DOES PHYSICAL INTERPRETABILITY OF OBSERVATION MAP IMPROVE THE
PHOTOMETRIC STEREO NETWORK?
Satoshi Ikehata1
1

National Institute of Informatics, Tokyo, Japan

ABSTRACTIn this paper, we revisit observation map which is the input representation for the deep photometric stereo networks
where pixelwise observations under different lights are protectively integrated to handle an arbitrary number of input images. Based on the hypothesis that the physical interpretability of observation map contributes to its performance, we empirically validate it by proposing two novel ideas; one is a pixelwise unified inverse rendering framework which accounts
the physical reasoning to recover the surface normals and theother is the network architecture that is equivariant/invariant
to the view-axis-around rotation of the pixelwise observation
map. By introducing these two ideas, our experimental evaluation on the public dataset indicated that more explicit physical reasoning of observation map improves the performanceof the photometric stereo task.
Index Terms— photometric stereo, observation map
1. INTRODUCTION
Photometric stereo [1] is a widely researched task for decades
which aims at recovering the surface normal map of an object
from images captured under different lights with a fixed camera. Since the classical physics-based photometric stereo algorithms [2, 3, 4] were hardly applicable to objects with complex non-convex geometry and non-Lambertian reflectance,
recent state-of-the-art photometric stereo algorithms adopt thedata-driven approach, using the deep neural networks [5, 6, 7,
8]. Unlike other computer vision tasks, the photometric stereo
networks must accept an arbitrary number of images as input
and pursuing the proper data aggregation strategy has been amajor interest in recent studies [7, 9, 10, 11, 12]. At current
time, one of the most promising strategy is based on observation map [8] due to its simplicity and effectiveness.
Observation map is a 2-d matrix where pixelwise observations under arbitrary number of directional lights are integrated. As shown in Fig. 1, each matrix cell corresponds to adiscretized light direction on a unit hemisphere, and each observation (i.e., pixel color) is projected to a point on the map
according to its light direction. In the photometric stereo task,
an observation map is individually encoded at each pixel andfed into the neural networks to predict its surface normal.
Though photometric stereo algorithms based on observation map [8, 13, 14, 15] have demonstrated the state-of-the-art
performance on the public benchmark [16], all the current algorithms have just let networks memorize the patterns of anobservation map and surface normal and there is no clear so-𝑥𝑥

Observation map at 𝑥

Fig. 1. Example of a physically interpretable observation map
at a nonconvex surface point, which reasonably encodes its
surface material and geometry.
lution when no identical pattern matches exist due to the huge
possible combinations of geometries, materials and lightings.
As for this point, we pay attention to the physical interpretability of observation map, which means that an observation map tells us the attributes at around a surface point.
For instance, Fig. 1 shows an observation map at a surface
point whose surface normal is pointing down left. The radially decreasing intensities clues that the light is reflected on
the rough dielectric surface (material). On the other hand, theabrupt change of values at right side evidences the presence of
cast shadows therefore the surface is non-convex (geometry).
Interpreted observation map in this way, we can also infer
the surface normal direction as a part of underling Bidirectional Reflectance Distribution Function (BRDF) [17] which
is visible in observation map as a reflectance lobe. This physical interpretability is definitely an advantage of observation
map, not in other strategies such as set-pooling [7, 9], graphconvolution [10] and self-attention [11, 12] in recent literature. Nevertheless, there were no studies that have examinedthe effects of physical interpretability of observation map.
The goal of this work is to take advantage of physical interpretabilty of observation map more explicitly to use it more
effectively. Following this motivation, we propose two ideas
to improve the photometric stereo networks using observation
map. First, we integrate the unsupervised inverse rendering
framework into the näive regression network. We ask neural networks to parse the observation map into the physicalintrinsic attributes (e.g., surface normal, surface roughness,
surface base color) and to integrate them in a physically plausible manner with the inverse rendering loss. Second, we explicitly account the reflectance isotropy [18], which implies
that the surface normal prediction should be equivariant to
the rotation of the observation map. In the prior work [8], thisproperty was enforced based on the external rotational data
augmentation. However, we will verify that the internal encouragement of the isotropy in neural networks also boost theencoder

input

n

n

decoderRot.

encoder

n
b
r
m

Rot.

targetinput
Direct Regression + Inverse Rendering (OURS)

Dense

1024 256 1or3

Dense

64

Encoder

Dense

Flatten64

Conv(3x3)

LeakyReLU

LeakyReLU

64

Convbn(3x3)

Convbn(3x3)32

LeakyReLU

LeakyReLU

32

Convbn(3x3)𝑤/4 × 𝑤/4

Convbn(3x3)

16

LeakyReLU

LeakyReLU

16

Convbn(3x3)

Convbn(3x3)4𝑤/2 × 𝑤/2

DecoderFig. 3. The architectures of our encoder and decoder.
performance even without using any data augmentation.
We validate our ideas on the DiLiGenT Photometric
Stereo Dataset [16] as well as compare them against recent
photometric stereo algorithms [6, 7, 8] and show that the neural networks can also successfully interpret observation map
in a physically plausible manner.
2. PRELIMINARIES
Problem Formulation: Photometric stereo is a problem torecover the unit normal vector n ∈ R3 for each pixel from acollection of observations I1≤j≤m ∈ R3 under m different
lighting directions l1≤j≤m ∈ R3 . Under the calibrated setup,pixelwise observations are normalized by corresponding lightintensities and the view direction is fixed by v = [0 0 1].
Henceforth, we rely on the classical assumptions of fixed,
linear orthographic camera and known directional lighting.
Observation Map: Observation map [8] is projectionsof appearances at a surface point onto a 2-D matrix based on
their light directions (l1≤j≤m ≜ [lxj lyj lzj ]⊤ ). Each elementof an observation map O ∈ Rw×w is defined as follow:
Oint(w(lxj +1)/2),int(w(lyj +1)/2) = αIj .

nencoder

Rot. Ave.
Pooling

decodernAverage
Pooling

decoderb,r,mRotational Pooling (OURS)

Fig. 2. We integrate the inverse rendering pipeline into theconventional Direct Regression framework for enhancing thephysical interpretability of observation map.
𝑤×𝑤

decoder

Rotational Data Augmentation (e.g., [8])Direct Regression (e.g., [8,13,14,15])
decoder
decoder
decoder
decoder

encoder(1)Here “int” is an operator to round a floating value to an integer. The size of the observation map (w) and the choice of thescaling factor (α) are arbitrary but were suggested to be set
by 32 and 1/max(I), respectively in [8] and we follow them.
3. METHOD
Our goal is to take advantage of physical interpretability ofobservation map to improve the photometric stereo network
taking it as input. First of all, we define the “baseline” net-

Fig. 4. Conventionally, the equivariance to rotation of observation map was realized by data augmentation [8]. On theother hand, we propose the pooling-based approach for realizing the rotation invariance/equivariance.
work and extend it according to our ideas. As shown in Fig. 2,
the simplest strategy to predict the surface normal from an observation map would be the “Direct Regression” where neural networks are trained only by the supervision of the surface normal. All the photometric stereo algorithms using observation map is based on this strategy [8, 13, 14, 15] withno physical reasoning behind. On the other hand, to explicitly utilize physical interpretability of observation map, we
integrate multiple decoders to predict surface attributes (e.g.,roughness, base color, metalness) with the baseline architecture and synthetically render the observation map based on
the predefined image formation model. Unsupervised inverse
rendering loss unifies all the attributes to confirm that they
are predicted in the physically plausible manner. We shouldnote that if we focus only on the normal prediction, we havethe decoder of same parameter size with the baseline, but we
expect that explicit physical reasoning by the additive inverse
rendering would improve the surface normal prediction.
In addition, we explicitly account the reflectance isotropy
and inherent equivariance/invariance of surface attributes torotation of observation map. As illustrated in Fig. 4, insteadof applying data augmentation as in existing works [8, 15],
we propose the pooling-based strategy which merges all
the feature maps from differently rotated observation maps
so that the prediction of surface normal is approximately
equivariant, and one of other surface attributes is invariant to
the rotation of observation map. We now describe details.
Inverse Rendering of Observation Map: We here detail the physical formation model of observation map for our
inverse rendering pipeline. We use a simplified version of
Principled BRDF [19] which has been commonly used forsynthesizing photometric stereo datasets [8, 12]. Assumingthat we don’t consider the anisotropic reflection, subsurface
scattering, sheen and clear coat which all are not dominant inreal-world isotropic materials, our BRDF (f ) is controlled bythree parameters, base color (b ∈ R3 ), roughness (r ∈ [0, 1])
and metalness (m ∈ [0, 1]) besides the surface normal andlighting. Please refer to [19] for mathematical descriptions,however we note that the specular lobe in Principled BRDF
is the Cook-Torrance microfacet BRDF model using the
GGX distribution as the microfacet normal distribution.
Based on this BRDF, the observation map (Ô) is numericallyTable 1. The ablation study of the rotation pooling.
Ba

K (Test) = 1
K (Test) = 5
K (Test) = 10
K (Test) = 20
K (Test) = 40
K (Test) = 90
K (Test) = 1
K (Test) = 5
K (Test) = 10
K (Test) = 20
K (Test) = 40
K (Test) = 1
K (Test) = 5
K (Test) = 10
K (Test) = 20
K (Test) = 40

Be

Bu Ca Co Go Ha Po1 Po2 Re
K (Train) = 20
10.4 13.0 16.4 17.3 18.2 16.7 27.9 12.6 14.0 21.3
3.0 4.7 8.0 5.0 5.7 7.5 14.7 5.2 5.8 11.4
2.6 4.0 7.8 4.5 5.5 7.2 14.2 4.9 5.5 10.4
2.5 4.0 7.6 4.5 5.4 6.9 14.1 5.0 5.3 10.5
2.5 3.9 7.6 4.5 5.4 6.9 14.0 5.0 5.3 10.4
2.6 3.9 7.6 4.5 5.4 6.9 14.0 5.0 5.3 10.6
K (Train) = 10
7.6 10.0 12.6 10.2 12.8 9.1 17.9 10.4 9.7 15.0
2.6 4.2 8.0 4.5 6.3 7.4 13.9 5.2 5.5 11.1
2.3 3.9 7.7 4.2 5.7 7.2 13.8 5.0 5.4 10.7
2.3 3.9 7.7 4.3 5.6 7.1 13.8 5.0 5.4 11.4
2.4 3.9 7.7 4.3 5.5 7.2 13.8 5.0 5.4 10.7
K (Train) = 1
2.8 4.1 8.0 4.6 7.1 7.6 14.0 5.3 5.7 11.4
13.2 15.2 16.1 12.3 16.9 16.9 20.1 14.0 15.0 21.3
15.5 16.9 17.8 13.9 18.6 18.7 21.7 15.6 16.7 22.6
16.5 17.7 18.6 14.6 19.4 19.6 22.6 16.2 17.7 23.5
17.2 18.3 19.1 15.1 19.8 20.1 23.1 16.7 18.3 23.9

Table 2. The evaluation on DiLiGenT dataset with 96 images.

Ave
16.8
7.1
6.7
6.6
6.6
6.6
11.5
6.9
6.6
6.6
6.6
7.1
16.1
17.8
18.6
19.2

formulated as follow:
Ô(l) = max{nT l ∗ f (n, b, r, m; l) ∗ S(θ), 0} ∀ l,

Ba
D.R.+I.R.L.+R.P. 2.3
D.R.+R.P.
2.0
D.R.+I.R.L.
2.8
D.R.+R.A.
2.1
D.R.
2.6
PS-FCN [7]2.8
Taniai [6]1.5
Woodham [1] 4.1

Be Bu Ca Co Go Ha Po1 Po2 Re
3.9 7.7 4.2 5.7 7.2 13.8 5.0 5.4 10.7
4.1 8.2 4.8 6.0 7.5 14.1 5.2 6.0 11.4
4.1 8.0 4.6 7.1 7.6 14.0 5.3 5.7 11.4
4.2 8.1 4.4 7.9 7.4 13.8 5.5 6.4 12.3
4.7 8.7 3.9 8.1 7.3 14.2 5.9 6.5 12.6
7.6 7.9 6.2 7.3 8.6 15.9 7.1 7.6 13.3
5.8 10.4 5.4 6.3 11.5 22.6 6.1 7.8 11.0
8.4 14.9 8.4 25.6 18.5 30.6 8.9 14.7 19.8

Ave
6.6
6.9
7.1
7.2
7.5
8.4
8.8
15.4

Table 3. The evaluation on DiLiGenT dataset with 10 images.
Ba Be Bu Ca Co Go Ha Po1 Po2 Re
D.R.+I.R.L.+R.P. 4.3 5.4 8.7 6.2 11.6 10.7 20.6 7.0 8.0 13.2
D.R.+R.P.
5.3 5.3 9.2 6.3 13.5 11.0 18.6 7.4 7.9 14.4
D.R.+I.R.L.
5.2 6.2 10.4 7.1 12.7 12.1 19.5 6.4 7.6 15.3
D.R.+R.A.
9.6 14.8 15.0 11.7 15.0 16.4 21.5 12.7 15.9 16.2
D.R.
10.6 16.8 15.4 12.5 16.8 17.4 22.0 13.4 16.9 16.8
PS-FCN [7]4.0 7.2 9.8 8.3 10.5 11.6 18.7 10.1 9.9 15.0
SPLINE-Net [13] 4.0 8.7 11.4 6.7 10.2 10.5 17.3 7.3 9.7 14.4
Minify-Net [14] 5.0 6.0 10.1 7.5 8.8 10.4 19.1 8.8 11.8 16.1

Ave
9.6
9.9
10.2
14.9
15.9
10.5
10.0
10.4

D.R.
+I.R.L+R.P.

(2)

where max operator accounts for the attached shadow andS(θ) is named observation mask which is actually a sampling
operator to sample light directions of input observation map.
Consideration of Reflectance Isotropy: As has beendetailed in [8], the surface reflectance isotropy implies thatthe surface normal prediction should be equivariant to therotation of observation map, which is formally described asr(g(x)) = g(r(x)) where x is the input observation map,g is the network and r is the rotation matrix which rotates
the lighting and surface normal directions (n, l) around theviewing direction. Similarly, predictions of other attributes(b, r, m) are rotation invariant as g(x) = g(r(x)). We canintuitively confirm them by rotating the actual observation
map in Fig. 1 and seeing the reflectance lobe.
Though rotational data augmentation in [8] had improved
the surface normal prediction, it was unclear if the improvement had really come from the care of reflectance isotropy
rather than from the larger training sample size. To clarify, we
propose the rotation equivariant/invariant networks without
relying on data augmentation but on rotational pooling. As
illustrated in Fig. 4, copies of input observation map arerotated at regular intervals (i.e., 360/K, K is the numberof rotations) and each copy is fed to the same encoder. The
rotation equivariance implies that the feature map from arotated observation map should also be rotated accordingly,
in other words, feature maps rotated backwards should beclose with each other. Based on this logic, we inversely rotate
and average feature maps of different rotation angles and pass
the result to the surface normal decoder. We call this process
as rotated average pooling. Since the pooling operation is anorder-agnostic operation, if all the features rotated by from 0
to 360 degrees are pooled together, theoretically the originalrotation angle of observation map doesn’t affect the output.
Similarly, when decoding surface material attributes, weapply the basic average pooling operation on all the encoded

D.R.+R.A.

lighting

Ball

Harvest

Pot2

Reading

Fig. 5. The qualitative comparison on DiLiGenT dataset with
10 images. The lighting configuration is presented.
feature maps and then feed the pooled feature to decoders
to encourage them to output the same values regardless of
the rotation angles of the input observation map (i.e.rotation
invariance). By introducing these operations, we expect both
encoder and decoder get constrained by reflectance isotropy
without explicitly increasing training samples.
Network Architecture and Training Loss: To clarify
our discussion point about the advantage of physical interpretability of observation map, we design the simple encoderand decoder. Please see details in Fig. 3 and we won’t go into
detail here due to the space limit.
Our training loss is comprised of the normal reconstruction loss LN and the scale-invariant inverse rendering loss LI .
The normal reconstruction loss LN penalizes for the distance
between the predicted and ground truth surface normals. In
our experiment, we used the standard ℓ2 distance for this loss
function. The unsupervised, scale-invariant inverse rendering
loss LI penalizes for the distance between the input and reconstructed observation maps.
LI = smoothℓ1 (O, β Ô).

(3)

Here O is the input observation map and Ô is given by Eq. (2).
smoothℓ1 (p, q) [20] is a scale-invariant ℓ1 function and its
scaling factor β is analytically computed by minimizing
β ← argminβ̂ ∥O − β̂ Ô∥22 .

(4)

Note that the scale-invariant loss is required since there is theambiguity of the intensity scale between O and Ô.

𝐧
𝐛

𝐧𝑔𝑡

r

m

Real

Reconstructed

Real

Reconstructed

observation maps (input/recovered) + BRDF (diffuse/specular/full)Harvest (96 images)
𝐧
𝐛

𝐧𝑔𝑡
r

m

Real

Reconstructed

Real

Reconstructed

observation maps (input/recovered) + BRDF (diffuse/specular/full)

Buddha (10 images)

Fig. 6. The surface attribute reconstruction results by our D.R.+I.R.L.+R.P. architecture. Top: Harvest with 96 images. Bottom:
Buddha with 10 images. From left to right: One of input images, recovered surface attributes, two reconstructed images undernovel lightings, reconstructed observation maps (input, reconstructed, diffuse/specular BRDFs and full BRDF).
4. EXPERIMENTS
Training and Test Details: We followed the exactly same
training strategy as [8] except that we trained our network
using CyclesPS+ dataset [12] which consists of 25 objects.
Each object provides 32-bit floating images with a resolutionof 256 × 256 under 740 different known lighting directions.
Our network was trained on 3× Nvidia Geforce GTX 1080Ti
with Adam optimizer for 20 epochs, a batch size of 256
and a learning rate of 0.0002. We evaluate our method on
DiLiGenT [16] which is a public benchmark dataset of 10
real objects (Ba:Ball, Be:Bear, Bu:Buddha, Ca:Cat, Co:Cow,
Go:Goblet, Ha:Harvest, Po1:Pot1, Po2:Pot2, Re:Reading).
Each object provides 16-bit integer images with a resolutionof 612 × 512 from 96 different known lighting directions andthe ground truth surface normal maps. For the evaluation, we
simply compute the mean angular error (MAE) of predicted
normal maps in degrees. Errors are averaged over 100 trials
when the number of images is ten. In our experiment, we
would like to show that baseline (i.e., Direct Regression:
D.R.) is enhanced with Inverse Rendering Loss (I.L.R.)
and Rotational Pooling (R.P.) that are more physicallyinterpretable architectures. Note that D.R. and D.R.+R.A.
(i.e., Rotational Augmentation) are equivalent to CNN-PS [8](i.e., w/o and w/ rotational augmentation) except that the
network architecture was slightly simplified.
Analysis on Rotational Pooling: First, we investigate
the effect of the number of rotation angles for the poolingin our D.R.+I.R.L+R.P architecture. Note that since there is
no learnable parameters in R.P., the values at training andtest could be different. As shown in Table 1, the optimal
results are basically obtained when the number of rotations in
training and test are close. Though there are multiple optimal
choices, we choose K(Train) = K(Test) = 10 in following
experiments simply Since they were also recommended in [8].
Quantitative Comparison on Dense Setup: We compared variants of our architectures on the dense setup (i.e.,
all 96 images are used) to validate the effects of the inverse
rendering loss and rotational pooling. Just for the reference,
we also lined up some recent deep-learning-based algorithmsof PS-FCN [7], Taniai and Maehara [6] and the conventional
Lambertian method [1]. The results are illustrated in Table 2.
Both I.R.L and R.P. enhanced the performance of D.R. as

expected. Interestingly, R.P. showed the obvious advantage
over R.A. which indicates that the R.P. could better consider
the surface isotropy rather than R.A. due to the explicit
enhancement of equivariance/invariance of the networks.
Quantitative Comparison on Sparse Setup: We also
evaluated our method on sparse photometric stereo setup(i.e., 10 images are used). Here we instead compared our
method against Minify-Net[13] and SPLINE-Net [14] thatare also based on observation map but tuned for the sparse
setup, as well as PS-FCN [7]. The results are illustrated
in Table 3 and Fig. 5. Though the tendency of the result issimilar with one of the dense setup, the numerical improvement was much bigger. This could be due to the fact that thesimple pattern matching in D.R. is more difficult in the sparse
observation map, making physical reasoning more important.
Analysis on the Rendered Observation Map: Finally,
we displayed the reconstructed surface attributes and rendering results of observation map in Fig. 6. Since there is notrue label, we can only qualitatively discuss the result but it
is obvious that the reconstructed observation map is almost
identical to the input observation map which means that our
inverse rendering pipeling behaved as intended. Since therecovered BRDF gives elements of observation map for all
the lighting directions, it can be used to render the image
under the novel lightings. The quality of synthesized image
also shows that our method reasonably worked to capture thephysics properties in observation map.
5. CONCLUSION
In this paper, we presented ideas to take advantages of thephysical interpretability of observation map for the photometric stereo task. Our evaluation demonstrated that integrating
the inverse rendering pipeline and the rotational pooling into
the basic direct regression networks improve the performance
without increasing the complexity of the network architecture
nor using augmented training examples.
In the future work, we are interested in testing our ideas
on more complicated photometric stereo networks that uses
observation map as input (e.g., [15]) to verify that this result
is general. In addition, we also want to discuss the physicalaspects in other aggregation strategies such as set-pooing [7]and self-attention [12] and discuss connections between datadriven approach and physics-based approach.

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