the optimization. However, this can sometimes leadto overfitting and local minima issues in gradient descent, leading to outliers
or noises in regions with fewer observations. We utilize standard regularization
terms that are commonly used in compressive sensing, including an L1 norm loss
and a TV (total variation) loss on our vector and matrix factors, which effectively
address these issues. We find that only applying the L1 sparsity loss is adequate

TensoRF: Tensorial Radiance Fields

11

for most datasets. However, for real datasets that have very few input images(like LLFF[30]) or imperfect capture conditions (like Tanks and Temples [22,26]
that has varying exposure and inconsistent masks), a TV loss is more efficientthan the L1 norm loss.
To further improve quality and avoid local minima, we apply coarse-tofine reconstruction. Unlike previous coarse-to-fine techniques that require uniquesubdivisions on their sparse chosen sets of voxels, our coarse-to-fine reconstruction
is simply achieved by linearly and bilinearly upsampling our XYZ-mode vector
and matrix factors.

5

Implementation details

We briefly discuss our implementation; please refer to the appendix for moredetails. We implement our TensoRF using PyTorch [38], without customized
CUDA kernels. We implement the feature decoding function S as either an MLP
or SH function and use P = 27 features for both. For SH, this corresponds to
3rd-order SH coefficients with RGB channels. For neural features, we use a small
MLP with two FC layers (with 128-channel hidden layers) and ReLU activation.
We use the Adam optimizer [21] with initial learning rates of 0.02 for tensor
factors and (when using neural features) 0.001 for the MLP decoder. We optimize
our model for T steps with a batch size of 4096 pixel rays on a single Tesla
V100 GPU (16GB). We apply a feature grid with a total number of N 3 voxels;
the actual resolution of each dimension is computed based on the shape of thebounding box. To achieve coarse-to-fine reconstruction, we start from an initial
low-resolution grid with N03 voxels with N0 = 128; we then upsample the vectors
and matrices linearly and bilinearly at steps 2000, 3000, 4000, 5500, 7000 with
the numbers of voxels interpolated between N03 and N 3 linearly in logarithmic
space. Please refer to Sec. 6 for the analysis on different total steps (T ), different
resolutions (N ), and different number of total components (3Rσ + 3Rc ).

6

Experiments

We now present an extensive evaluation of our tensorial radiance fields. We
first analyze our decomposition techniques, the number of components, grid
resolutions, and optimization steps. We then compare our approach with previous
and concurrent works on both 360◦ objects and forward-facing datasets.
Analysis of different TensoRF models. We evaluate our TensoRF on theSynthetic NeRF dataset [31] using both CP and VM decompositions with different
numbers of components and different numbers of grid voxels. Table 2 shows theaveraged rendering PSNRs, reconstruction time, and model size for each model.
We use the same MLP decoding function (as described in Sec. 5) for all variants
and optimize each model for 30k steps with a batch size of 4096.
Note that both TensoRF-CP and TensoRF-VM achieve consistently better
rendering quality with more components or higher grid resolutions. TensoRF-CP

12

A. Chen, Z. Xu et al.

achieves super compact modeling; even the largest model with 384 components
and 5003 voxels requires less than 4MB. This CP model also achieves the best
rendering quality in all of our CP variants, leading to a high PSNR of 31.56,
which even outperforms vanilla NeRF (see Tab. 1).
On the other hand, because it compresses more parameters in each component,
TensoRF-VM achieves significantly better rendering quality than TensoRF-CP;
even the smallest TensoRF-VM model with only 48 components and 2003 voxels
is able to outperform the best CP model that uses many more components andvoxels. Remarkably, the PSNR of 31.81 from this smallest VM model (which
only takes 8.6 MB) is already higher than the PSNRs of NeRF and many otherprevious and concurrent techniques (see Tab. 1). In addition, 192 components
are generally adequate for TensoRF-VM; doubling the number to 384 only leads
to marginal improvement. TensoRF-VM with 3003 voxels can already lead tohigh PSNRs close to or greater than 33, while retaining compact model sizes(<72MB). Increasing the resolution further leads to improved quality, but also
larger model size.
Also note that all of our TensoRF models can achieve very fast reconstruction.
Except for the largest VM model, all models finish reconstruction in less than 30
min, significantly faster than NeRF and many previous methods (see Tab. 1).
Optimization steps. We further evaluate our approach with different optimization steps for our best CP model and the VM models with 3003 voxels. PSNRs
and reconstruction time are shown in Tab. 3. All of our models consistently
achieve better rendering quality with more steps. Our compact CP-384 model(3.9MB) can even achieve a PSNR greater than 32 after 60k steps, higher than
the PSNRs of all previous methods in Tab. 1. On the other hand, our VM models
can quickly achieve high rendering quality in very few steps. With only 15k steps,
many models achieve high PSNRs that are already state-of-the-art.
Comparisons on 360◦ scenes. We compare our approach with state-of-the-art
novel view synthesis methods, including previous works (SRN[46], NeRF[31],
NSVF[26]), SNeRG[17], PlenOctrees[59]) and concurrent works (Plenoxels [43],
DVGO[47]). In particular, we compare with them using our best CP model andour VM models (3003 voxels) with 48 and 192 components. We also show a
192-component VM model with spherical harmonics shading function. Table
1 shows the quantitative results (PSNRs and SSIMs) of ours and comparison
methods on three challenging datasets, where we also show the corresponding
batch size, optimization steps, time, and final output model size for each model, tocompare all methods from multiple perspectives. Note that all of our CP and VM
models can outperform NeRF on all three datasets while taking substantially less
optimization time and fewer steps. Our best VM-192 model can even achieve thebest PSNRs and SSIMs on all datasets, significantly outperforming the comparison
methods. Our approach can also achieve qualitatively better renderings withmore appearance and geometry details and less outliers, as shown in Fig. 4.
Our models are highly efficient, which all require less than 75MB space and canbe reconstructed in less than 30 min. This corresponds to more than 70x speed up

TensoRF: Tensorial Radiance Fields

MethodBatchSize StepsSRN [46]
NSVF [26]8192
150k
NeRF [31]4096
300k
SNeRG [17]
8192
250k
PlenOctrees [59]
1024
200k
5000
128k
Plenoxels [43]
DVGO [47]5000
30k
Ours-CP-384
4096
30k
Our-VM-192-SH
4096
30k
Ours-VM-48
4096
30k
Ours-VM-192
4096
15k
Ours-VM-192
4096
30k

13

Synthetic-NeRF
NSVF
TanksTemples
Time ↓ Size(MB)↓ PSNR↑ SSIM↑ PSNR↑ SSIM↑ PSNR↑ SSIM↑
>10h
22.26
0.846
24.33
0.882
24.10
0.847
>48∗ h
31.75
0.953
35.18
0.979
28.48
0.901
∼35h
5.00
31.01
0.947
30.81
0.952
25.78
0.864
∼15h
1771.5
30.38
0.950
∼15h
1976.3
31.71
0.958
27.99
0.917
11.4m
778.1
31.71
0.958
27.43
0.906
15.0m
612.1
31.95
0.957
35.08
0.975.
28.41
0.911
25.2m
3.9
31.56
0.949
34.48
0.971
27.59
0.897
16.8m
71.9
32.00
0.955
35.30
0.977
27.81
0.907
13.8m
18.9
32.39
0.957
35.34
0.976
28.06
0.909
8.1m
71.8
32.52
0.959
35.59
0.978
28.07
0.913
17.4m
71.8
33.14
0.963
36.52
0.982
28.56
0.920

Table 1: We compare our method with previous and concurrent novel view
synthesis methods on three datasets. All scores of the baseline methods are
directly taken from their papers whenever available. We also report the averaged
reconstruction time and model size for the Synthetic-NeRF dataset. NVSF
requires 8 GPUs for optimization (marked by a star), while others run on asingle GPU. DVGO’s 30k steps correspond to 10k for coarse and 20k for finereconstruction.

TensoRF-CP

TensoRF-VM

#Comp

2003

3003

5003

48
96
192
384
48
96
192
384

27.98/09:29/0.74
28.50/09:57/0.88
29.50/11:09/1.08
30.47/14:41/1.59
31.81/11:29/08.6
32.33/11:54/16.5
32.63/13:26/32.3
32.69/17:24/63.4

28.24/11:45/1.09
28.83/12:12/1.29
29.99/13:41/1.59
31.08/18:09/2.33
32.39/13:51/23.5
32.86/14:08/37.3
33.14/17:36/76.7
33.21/25:14/143.

28.38/14:20/1.85
29.06/15:27/2.18
30.33/18.03/2.66
31.56/25:11/3.93
32.63/18:17/55.8
33.06/20:11/105.
33.31/27.18/204.
33.39/43:19/397.

Table 2: We compare the averaged PSNRs / optimization time (mm:ss) / model
sizes (MB) of CP and VM TensoRF models on Synthetic NeRF dataset [31] withdifferent numbers of components and grid resolutions, optimized for 30k steps.
5k 15k 30k 60k 5k 15k 30k 60k
CP-384 28.37 30.80 31.56 32.03 03:03 11:30 25:11 51:47
VM-48 29.28 31.80 32.39 32.68 01:57 06:21 13:51 27:20
VM-96 29.65 32.26 32.86 33.17 02:01 06:41 14:08 28:57
VM-192 29.86 32.52 33.14 33.44 02:16 08:08 17:37 35:50
VM-384 29.95 32.62 33.21 33.52 02:51 11:30 25:14 52:50

MethodTime ↓ Size PSNR↑ SSIM↑NeRF [31]36h 5.00M 26.50 0.811
Plenoxels [43] 24:00m 2.59G 26.29 0.829
Ours-VM-48 19:44m 90.4M 26.51 0.832
Ours-VM-96 25.43m 179.7M 26.73 0.839

Table 3: PSNRs and time of CP and VM
models with different training steps on
the Synthetic-NeRF dataset [31].

Table 4: Quantitative comparisons ofour method with NeRF and Plenoxels
on forward-facing scenes [30].

compared to NeRF that requires about 1.5 days for optimization. Our CP model
is even more compact than NeRF. Moreover, SNeRG and PlenOctrees require
pre-training a NeRF-like MLP, requiring long reconstruction time too. DVGO
and Plenoxels are concurrent works, which can also achieve fast reconstructionin less than 15 min. However, as both are voxel-based methods and directly
optimize voxel values, they lead to huge model sizes similar to previous voxelbased methods like SNeRG and PlenOctrees. In contrast, we factorize feature
grids and model them with compact vectors and matrices, leading to substantially

14

A. Chen, Z. Xu et al.

Chair

Lego
NeRF

Plenoxels

DVGO

NSVF

Ours

GT

Fig. 4: Qualitative results of our VM-192-30k model and comparison methods(NeRF[31], plenoxels [43], DVGO [47], NSVF [26]) on two Synthetic NeRF scenes.
smaller model sizes. Meanwhile, our VM-192 can even reconstruct faster than
DVGO and Plenoxels, taking only 15k steps, and achieving better quality inmost cases. In fact, Plenoxels’ fast reconstruction relies on quickly optimizing
significantly more steps (> 4 times our steps) with their CUDA implementation.
Our models are implemented with standard PyTorch modules and already achievemuch better rendering quality with fewer steps taking comparable and even lessreconstruction time than Plenoxels. Note that our SH model essentially represents
the same underlying feature grid as Plenoxels but can still lead to more compact
modeling and better quality with fewer steps, showing the advantages of ourfactorization based modeling. In general, our approach enables fast reconstruction,compact modeling, and photo-realistic rendering simultaneously.
Forward-facing scenes. Our approach can also achieve efficient and high-quality
radiance field reconstruction for forward-facing scenes. We show quantitative
results of our two VM models on the LLFF dataset [30] and compare with NeRF
and Plenoxels in Tab. 4. Our models outperform the previous state-of-the-art
NeRF and take significantly less reconstruction time. Compared with Plenoxels[43], our approach leads to comparable or faster reconstruction speed, better
quality, and substantially smaller model sizes.

7

Conclusion.

We have presented a novel approach for high-quality scene reconstruction and rendering. We propose a novel scene representation – TensoRF which leverages tensor
decomposition techniques to model and reconstruct radiance fields compactly
as factorized low-rank tensor components. We hope our findings in tensorized
low-rank feature modeling can inspire other modeling and reconstruction tasks.

TensoRF: Tensorial Radiance Fields

15

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