Efficient Simulation of Furniture LayoutTaking into Account Lighting Environment

Takuya Yamakawa Yoshinori Dobashi Tsuyoshi Yamamoto

Hokkaido University

AbstractFurniture layout design is a challenging prob-lem, and several methods have recently beenproposed. Although the lighting environmentin a room has a strong relationship with thefurniture functionality, the previous methodscompletely overlooked it in designing furniturelayout. This paper addresses this problem; wepropose an efficient method for computing fur-niture layout taking into account the lighting en-vironment. We propose a new cost function thatevaluates the lighting environment taking intoaccount inter-reflections of light. A fast methodfor evaluating the cost function is also proposed.We demonstrate that our method improves thequality and usability of furniture layout bytaking into account the lighting environment.

Keywords: interior design, furniture arrange-ment, lighting design, lighting environment

1 Introduction

When you are moving into a new home, youneed to arrange the living room furniture. Youmay have a sofa, armchairs, coffee tables, diningtables, dining chairs and so on. Furniture place-ment is challenging because it requires coherentoptimization of a variety of functional and vi-sual criteria. The lighting environment, such aspositions and colors of light sources, is also animportant factor that significantly influences theatmosphere of a room and of human activitiesthat take place in the room. For example, a din-ing table should be placed in a bright spot in aroom. We should take account of the lightingenvironment as well as the functionality in de-signing the furniture layout.

Most people responsible for furnishing a newhome have no training in interior and lightingdesigns. The lighting equipment in a new homeis usually fixed and cannot be moved. Deter-mining the furniture layout considering the fixedlighting environment is therefore a difficult taskfor those who have no knowledge of interiorand/or lighting designs. There are some pre-vious methods that automatically determine thelayout of furniture [1] [2]. Although these meth-ods can suggest an acceptable layout for furni-ture taking into account functionality and visualquality, no attention has been paid to the lightingenvironment.

In this paper, we propose a furniture layoutsimulation method that also takes the lightingenvironment into account. We propose a newcost function using average illuminance, takinginto account inter-reflections of light, in order toevaluate the quality of the lighting. We also pro-pose a fast method for estimating the cost func-tion. Our method calculates the average illumi-nance accurately, both for direct light and forlight reflected from the walls. The average il-luminance due to higher-order reflections is es-timated by our example-based approach. Thefurniture layout is then determined by using theMetropolis Hasting algorithm proposed by Mer-rell et al [1]. We incorporate our new cost func-tion into their framework. Our method can de-termine a suitable furniture layout within a fewseconds, while taking into account the lightingenvironment.

2 Related Work

In this section, we briefly discuss some of theprevious methods related to the layout and light-

ing design problems.One of the earliest works that dealt with the

layout problem was proposed by Harada et al[3]. They developed an interactive system forcreating floor plans in buildings [3]. However,this method does not treat the furniture layoutproblem. Xu et al. proposed a constraint-basedautomatic placement system for computing thelayout of large spaces, including multiple piecesof furniture [4]. This method uses a set ofplacement constraints, physics, and a semanticdatabase to determine the layout. Germer andSchwarz focused on the furniture arrangementproblems of larger spaces for real-time walk-throughs [5]. They proposed a procedural tech-nique to determine the best arrangement. Mer-rell et al. presented an interactive system for fur-niture layout based on interior design guidelines[1]. This method can interactively suggest mul-tiple layouts by minimizing a cost function de-rived from the interior layout guidelines. Yu etal. proposed a different approach to the furniturelayout problem [2]. This method learns the re-lationship between furniture items from a set ofexample layout patterns. Fisher et al. also useda large database of example layout patterns [6].More recently, researchers tried to learn stylesof furniture layout from crowd-sourced datasets[7] [8]. All of these existing methods are use-ful for solving the problem of furniture layout,but none of them takes into account the lightingenvironment of the space where the furniture issited.

Lighting design has been an active researcharea in computer graphics. There are two pri-mary approaches: the forward and the inverseapproaches. We will now briefly discuss theinverse lighting approach. One of the earliestworks is the system developed by Schoenemanet al [9]. The system computes the intensi-ties of light sources that closely match the tar-get image painted by the user. Simultaneously,Kawai et al. proposed a method for optimiz-ing lighting parameters, including light sourceintensity, object reflectivity, and spotlight direc-tionality [10]. After these pioneering works,many methods have been developed to solve theinverse lighting problem under various settings.Shacked and Lischinski developed an automaticmethod for computing various lighting param-eters by optimizing a perception-based image

quality objective function [11]. Okabe et al.proposed a method for inversely computing en-vironmental lighting from the desired intensi-ties painted by the user [12]. Shesh and Chendeveloped a system to determine the layout ofpoint light sources so that sketched lighting ef-fects are achieved [13]. More recently, Schwarzand Wonka presented a system for the lightingdesign of procedurally modeled buildings [14].These methods are useful for lighting design,but none of them focuses on the layout of fur-niture under fixed lighting parameters.

3 Furniture Layout Optimization

This section provides a brief overview of themethod proposed by Merrell et al [1], which ourmethod is based on. Please refer to [1] for moredetails.

The method determines the furniture layoutby minimizing a cost function that evaluates thequality of a layout. The cost function is designedby referring to furniture layout guidelines andconsists of two criteria : functional and visualcriteria. Minimizing the cost function is a high-dimensional and nonlinear problem; there areno analytical solutions to it. It is also difficultto solve the problem by using standard nonlin-ear solvers, such as the gradient descent method.The method therefore employs a Markov chainMonte Carlo sampler to explore the function,and produces multiple optimized samples. Morespecifically, they used the Metropolis-Hastingalgorithm to find a set of optimized layout pat-terns.

The method searches for a good furniture lay-out by iteratively updating positions and orien-tations of furniture items. The method first per-turbs the position of a randomly selected furni-ture item by adding random numbers obeying aGaussian distribution. The orientation of a ran-dom furniture item is also perturbed in a sim-ilar way. Next, the method swaps the positionsand orientations of two randomly selected items.These processes are repeated until the cost func-tion is minimized.

Although their method can find a pleasing fur-niture layout, they do not take into account thelighting environment. Therefore, their methodmay place, for example, a dining table at a posi-

Room function Recommend illluminance (lx)Studying, Reading 1000-500Eating, Cooking 500-220Playing, Gatherings 220-150General lighting 150-75Sleeping, Private room 10-30

Table 1: Function of the room based on LightingEnvironment

tion where there are no light sources around it.We use the same algorithm to find a good furni-ture layout, but we introduce a new cost functionthat takes into account the lighting environmentof the room.

4 Proposed Method

This section describes our method for comput-ing furniture layout taking into account the light-ing environment. We first propose our new costfunction to evaluate the lighting environment fora furniture layout. We use the average illumi-nance for each furniture item. Then, we pro-pose a fast method for computing the cost func-tion. We compute the cost function taking intoaccount the interreflections of light.

4.1 Cost Function for LightingEnvironment

There is a close relationship between the func-tion of a room and the lighting environment.For example, a living room would be generallybright while a bed room would be darker. Ta-ble 1 shows a list of recommended illuminancefor different room functions found in a typicallighting design guideline. The recommended il-luminance changes depending on the human ac-tivities that are undertaken in the room. We de-signed our cost function, cL, according to thistable. In the following, we denote each furnitureitem by f and a furniture layout by F .

We define the cost function cL for a given fur-niture layout F as:

cL(F ) = −∑f∈F

rf ·t(L(f), Lm(f), LM (f), α),

(1)where rf = 1 for furniture item f that requiresevaluation of average illuminance and 0 other-

light source

direct

light

higher order

reflections

reflection

from wall

Figure 1: Calculation of average illuminance.

Lmin

Lmax

t

L

Figure 2: t(L,Lm, LM , α)

wise, L(f) is a function that evaluates the av-erage illuminance of furniture item f , (Lm(f),LM (f)) is the range for the recommended illu-minance for f , and α is a user-specified param-eter for function t. We use the same function fort as that suggested by Merrell et al [1], that is,

t(L,Lm, LM , α) =

( LLm

)α (L < Lm)

1 (Lm ≤ L ≤ LM )

(LML )α (L > LM )

.

(2)This function, as illustrated in Figure 2, is de-

signed to plateau when L is within the recom-mended range (Lm, LM ) and to gradually de-crease as L goes below Lm or above LM . Thedegree of the decay is controlled by the parame-ter α.

The illuminance function L evaluates the av-erage illuminance on the top surface of each fur-niture item. Figure 1 shows different light pathscontributing to the average illuminance. Themost significant contribution to the average il-luminance is obviously the direct light from thelight source. The next contribution is the lightreflected from the walls, which often work aslarge-area light sources. The rest of the higher

order inter-reflections make some contribution,but these are relatively small. We thus define Lby the following equation.

L(f) = Ld,w(f) + Lamb(f), (3)

where Ld,w represents the illuminance due to thedirect light and light reflected from the walls.Lamb is the contribution from the higher orderinter-reflections.

A straight forward way to compute the abovecost function is as follows. Ld,w is obtainedby calculating the sum of the intensities of lightcoming from light sources and the light reflectedat every point on the walls [15]. Lamb is ob-tained by solving the rendering equation [16].Assuming diffuse surfaces, we compute Lamb

by solving the radiosity equation, or a linear sys-tem of equations by using the method describedin [17].

Although the above method can evaluateL accurately, its computational cost is pro-hibitively expensive for interactive optimization.We therefore propose fast methods for approxi-mately evaluating Ld,w and Lamb. Since Ld,w

has a significant contribution, it is evaluatedmore accurately than Lamb. The details are de-scribed in the next subsection.

4.2 Fast Evaluation of IlluminanceFunction

We develop two methods for the efficient eval-uation of each term of the illuminance functionL defined by Equation 3. For a typical furnitureitem that requires the evaluation of its averageilluminance, such as a dining table, the shape ofthe top surface is usually rectangular. In the fol-lowing, we therefore assume that the top surfaceis rectangular in shape. For a non-rectangularshape, we approximately compute the averageilluminance by using the bounding rectangle ofthe top surface.

4.2.1 Evaluation of Ld,w

Let us consider the computation of Ld,w for fur-niture item f placed at Pf with orientation θf , asshown in Figure 3. Obviously, Ld,w is a functionof Pf and θf . Point P on the top surface (rectan-gle) is parameterized by two parameters, s and t

θf

Pf

P

s

S T

t

uv

(=A)

B

C

D

room

Figure 3: Calculation of Ld,w

(0 ≤ s ≤ S, 0 ≤ t ≤ T ; see Figure 3). Ld,w isthen expressed by the following equation.

Ld,w(Pf , θf ) =1

ST

∫ S

0

∫ T

0ϕ(s, t, θf )dsdt,

(4)where,

ϕ(s, t, θf ) = ϕd(s, t, θf ) + ϕw(s, t, θf ). (5)

ϕd and ϕw are the illuminances at point P dueto the direct light and the light from the walls,respectively. We do not provide detailed expres-sions for ϕd and ϕw but ϕd represents the sum ofthe contributions from all the light sources andϕw is that from all points on the walls. Thatis, ϕd and ϕw themselves are expressed in theform of double integrals over the surfaces oflight sources and walls. Ld,w is thus expressedin the form of a quadruple integral.

Since the computation of Ld,w is very ex-pensive, we employ a precomputation-based ap-proach. The basic idea is to precompute Ld,w

for all possible positions Pf and orientations θf .However, this results in a long precomputationtime since we need to compute the quadrupleintegral at each sampling position and orienta-tion. We address this problem by using the ideaof summed area tables [18].

Let us now assume a horizontal plane onwhich the top surface of furniture item f lies anddefine a coordinate system (u, v) on this planeso that furniture item f is aligned with the lo-cal coordinate axes (see Figure 3). The origin

of this coordinate system is at one of the cornersof the bounding box of the room. We define thefollowing function using this coordinate system.

Φ(u, v, θf ) =

∫ u

0

∫ v

0ϕ(u∗, v∗, θf )du

∗dv∗

(6)We precompute Φ(u, v, θf ) numerically by sam-pling (u, v, θf ) at regular intervals and store theresult in a lookup table. The precomputation ofΦ is efficient, since it can be computed incre-mentally for a certain orientation θf . Let us as-sume that both u and v are sampled at an intervalof ∆. We first compute the following function.

Φv(i, j, θf ) =i−1∑k=0

ϕ(k∆, j, θf )∆

= Φv(i− 1, j, θf ) + ϕ(i∆, j, θf )∆. (7)

Then, Φ is computed by the following equation.

Φ(i, j, θf ) =

j−1∑l=0

Φv(i, j, θf )∆

= Φ(i, j − 1, θf ) + Φv(i, j, θf )∆. (8)

Once Φ is tabulated, Ld,w can be obtainedvery efficiently by using the lookup table for Φ,that is,

Ld,w(Pf , θf ) =1

ST(Φ(uA, vA, θf ) + Φ(uC , vC , θf )

−Φ(uB, vB, θf )− Φ(uD, vD, θf )),

(9)

where (uA, vA), (uB, vB), (uC , vC), (uD, vD)are the coordinates of the four corners of therectangular top surface of furniture item f (seeFigure 3).

4.2.2 Evaluation of Lamb

In order to evaluate Lamb accurately, we needto compute the inter-reflections of light betweenwalls and furniture objects, which is very time-consuming. Precomputing Lamb is not a practi-cal solution in this case, since we have to com-pute Lamb for all possible furniture layout pat-terns. Our solution is to approximate Lamb asa constant ambient term. Since Lamb representsthe average illuminance due to higher order re-flections, we assume that Lamb can be approxi-mated by a constant for each furniture item f .

Figure 4: Illuminance due to higher order inter-reflections (scaled by 5). The inset im-ages show the full radiosity solutions.

This idea is inspired by the constant ambientterm that is often used in the traditional shad-ing model [19]. We determine Lamb from a setof examples.

In a pre-process, we prepare a set of randomlayout patterns and compute the interreflectionsfor each of them. We use the progressive radios-ity algorithm to compute the inter-reflections[17]. The shape of each furniture item is re-placed by its bounding box to approximatelycompute the inter-reflections. For each layoutpattern i, we first compute the average illumi-nance Ls

i (f) due to single reflected light. Theaverage illuminance Lm

i (f) obtained from thefull radiosity solution is also computed. Then,our ambient term for furniture item f is givenby:

Lamb(f) =1

N

N−1∑i=0

(Lmi (f)− Ls

i (f)), (10)

where N is the number of randomly generatedlayout patterns.

We examine our constant ambient term as-sumption by using the scene shown in Section5. We randomly generate thirty sets of layoutpatterns and computed the inter-reflections foreach of them. Figure 4 shows the illuminancedue to higher order inter-reflections for two dif-ferent layout patterns. The inset images showthe full radiosity solutions. We can observe thatthe intensities on the top surfaces are uniformand have similar values even if the furniture lay-

(a) coffee table

(b) dining table

0

10

20

30

40

0 5 10 15 20 25 30

0

10

20

30

40

0 5 10 15 20 25 30

Figure 5: Average luminance Lamb for coffee(a) and dining (b) tables.

out is different. Figure 5 shows the average il-luminance due to higher order inter-reflections,Lamb, computed for the coffee and dining ta-bles. These are typical furniture items that re-quire the evaluation of their illuminance. Thebrown boxes in Figure 4 correspond to thesefurniture items. The horizontal axis indicatesthe index of the layout pattern and the verticalaxis is the average illuminance. The plot in-dicates that the average illuminance for thesefurniture items can be approximated very wellby a constant. In this particular case, the av-erage/standard deviations of Lamb for the coffeeand the dining tables are 34.7/3.0 lx and 32.8/1.2lx, respectively. These values also support ourassumption.

5 Results

This section shows some experimental resultsthat were obtained using the proposed method.We use a desktop PC with an Intel Core i7-2600k CPU and GeForce GTX 580. We set theintensities of the light sources to those of actualexisting lighting equipments by referring to acommercial catalog. We use a rectangular roomand five furniture objects shown in Figure 6. The

armchair x 2

chair x 2 dining table

coffee tablesofa

Figure 6: Furniture objects used in our experi-ment.

Figure 7: Our interactive optimization system

reflectances of the room and furniture items arealso chosen from the existing ones. The dimen-sion of the room is 7.5m × 5.0m × 2.5m. Inour experiment, we use eight light sources onthe ceiling. Three of them are downlights, mod-eled by spotlights, that are often used to illumi-nate furnitures. The other five light sources areused to illuminate the whole room and we usepoint light sources for them. We set the intensi-ties of these light sources to those of actual ex-isting lighting equipments by referring to a com-mercial catalog. We use our system to find furni-ture layouts so that the average illuminances onthe coffee and the dining tables are within therange of the recommended illuminance shownin Table 1. For (Lm, LM ) in Equation 1, we use(220, 500) and (150, 220) for the dining and thecoffee tables, respectively. However, in Figure9(c), we use (10, 30) for the coffee table. Theaverage illuminances on these tables in the fol-lowing examples are summarized in Table 2.

A simple OpenGL based renderer has beendeveloped to visualize the furniture layout dur-ing the optimization process as shown in Figure7. During the optimization process, our systemretains the top three layouts found so far anddisplays them to the user. We do not use theGPU for computing the layout; it is only used

(a) without our method (b) with our method (c) direct light only

Figure 8: Comparison. The layout in (a) is determined without taking into account the lighting envi-ronment. The layout in (b) takes into account the lighting environment. (c) shows the samelayout as (b) but is rendered with direct light only.

(a) bright spot on the right (b) bright spot in the center (c) indirect lighting

Figure 9: Furniture layout for different lighting conditions.

for the visualization of the layout. The layoutcomputation is done on the CPU. The user canstop the computation when the user is satisfiedwith the suggested layout. The user can also re-start the computation with a different seed num-ber for the random number generator. After theoptimization has finished, we render the sceneusing Autodesk Maya 2015 to verify the results.The following images are rendered by using Au-todesk Maya 2015.

The first experiment shown in Figure 8 is todemonstrate the importance of the lighting en-vironment in the furniture layout. Figure 8 (a)shows the result without our method. 3,241 it-erations were used to arrive at this layout, andit took 0.795 sec. As shown in Figure 8 (a), thedining table is placed in the darker region of theroom. The illuminance on the dining table iscompletely out of the suitable range suggestedby the guidebook for eating (see Table 2). Al-though the appearance of the layout itself looksgood, this is actually a bad layout in terms of thelighting environment. Figure 8 (b) shows the re-sult obtained with the proposed method. Thislayout was found after 3,551 iterations, whichtook 1.589 sec. The illuminance on the din-ing and the coffee tables are both in the desiredrange for eating and playing.

Figure 8 (c) shows the same layout as (b)

dining table coffee tableFigure 8(a) 97 (220-500) 174 (150-220)Figure 8(b) 263 (220-500) 197 (150-220)Figure 8(c) 121 (220-500) 128 (150-220)Figure 9(a) 346 (220-500) 159 (150-220)Figure 9(b) 298 (220-500) 153 (150-220)Figure 9(c) 244 (220-500) 22 (10-30)

Table 2: Average illuminance [lx] on the diningand coffee tables. In the brackets arethe desired ranges of the average illu-minance.

but the image is rendered with direct light only.Without the inter-reflections of light, the diningand the coffee tables become darker, indicatingthe importance of the inter-reflections. Withoutthe inter-reflections, we found that the averageilluminance was always out of the range unlesswe used lighting equipments with very high in-tensities, such as spotlights.

Figure 9 shows the results obtained by usingour method for different lighting conditions. InFigure 9 (a), the light source is placed so thatthe bright spot appears on the right of the room.In Figure 9 (b), the bright spot appears in thecenter of the room. These layouts were obtainedwithin a few seconds. Our method can success-fully find a layout in which the illuminance on

the dining and the coffee tables are in the desiredrange. Figure 9(c) shows an example where in-direct lighting is dominant. The left side in theroom is mainly illuminated by indirect light dueto the light source on the left wall, which illu-minates the wall only. This room is designedto foster a ’private’ atmosphere. Even with thisindirect lighting environment, our method suc-cessfully places the dining and coffee tables sothat the illuminance is in the desired range.

6 Discussion

In our current implementation, we do not takeinto account shadows in computing Ld,w. Thisis because it is rare that shadows of a furnitureitem have significant influence on other items.In a typical furniture layout, tall furniture items,such as a closet, are often placed around thewalls of a room and therefore they do not castshadows on other objects. Although shadowscan easily be incorporated by shooting shadowrays from the furniture objects to light sources,this increases the computational cost for Ld,w

and we do not think that the effects of shadowsare worth the costs involved.

7 Conclusions

We have proposed a method for computing fur-niture layout that takes lighting conditions intoaccount. Based on the lighting design guide-lines, we introduced a new cost term using theaverage luminance. We proposed a fast methodusing the idea of summed area tables for com-puting the average luminance. The effectivenessof our method has been demonstrated throughthe set of experiments shown in Section 5. Byusing our method, the user can design his/herown furniture layout that is optimal in terms offunctionality, visual composition, and lightingenvironment.

As for the future work, we would like to ex-tend our method to taking into account naturallight sources, i.e., the sunlight and the skylight.We need to compute the inter-reflections takinginto account windows to achieve this. It is alsointeresting to optimize the furniture layout andthe lighting parameters simultaneously.

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