Olfactory Signal Processing

Olfacto ry Signal Pro cessi ng ✩ Kush R. V arshney Mathematic al Scienc es and Analytics Dep artment IBM Thomas J. Watson R esea r ch Center Y orktown Heights, NY 10598 USA Lav R. V arshney Dep artment of Ele ctric al and Computer Engine ering University of Il linois at Urb ana-Champ aign Urb ana, IL 6180 1 USA Abstract Olfaction, the sense of smell, has received scant atten tion from a signal pro- cessing p ersp ective in co mparison to a udition a nd vision. In this pape r , w e develop a sig nal pro cessing par adigm for olfactory signa ls based on new sci- ent ific discoveries including the psyc hophysics concept of olfactory white . W e describ e a framework for predicting the per ception of o dorant compo unds from their ph ys ico ch emica l features and use the prediction as a foundation for sev- eral downstream pro cessing task s. W e detail formulations for o dor cancellation and fo o d steganogr aphy , and pr ovide rea l-world empirical examples for the two tasks. W e a ls o discuss ada ptiv e filtering and other olfacto ry signal pro cess ing tasks at a high level. Keywor ds: Adaptive filtering , foo d steg anogra ph y , noise cance lla tion, odo r cancellation, olfactory signa l pro ces s ing, p erceptio n, structured sparsity 1. In tro duction Audition, visio n, and olfaction are the three wa ys that p eople remotely sense stimu li; mu ch signal pro cessing resea rch has dealt with audio and video signals, ✩ Po r tions of the material in this paper were first presented in [ 1] and [2]. 1 but study o f olfactory sig nal pro cessing has b een neglected. One reason is the difficult y in compac tly specifying the fundamental inputs to the h uman per ceptual system. Whereas vibratio n a nd lig ht signals interacting with the ears and eyes ar e compactly par a meterized by amplitude, phas e, and frequency , olfactory signals in ter a cting with the nose manifest as collections of chemical comp ound molecules drawn from a very large set. Despite the p ossible input set having very large ca rdinality , recent evidence suggests that the space of olfactory per ception is fair ly low-dimensional [3 , 4, 5]. The most bas ic dimension, ak in to the DC co mponent of a wav eform, is ple asantness [6, 7]. Another recent finding shows the exis tence of olfactory white in human psychoph ysics with similar per ceptual prop erties as white light and white audio signals [8]. In this pa per , w e inv es tig ate olfactor y signals and systems at a le vel of a b- straction r e moved from the ph ysical s ensing and actuation o f chemical com- po unds. Prior work in olfacto ry signal pro cessing at the lower physical le vel includes the following. One long- standing a rea of rese a rch has been developing chemical sensors and so-called ele ctr onic noses , see e.g. [9, 10, 11], references therein a nd thereto . A v ariety o f s ensing technologies including chemical gas sensors, o ptical sensor sys tems, infra red sp ectroscopy , and micro electr o mechani- cal senso rs have b een developed, and the signa l pro cessing and machine learning challenge is using r aw sensor data to identify the sp ecific comp os itio n o f com- po unds pr esent, cf. [12, 13, 1 4]. Gas c hro matography mass s pectr ometry is considered the gold sta ndard in labo ratories , but the goal is to make p ortable, low-pow er, and low-cost systems with similar per formance. In all such sy s tems, hu man p erception is not cons idered and the goa l is simply to clas sify accor ding to ph ysico chemical pr op erties [15]. Moving from senso r s to a ctuators, ph ysical de v ices used for a c tiv ely pro- ducing o dor sig nals ar e called virtual ar oma synthesizers [16] and function b y mixing comp ounds from several car tridges into an airstream, m uch like how inkjet prin ter s pro duce arbitrary co lors. These devices hav e b een put together in a v ariety of o ld and new o dor communication technologies [17, 18]. Classi- cal examples like Aroma Rama and Smell-O-Visio n attempted to e nhance the 2 exp erience of cinema v iew er s thr ough a greater deg r ee o f immersio n, where as mo dern e xamples like oP hone aim to enable multi-odiferous messa g es trans mit- ted to individuals. With the practice of olfac to ry communication, there is also an informa tion theor y of olfaction concerne d with b ounding the human capacity to per ceive and differentiate odo rs [1 9, 20]. There has been muc h new under standing of olfactory p erception and many new developmen ts in the sc ie nc e of smell, see e.g . [2 1], whic h in cont r ast to the low-level signal pr o c essing describ ed ab ov e, is what we build up on in this work. An imp ortant finding is that the full gam ut of odor perception fo r a comp ound or mixture of co mpounds (including plea santness and whiteness) can be predicted from the physicochemical proper ties of the molecules [22], in part via information proces sing models of cor tex [23]. Moreov er , h uman olfactory p erception is primar ily synt hetic rather than ana lytic. What this means is that when p e ople smell mixtures of comp ounds, they do not p er c eive a mixture of individua l c o mpo und p ercepts. Instead, they perc eive a single ph ys ico chem ica l ob ject all a t o nce, where that single physico chemical ob ject is a weigh ted co m binatio n of the individual physico ch emica l features of the comp ounds in the mixture. One may ask how the o lfactory p er c eption space is r epresented. Ex p er iments hav e human sub jects des c rib e the smell of pure chemical comp ounds in words— tolualdehyde smelling ‘fr agrant,’ ‘ar omatic,’ ‘almond,’ and ‘sweet;’ o r v aleric acid smelling ‘r a ncid,’ ‘sweaty ,’ ‘putrid,’ ‘fecal,’ and ‘sick ening’ [24 ]—resulting in a p erceptual spac e who se dimens io ns are these o dor des criptors. B y averaging o dor desc r iptor judgements ov er several sub jects, each comp ound c an b e placed as a point in a rea l- v alued p erceptual space where the coo rdinate v alue is a function of the p ercentage of sub jects who us e a des criptor for a comp ound. Such exper imen ts hav e only b een conducted on a small se t of comp ounds, but we estimate the per ception of uncharacterized comp ounds and mixtures fr om their physicochemical structure represented as a vector o f features describing the molecule. In pa rticular, we learn the mapping b etw een the physico chemical description 3 of o do rant comp ounds and their p erceptua l descriptions fro m a small amount of training data so it g eneralizes to a ll comp ounds. W e p ose the learning problem as one of multiv ar ia te regr ession. Our training set includes o do r descriptor data (lab els) and ph ys ico ch emica l data (features ) from the small subset of comp ounds for which exp erimentally-determined o dor descriptors exis t. As it is b elieved ol- factory per ception is fairly low-dimensional, w e use nuclear no r m r egularizatio n to k eep the r a nk of the estimated mapping op erator small [2 5]. An a sp ect of o dor p erception not yet fully r e solved in the literatur e is how per ceived o dor intensity is determined by the c oncentration and molecula r pro p- erties of comp ounds and the medium in which they are suspended [26]. W e use straightforward concentration a s an acceptable first-o rder approximation [27]. Given recent s cient ific pro gress on understanding olfactory p erception, the time is r ipe to develop engineer ing theories and technologies that build up on the science for applications in indo or air quality [28, 1], virtual reality [17, 18], culinary arts 1 [30, 2, 31, 32], hun ting [33], and numerous others. In this work, our co n tribution is to take a statistical signa l pro c e ssing p ersp ective on systems inv olving olfaction and develop the basic too ls needed to eng ineer them. Upon showing how to lear n the olfacto r y physicochemical–p erceptual mapping, we develop sp ecific example designs for pr oblems o f active o dor c anc el lation [1] and fo o d ste gano gr aphy [2], a nd discuss many other olfac to ry op era tions a t a high level, including filter ing a nd s mo o thing, enhancement, los sy compressio n, communication and storag e , and retriev al. As far as we know, ther e is no prior work on active o dor ca ncellation or olfactory steganogr aphy . The remainder of this pap er is orga nized as follows. In Section 2, we descr ib e the common fir st step fo r olfactor y s ignal pro cessing of learning the mapping betw een the physico chemical and per ceptual spaces. W e detail the formulation of one example o lfactory signal pro cessing system, active o dor cancella tion, in Section 3 and another, fo o d steganog raphy , in Section 4. W e provide high-level 1 The prim ary contributors to human fla vor p erception are r etronasal and orthonasal smell [29]. 4 views on se veral other systems inv olv ing o lfactory signals in Se c tion 5. Section 6 presents e mpir ical studies on lear ning, cancellation, and steg a nography . Finally , Section 7 summarizes the work a nd pr esents an outlo ok of future work in this area. 2. Learning the M apping Betw een Ph ysico c hemical and Perceptual Spaces The g uiding principle of psychoph ysics , verified ov er centuries of exp eriments with human sub jects, is that the ph ysic al prop erties of a stimulu s largely de- termine its percept. F or olfactory signals, we a ssume there is some general nonlinear ma pping from the physicochemical attributes of a comp ound to its per ceptual o do r desc r iption. In this section we develop a statistical metho dol- ogy to learn a g eneralizable mapping from molecular structure of comp ounds to their p ercept. The goal is to es tima te the per ceptual repre s en tatio n o f com- po unds and mixtur es of comp ounds for which no exper imen tal ground truth on per ception exists, but for which physico chemical prop erties a re readily av aila ble. Human o lfa ctory p erception is difficult to pin down pr ecisely; the mos t com- mon technique us e d in the psychology and s cience liter atures is to present an observer with a list of o do r descriptor words or concepts and have him or her ev aluate whether a given c hemical’s smell matches ea c h o do r descriptor. Aver- aging over ma n y individual o bservers yields a real-v alued o do r des c r iptor space in which ea ch chemical compo und has co ordinates. The physico chemical pro p- erties we c o nsider a re also numerical, so o ur go a l is to lear n a functional mapping betw een the t wo spaces. In this work, w e restr ic t o urselves to linear ma pping s, the v alidity of which is sugges ted by human o lfaction studies [6]. (More com- plex mappings , including p olynomial mappings suggested in [4] a nd kernel-based mappings [34], can be a ccomo dated in the same t yp e o f linear mo del describ ed below.) Thu s , we are given a set of tr aining samples { ( x 1 , y 1 ) , . . . , ( x n , y n ) } where the x i ∈ R k are physicochemical features of comp ounds and the y i ∈ R l are 5 the p erceptual vectors in the o dor des c r iptor space. Desiring a low-dimensional mapping, we use nuclear norm-r egularized multiv ar iate line a r reg ression to learn a matrix A ∗ ∈ R l × k that maps uns een comp ounds from the chemical to the per ceptual space. In pa r ticular, if w e concatenate all the training samples into matrices X ∈ R k × n and Y ∈ R l × n , the problem to so lve is : A ∗ = ar g min A k Y − AX k F + λ k A k ∗ (1) where λ trades data fidelity for sparsity of the singula r v alues of A ∗ . This problem is co n vex and can b e solved by interior p oint metho ds and a v aria n t of Nesterov’s smo oth metho d [25]. Note that Euclidean norms make sense as bo th optimization ob jectives (F rob enius norm) and characteriza tions of system p erformance (RMSE), since they are used in olfactory psychophysics studies with h uman sub jects fr o m sev- eral different lab ora tories [4, 5, 22, 35], and in the rece n t DREAM O lfa ction Prediction Challenge, a part of the Ro ck efeller University Smell Study . 2 3. Activ e Odor Cancellation Noise cance llation is one of the mo st basic of sig nal pro cessing tas k s [36, 37], and thus we use it as the firs t task within which to describ e olfactory signa l pro cessing. The r e are o ften s e ttings where c hemical signals should b e canceled: po o r indo or air qualit y a nd ma lo do rs a re not only a n uisance and source o f dissatisfaction, but can decrease the pro ductivity o f office workers s ix to nine per cent [28]. F o ur genera l ca tegories of techniques are currently used for r educ- ing o r eliminating o dors: m asking , which a ttempts to ‘ov erp ow er’ the offending o dor with a single pleasant o dor; absorbing , which uses a ctive ingr edients like baking so da and activ ated car bo n; eliminating , in which chemicals react with o dor molecules to turn them into iner t, o do rless compounds; and oxidi zing , which accelerates the break- down of malo doro us comp ounds. Instead, here we 2 https:// www.synapse.org / #!Synapse:syn2811262 6 develop a s tatistical signal pro cessing metho d fo r per forming a ctive o do r cancel- lation, with s ome rese mblance to active noise and vibr ation canc e llation [38, 3 9]. W e approach the problem by taking adv antage of the psychoph ysica l pr op- erties of human e nd-consumers of o do r. In par ticular, there is a rec e ntly dis- cov ered percept called olfactory white, which is the neutral smell generated by eq ual-intensit y s timuli well-distributed across the ph ys ico ch emica l space [8], m uch like white light o r auditor y noise. Mor e sp ecifically , the set of all o dor- ant comp ounds spans a particular subspace of physico chemical attribute v alues; when mo re than approximately thir t y comp ounds, all diluted to a concentration having equal p erceived intensit y , are s melled together in a mixture , the result- ing p ercept is a neutral o do r that is the same no matter which compounds are included in the mixture, but only if the comp ounds in the set are fairly uni- formly distributed in the subspa c e [8]. Whiteness is a central concept in a ctive signal cancellation generally sp eaking [37]. Our goal in this section is to sense an existing malo dor and to o utput a comp ound mixture from a virtual ar oma synthesizer such that the r esulting combined o dor is white. In the active o do r c ancellation a pplications of interest to us, several differ en t malo dors may b e sensed and canceled by the same virtual aro ma synth es izer. Therefore, in addition to providing excellent cancellation p erforma nce, we also desire the car dinality of the comp ound set in the sys tem to be minimized be- cause it is costly to hav e many cartridges. T ow ard this goal, w e use the g roup lasso or sim ultaneous sparsity-inducing ℓ 1 /ℓ 2 norm [40]. W e also requir e a non- negativity c onstraint b ecause optimized co mpo und mixtures ca n o nly b e output int o the air, not subtra cted [41]. Due to the synthetic nature o f human olfac tion, the g e nerally nonlinear p erceptual mapping (simplified to linear in this pap er) is applied to the physico chemical representation of mixtures o f comp ounds exha led by the system. As a star ting point, we co lle c t a set of n co mpo unds that co uld po ssibly b e used in the aro ma synthesizer. Let the physicochemical repr esentation of this dictionary b e X dict ∈ R k × n . W e would like to design the system to optimally cancel m different malo dor s with p erceptual representations Y mal ∈ R l × m . W e 7 would like to determine a simultaneously spa r se s e t o f non-nega tive co efficients W ∗ ∈ R n × m + that minimize: 1 2 k Y mal + A ∗ ( X dict W ) k 2 F + µ k W k 1 , 2 , s. t. W ≥ 0 , (2) where µ is a regular iz a tion para meter, a nd the ℓ 1 /ℓ 2 norm takes ℓ 2 norms o f each of the n length- m rows of W firs t and then takes the ℓ 1 norm of the r e- sulting leng th- n vector. The physicochemical-to-p erceptual mapping A ∗ comes from the learning problem describ ed in Section 2. The data fidelity term is k Y mal + A ∗ ( X dict W ) k 2 F bec ause the all-zero s vector in the per ceptual space is an olfactory white. A more general formulation for the o ptimiza tion can inco rp orate the fac t that there is not a single olfac to ry white, but a family of them. The mor e general optimization pr oblem is then a minimization over the co efficients and the particular olfactory whites in the family: 1 2 k Y mal + A ∗ ( X dict W ) − Y white k 2 F + µ k W k 1 , 2 , s. t. W ≥ 0 , Y white ∈ Y white . (3) The family Y white can be sp ecified as the s et of matrices with all rows within a co lumn having the same v alue, i.e., of the form 1c T , where 1 is the length l all-ones vector and c is a leng th m vector that is free to b e decided up on. 4. F o o d Steganograph y Many children (and adults) are picky eaters to whom junk fo o d is mor e at- tractive than healthy fo o d. This instinct was use ful fo r hun ter - gatherers tha t depe nded heavily on their senses to decide what to eat: in nature, sweet foo ds are almo st always safe and nutritious where a s fo o ds that smell o dd ar e p o ten- tially toxic or sp oiled. In mo der n en viro nmen ts, this same instinct o ften serves to make pe ople ob ese a nd chronically ill. Hiding a n utritio us , av erse fo o d in a delectable fo o d may therefor e aid p eople in eating healthier . A second problem, closely related to ac tive o dor cance lla tion, is hiding the flav or of one foo d inside the flav or of ano ther fo o d thro ug h the use of an a ddi- 8 Figure 1: Depiction of foo d steganograph y in the p erceptual domain, where macaroni & c heese is delectable, cauliflow er i s av erse, and the white p o wder is the additive. tive: fo o d ste gano gr aphy . Steg anogra ph y is the very old conce pt of imp ercep- tibly hiding a signal into a cov er medium [42, 43, 4 4, 45], which has a s ig nal pro cessing flavor in appro a ches like sprea d sp ectrum image s teg anogra ph y [46]. W e demonstrate a statistical signal pro ces sing approa ch to optimally des ign a fo o d additive (either using pure comp ounds o r natural ingredients) to ac t as a steganog r aphic key for this fo o d stega nography problem. The stega no graphic per cepts a re de pic ted in Fig. 1, illustrated using the hiding of ca uliflow er inside of macaro ni a nd c heese as an example. Note that ther e are many p os s ible goals in stegano g raphy; her ein the goa l is not for the receiver to dec ipher a hidden message, but only to make imper ceptible a fo o d to which the receiver is av erse (and whic h may hav e g o o d nutritional prop erties). A fo o d additive (steganog raphic key) combines with the averse fo o d (hidden signal), and the delectable fo o d (cover medium) such that the co m binatio n is p erceived a s only the delectable fo o d’s flavor; the olfacto ry white signa l is used as a mathema tica l intermediary . The fo o d additive may b e comp osed of some weigh ted mixture of pure c ompo unds or some weight ed mixture of fo o d ingredients from a dictiona ry . W e may also want to regularize the pr oblem by including a spars ity or o ther cost-related pena lt y on the fo o d additive. There ar e t ypica lly tens to hundreds of differe n t chemical comp ounds con- tributing to flavor per fo o d ingredient. Using da ta on the concen tra tions o f comp ounds in fo o ds, we take a weigh ted co m bina tio n of the physicochemical vectors of the c onstituent comp ounds of a fo o d to determine its p erceptual rep- resentation using the mapping lear ned in Section 2. Next, we so lve a re gularized 9 inv erse pr oblem with a non- ne g ativity cons tr aint to find comp ounds or fo o ds and their coe fficien ts required to synthesize an additive that pr o duce s olfactor y white when combined with an averse fo o d of in ter est. Let X cov be the physico chemical re presentation of the cov er medium’s com- po unds and w cov be the concentrations of the cov er medium’s co mpo unds . Like- wise let us in tro duce X hid and w hid for the hidden data. Let X dict be a dictio- nary of n p oss ible co mp ounds from which we c an cons truct the steganogr a phic key (fo o d additive) along with its weigh t vector w dict , which is the sub ject of design. First, with a ge neral nonlinear physico chemical-to-pe r ceptual ma pping A ∗ ( · ), the p erceptual hiding we w ant to p erfor m is to c ho os e w dict to satisfy: A ∗ ( X cov w cov + X hid w hid + X dict w dict ) ≈ A ∗ ( X cov w cov ) . (4) With the linear mapping that we a re as s uming in this work, the ob jective simplifies to: A ∗ X hid w hid ≈ − A ∗ X dict w dict . (5) The determination of the s teganogra phic key do es not dep end on the co ver medium, and is simply the o dor ca nce llation signal from Section 3 . Sp ecifically , we solve the following optimization problem: min w dict k A ∗ X hid w hid + A ∗ X dict w dict k 2 2 + ν J ( w dict ) (6) s. t. w dict ≥ 0 where J ( · ) could b e one o f a num b er o f po ssible regula rization terms mea nt to promote secondar y o b jectiv es s uc h a s mo netary fruga lit y , sparsity , or nutrition. In the case tha t we only wan t to use a s e t of n ′ fo o d ingredients to co mp ose the additive, we first use data o n the known concentrations of comp ounds in fo o d ingredien ts to construct an n × n ′ weigh t matrix W ingr that multiplies X dict to obtain a dictionar y of fo o d ingredient physico chem ica l features. The weigh t v ec to r to b e solved for is then an n ′ × 1 vector w ingr : min w ingr k A ∗ X hid w hid + A ∗ X dict W ingr w ingr k 2 2 + ν J ( w ingr ) (7) s. t. w ingr ≥ 0 . 10 5. Other Sys tems W e hav e detailed tw o sp ecific olfactory signal pr o c essing sys tems in the pr e- vious tw o sectio ns. Ho wev er , the v ar iet y of p os sible tasks one may wan t to per form in the olfacto ry mo dality is as broad as in other mo dalities. In this section, we dis cuss formulations for several o f those other p ossible tas k s. This section is not meant to b e a n exha ustive cov er age o f all p ossible olfactor y signa l pro cessing task s, but is meant to show case the realm o f p ossibilities and spur future resear ch. The main terms in the ob jectives of ca nce llation and steganog raphy (2), (6), and (7) a re as they a re b ecause we desir e a neutral all-zer os pe r ception a s output in those task s. How ever, there is nothing preven ting us from inserting a desired tar g et or output o do r p ercept into the ob jective (as we did for the full family o f olfacto r y w hites ), which would allow us to p erform general filtering or equa lization op erations. With a given desired p e rceptual output y des , the problem is: min w k A ∗ x in + A ∗ X dict w − y des k 2 2 + µJ ( w ) , s. t. w ≥ 0 , (8) Olfactory cancellation, filtering , or equaliz a tion will, in g eneral, ta ke place in dynamic rather than static environmen ts. F or example, think of the indo or air quality of an auto mobile traveling from a chemical plant to an urban envi- ronment via a garba ge dump. Pro blem (8) ca n b e extended to include a time v ariable t that applies to x in , w , and p os s ibly y des . The dynamic version o f the pro blem can b e a ddressed using the theory o f adaptive filtering; o ne sp ecific wa y to formulate the adaptive version is through a v aria tion o n a regular ized LMS algor ithm with nonnega tivit y constr aint [4 7, 4 8, 49]. The differe nce fro m the s tandard ada ptiv e linea r combiner here is tha t w t is m ultiplied by the dictionar y o f av ailable co mpo unds and x in ,t is an a dditiv e term; w t and x in ,t are no t m ultiplied o r co nv olved with each other . The up date 11 rule for an LMS-like adaptive filter is: w t +1 = w t − 2 η diag { w t } ×  ( A ∗ X dict ) T ( A ∗ x in ,t + A ∗ X dict w t − y des ,t ) + µ∂ J ( w t )  , (9) where η is the step size of the LMS alg orithm. Virtual reality applica tions may requir e a ‘smelltrack’ similar to a sound- track to acco mpany mo tion pictures. In s uc h applicatio ns, we can a ssume that the desired olfactory p erception signa l ov er the entire time pe r io d is known in adv ance and that there is no ambien t o dor to ov er c o me. Therefore , we may be able to do b etter than adaptive filtering. The difficulty is that comp ounds exhaled into a n environmen t ling er for s o me duration. If we develop a sto chastic mo del for this p ersis tence, p erha ps using a Gauss ian puff mo del [50, 51] com- po sed with a psychoph ysica l sniff mo del [52, 5 3], then we can use a ppropriate extensions o f the Rauch-T ung-Strieb el smo other to obta in an optimal control strategy of virtual ar oma synthesizer a ctuation. Suc h an approa ch can also a l- low the comp oser to o nly s pecify ‘key frames’ of smell with the s ig nal pro cessing algorithm in ter po la ting the rest. An alternative to sp ecifying a desired output s ignal y des , as in adaptiv e filtering, is to specify the desir ed behavior a nd r e quirements o f the filter to b e designed without using a sp ecific input sig nal realizatio n in the ob jective [5 4]. F or example, a desir ed b ehavior might b e to allow pleasa nt o dor comp onents to pa ss throug h the filter unchanged and to cancel unpleasant o dor comp onents (similar to a low-pass filter), or to allow all o dors except fo r the odor des criptor ‘vomit’ to pas s (similar to a notch filter ). Suc h filter s do not dep end on the input signal, but hav e the sa me multiplicativ e or conv olutive behavior for all inputs. Unfortunately in o ur s ignal pro ce ssing approa c h, we affect the synthesized per ceptual repre s en tatio n o f the output through the sup erp osition of the input and a set of comp ounds we design, not by physically filtering different com- po unds or t yp es of comp o unds. This means, as previo usly noted, that the sp ecification w do es not multiply (or c o nv olve) the input s ig nal x in , but adds 12 to it after b eing mo dulated by the dictio nary X dict . This implies filters imple- men ting multiplicativ e b ehavior a re not po ssible for o lfactory signals . Letting y in = A ∗ x in and y w = A ∗ X dict w , we are saying that it is not p ossible to choos e a y w such that y in + y w = 0 for all y in . F urthermore, we cannot define general filterbank deco mpo sitions either. What is not pr ecluded, howev er, is de c o mpo sing sp ecific rea liz ations of o dor s ignals. F or example, extending the idea of Kisstixx lip balm, 3 we ca n de c o mpo se a well-recognized p erceived o dor into tw o different p erceived o dor s which a re each well-recognized separately , just like decomp osing into parts with no n-negative matrix factorizatio n [55]. The learned p erceptual mapping A ∗ allows us to de fine a distor tion function betw een comp o und mixtures that can b e used for a v ariety of pro ce ssing tasks including los sy compress ion (i.e. choo sing a different, les s cos tly set of co m- po unds to approximately r epro duce the o do r of the origina l set of co mpo unds ), denoising, hashing, a nd retriev al. Moreov er , the sto r age and communication of olfactory signal data only needs to b e in the synth etic p erceptual domain; data on the analytic physical co mpo unds o f smells is not required. Going one step further, since pleasa n tness is the main comp onent of o dor signals, storing and comm unica ting only the pleasantness scala r v alue or scalar time series is often the only desir e [56]. F or ex ample, pleasantness and not any higher-or der odo r dimensions, is o ne o f the tw o main criteria in the selection step of a successful computational crea tivit y system for culinary r ecip es [30, 32, 31]. Finally , w e co njectur e that o dor enhancement ca n b e appr oached in muc h the same wa y as imag e enhancement via hue and satur ation. As we noted earlier, olfactory white emerges only with a lar ge num b er of comp ounds that s pa n the per ceptual space. Olfactor y white can be viewed a s a fully desatur ated o do r . This reasoning implies that we can start with an o dor a nd decr ease its satura tion by gra dually adding many comp ounds fr om different parts o f the space, a nd increase its saturatio n by adding to the concentration of a few comp ounds fr om 3 http://k isstixx.com/ 13 one part of the spa ce. W e can change the hue of an o dor , e.g. start with the smell of a rose and make it ‘fishier’ [57], by adding appropria te co mpo unds while ensuring not to alter the sa tur ation. 6. Empirical R esults In this s ection, we pr esent illustrative empir ical results on learning the map- ping from physico chemical features to o do r des criptors and using this mapping for active o dor cancellatio n in a setting that may ar ise in the break r o om o r lunch ro om of a s mall office. W e als o illustrate the use of this mapping for fo o d steganogr aphy where the hidden foo d is co oked bro ccoli, which has many po sitive nutritional q ua lities but to which many have av ers ion. 6.1. L e arning the Mapping The fir st step in o ur empirica l study is to lea r n the mapping A ∗ from physic- o chemical prop erties of comp ounds to the olfactory p erception of thos e com- po unds. W e collec t a ( k = 18 )-dimensional ph ysico chemical feature vector for each of 143 different chemical co mpounds that hav e b een judged by human ob- servers a gainst l = 14 6 different o dor des c riptors a s diverse as ‘almo nd,’ ‘cat urine,’ ‘soa py ,’ ‘stale to bacco smo ke,’ a nd ‘vio lets.’ The 18 ph ys ico ch emica l fea- tures a re obtained from the Nationa l Center for B io technology Informa tion’s PubChem Pro ject a nd include among others : top ologica l p olar surfa ce area, partition co efficient pr ediction (XLog P), molecular weigh t, complexity , heavy atom count, hydrogen b ond donor count, and ta utomer count. The feature v al- ues a re the pr op erties exhibited by a single molecule of the co mpo und, e.g., for eth yl pyrazine, top ologic a l p olar surfac e a rea = 25.8, XLogP = 0.7 , molecular weigh t = 10 8.1411 2, a nd so o n. The human judgements on o dor de s criptors are obtained fr om the Atlas o f Odor Char a cter P rofiles (A O CP) [24], which po o ls da ta from a panel of hundreds of flav or/fr a grance exp erts. 4 The tw o data 4 W e use the p ercent age of applicability data f rom A OCP . F or eac h compound, many human experts ev aluate it against each of the 146 descriptors on a zero to fiv e scale. The p ercent age 14 10 1 10 2 10 3 10 4 10 5 10 6 10 7 4 6 8 10 λ average RMSE Figure 2: Five-fold cross- v alidation testing ro ot mean squared error of the mapping b et ween ph ysico che mi cal and p erceptual spaces av eraged across the 146 p erceptual di mensions. sets are matched and joined using Che mica l Abstracts Service (CAS) Registry nu mber s. Using the p ercept matrix Y ∈ R 146 × 143 from AOCP and the physicochem- ical matrix X ∈ R 18 × 143 from PubChem, we lea rn the mapping b y solving the nuclear norm-r egularized m ultiv aria te linear regr ession pro blem discussed in Section 2 using the method of [25]. W e conduct five-fold cr oss-v alidation to determine the b est v alue of λ . As a figure of merit, we consider the ro ot mean squared err or (RMSE) av era ged over the 146 dimensions; Fig . 2 shows the cross- v alidation testing average RMSE as a function of λ . The erro r is minimized at approximately λ = 10 4 and is the v a lue we use go ing for ward. 6.2. A ctive Odor Canc el lation W e consider m = 4 different offending o dors that we wish to cancel with the same, small-c a rdinality set of olfac tory comp ounds. The four s mells are: durian (Durio zib ethin us), onion (Allium cepa L.), k atsuo bus hi (dried bo nito), and sa uerkraut. With a n optimal solution to the problem (2), w e can cr eate a device with minimal complex ity that sense s the current o dor and outputs the appropria te concentrations of comp ounds to ca nce l it. When placed in a lunch of applicabili ty f or a giv en descriptor ranges fr om zero to one hu ndred and is the geometric mean of the percentage of experts who give a score greater than zero and the ratio of the sum of the scores and fiv e times the num b er of exp erts as a p ercen tage [24]. When in terpreting accuracy results in this section, the definition of p ercent age of applicability should be kept in mind. 15 ALCOHOLIC ALMOND AMMONIA ANIMAL ANISE (LICORICE) APPLE (FRUIT) AROMATIC BAKERY (FRESH BREAD) BANANA BARK, BIRCH BARK BEANY BEERY BITTER BLACK PEPPER BLOOD, RAW MEAT BURNT CANDLE BURNT MILK BURNT PAPER BURNT RUBBER BURNT, SMOKY BUTTERY, FRESH BUTTER CADAVEROUS (DEAD ANIMAL) CAMPHOR CANTALOUPE, HONEY DEW MELON CARAMEL CARAWAY CARDBOARD CAT URINE CEDARWOOD CELERY CHALKY CHEESY CHEMICAL CHERRY (BERRY) CHOCOLATE CINNAMON CLEANING FLUID CLOVE COCONUT COFFEE COLOGNE COOKED VEGETABLES COOL, COOLING CORK CREOSOTE CRUSHED GRASS CRUSHED WEEDS DILL DIRTY LINEN DISINFECTANT, CARBOLIC DRY, POWDERY EGGY (FRESH EGGS) ETHERISH, ANAESTHETIC EUCALIPTUS FECAL (LIKE MANURE) FERMENTED (ROTTEN) FRUIT FISHY FLORAL FRAGRANT FRESH GREEN VEGETABLES FRESH TOBACCO SMOKE FRIED CHICKEN FRUITY, CITRUS FRUITY, OTHER THAN CITRUS GARLIC, ONION GASOLINE, SOLVENT GERANIUM LEAVES GRAINY (AS GRAIN) GRAPE JUICE GRAPEFRUIT GREEN PEPPER HAY HEAVY HERBAL, GREEN, CUT GRASS HONEY HOUSEHOLD GAS INCENSE KEROSENE KIPPERY (SMOKED FISH) LAUREL LEAVES LAVENDER LEATHER LEMON LIGHT MALTY MAPLE SYRUP MEATY (COOKED, GOOD) MEDICINAL METALLIC MINTY, PEPPERMINT MOLASSES MOTHBALLS MOUSE MUSHROOM MUSK MUSTY, EARTHY, MOLDY NAIL POLISH REMOVER NEW RUBBER NUTTY (WALNUT ETC) OAK WOOD, COGNAC OILY, FATTY ORANGE PAINT PEACH (FRUIT) PEANUT BUTTER PEAR PERFUMERY PINEAPPLE POPCORN PUTRID, FOUL, DECAYED RAISINS RANCID RAW CUCUMBER RAW POTATO ROPE ROSE SAUERKRAUT SEASONING (FOR MEAT) SEMINAL, SPERM SEWER SHARP, PUNGENT, ACID SICKENING SOAPY SOOTY SOUPY SOUR MILK SOUR, VINEGAR SPICY STALE STALE TOBACCO SMOKE STRAWBERRY SULFIDIC SWEATY SWEET TAR TEA LEAVES TURPENTINE (PINE OIL) URINE VANILLA VARNISH VIOLETS WARM WET PAPER WET WOOL, WET DOG WOODY, RESINOUS YEASTY Durian Katsuobushi Sauerkraut Onion −5 0 5 10 Figure 3: Pe rceptual pro jection of the mixture of comp ounds con tained in du r i an, k atsuobushi, sauerkraut, and onion. ro om, the device will b e able to cancel these four o dor s , but also ma ny others. The p erce ptual repres e n tation of the four o dor mixtur es of interest can b e predicted from the learned mapping. First, in the same spir it a s the s ynt hesis that takes place in human olfactor y p erceptio n, we take a linear combination of the physicochemical features o f the comp onents of the o dor and then map the res ulting physicochemical vector to p erceptual space. W e o btain the set of olfactory co mpounds present in the four o dors and their concentrations from the V olatile Comp ounds in F o o d 14.1 database (VCF) and obtain physico chemical features o f those comp ounds fro m PubChem. The re sulting predicted p ercep- tions o f duria n, k atsuobushi, saue r kraut, and onion are shown in Fig. 3. F o r example, it can b e seen in the figur e that sauerkra ut is p erceived most lik e the ‘oily , fa tt y’ descriptor and lea st like the ‘fruity , citrus’ descriptor. The o dor de- scriptors with la rgest p ositive co efficients fo r the other three malo dor s a re ‘sick- ening,’ ‘fr agrant,’ and ‘sweet,’ resp ectively . The o dor descriptors with larg est negative co e fficien ts for the other three malo dors are ‘fr a grant,’ ‘chemical,’ a nd ‘chemical,’ resp ectively . Having pr edicted the per ception of the four o dors of interest, the next step is to find comp ounds that can be used to cancel their smells per ceptually . T oward this end, we first construct a dictiona ry of comp ounds from which we can find the cancellation set. W e ex tract n = 5736 co mpo unds from V CF found nat- urally in fo o d a nd find their physico chemical prop er ties from PubChem. This dictionary , with members only from natural edible pro ducts has certain lim- itations, which we comment on later . W e use the non-neg ative simult a neous 16 0 1000 2000 3000 4000 5000 0 0.2 0.4 Durian 0 1000 2000 3000 4000 5000 0 2 4 Katsuobushi 0 1000 2000 3000 4000 5000 0 5 Sauerkraut 0 1000 2000 3000 4000 5000 0 1 2 Onion Figure 4: Di ctionary coefficient v alues in optimal cancellation solution wi th µ = 1. ALCOHOLIC ALMOND AMMONIA ANIMAL ANISE (LICORICE) APPLE (FRUIT) AROMATIC BAKERY (FRESH BREAD) BANANA BARK, BIRCH BARK BEANY BEERY BITTER BLACK PEPPER BLOOD, RAW MEAT BURNT CANDLE BURNT MILK BURNT PAPER BURNT RUBBER BURNT, SMOKY BUTTERY, FRESH BUTTER CADAVEROUS (DEAD ANIMAL) CAMPHOR CANTALOUPE, HONEY DEW MELON CARAMEL CARAWAY CARDBOARD CAT URINE CEDARWOOD CELERY CHALKY CHEESY CHEMICAL CHERRY (BERRY) CHOCOLATE CINNAMON CLEANING FLUID CLOVE COCONUT COFFEE COLOGNE COOKED VEGETABLES COOL, COOLING CORK CREOSOTE CRUSHED GRASS CRUSHED WEEDS DILL DIRTY LINEN DISINFECTANT, CARBOLIC DRY, POWDERY EGGY (FRESH EGGS) ETHERISH, ANAESTHETIC EUCALIPTUS FECAL (LIKE MANURE) FERMENTED (ROTTEN) FRUIT FISHY FLORAL FRAGRANT FRESH GREEN VEGETABLES FRESH TOBACCO SMOKE FRIED CHICKEN FRUITY, CITRUS FRUITY, OTHER THAN CITRUS GARLIC, ONION GASOLINE, SOLVENT GERANIUM LEAVES GRAINY (AS GRAIN) GRAPE JUICE GRAPEFRUIT GREEN PEPPER HAY HEAVY HERBAL, GREEN, CUT GRASS HONEY HOUSEHOLD GAS INCENSE KEROSENE KIPPERY (SMOKED FISH) LAUREL LEAVES LAVENDER LEATHER LEMON LIGHT MALTY MAPLE SYRUP MEATY (COOKED, GOOD) MEDICINAL METALLIC MINTY, PEPPERMINT MOLASSES MOTHBALLS MOUSE MUSHROOM MUSK MUSTY, EARTHY, MOLDY NAIL POLISH REMOVER NEW RUBBER NUTTY (WALNUT ETC) OAK WOOD, COGNAC OILY, FATTY ORANGE PAINT PEACH (FRUIT) PEANUT BUTTER PEAR PERFUMERY PINEAPPLE POPCORN PUTRID, FOUL, DECAYED RAISINS RANCID RAW CUCUMBER RAW POTATO ROPE ROSE SAUERKRAUT SEASONING (FOR MEAT) SEMINAL, SPERM SEWER SHARP, PUNGENT, ACID SICKENING SOAPY SOOTY SOUPY SOUR MILK SOUR, VINEGAR SPICY STALE STALE TOBACCO SMOKE STRAWBERRY SULFIDIC SWEATY SWEET TAR TEA LEAVES TURPENTINE (PINE OIL) URINE VANILLA VARNISH VIOLETS WARM WET PAPER WET WOOL, WET DOG WOODY, RESINOUS YEASTY Durian Katsuobushi Sauerkraut Onion −5 0 5 10 Figure 5: Percept ual representat i on of residual odor after cancellation of duri an, k atsuobushi, sauerkraut, and onion with µ = 1. sparsity formulation given in Section 3 with this dictionary to find the o ptimal sparse set of comp ounds for active o dor cancellatio n with different v alues o f the regular iz ation parameter µ . W e use SDPT3 to solve the optimization problem [58]. The set of co efficients W found for µ = 1 is shown in Fig . 4. Ther e ar e 22 comp ounds with p ositive co efficient v alue in at leas t one of the four cancellatio n additives. The res idual o dor remaining after ca ncellation is shown in Fig. 5. The F rob enius norm of the res idual is 17 .1 3 a nd the ℓ 2 norms of the individual o dors are 1.41 fo r durian, 4.38 for k atsuo bus hi, 16.30 for sauerkr aut, and 2.50 for onion. By reducing µ , we can improve the ca ncellation at the exp ense of increas ing the num b er of comp ounds used. The co efficients in the optimal solution for µ = 0 . 25 are shown in Fig. 6 and the r esidual p erception in Fig. 7. In this solution, 38 comp ounds have po sitive co efficients and the F rob enius norm of the residual is 2 .30. Residual ℓ 2 norms of individual o dors are: durian 0.04, 17 0 1000 2000 3000 4000 5000 0 0.2 0.4 Durian 0 1000 2000 3000 4000 5000 0 1 2 Katsuobushi 0 1000 2000 3000 4000 5000 0 5 10 Sauerkraut 0 1000 2000 3000 4000 5000 0 0.5 1 Onion Figure 6: Di ctionary coefficient v alues in optimal cancellation solution wi th µ = 0 . 25. ALCOHOLIC ALMOND AMMONIA ANIMAL ANISE (LICORICE) APPLE (FRUIT) AROMATIC BAKERY (FRESH BREAD) BANANA BARK, BIRCH BARK BEANY BEERY BITTER BLACK PEPPER BLOOD, RAW MEAT BURNT CANDLE BURNT MILK BURNT PAPER BURNT RUBBER BURNT, SMOKY BUTTERY, FRESH BUTTER CADAVEROUS (DEAD ANIMAL) CAMPHOR CANTALOUPE, HONEY DEW MELON CARAMEL CARAWAY CARDBOARD CAT URINE CEDARWOOD CELERY CHALKY CHEESY CHEMICAL CHERRY (BERRY) CHOCOLATE CINNAMON CLEANING FLUID CLOVE COCONUT COFFEE COLOGNE COOKED VEGETABLES COOL, COOLING CORK CREOSOTE CRUSHED GRASS CRUSHED WEEDS DILL DIRTY LINEN DISINFECTANT, CARBOLIC DRY, POWDERY EGGY (FRESH EGGS) ETHERISH, ANAESTHETIC EUCALIPTUS FECAL (LIKE MANURE) FERMENTED (ROTTEN) FRUIT FISHY FLORAL FRAGRANT FRESH GREEN VEGETABLES FRESH TOBACCO SMOKE FRIED CHICKEN FRUITY, CITRUS FRUITY, OTHER THAN CITRUS GARLIC, ONION GASOLINE, SOLVENT GERANIUM LEAVES GRAINY (AS GRAIN) GRAPE JUICE GRAPEFRUIT GREEN PEPPER HAY HEAVY HERBAL, GREEN, CUT GRASS HONEY HOUSEHOLD GAS INCENSE KEROSENE KIPPERY (SMOKED FISH) LAUREL LEAVES LAVENDER LEATHER LEMON LIGHT MALTY MAPLE SYRUP MEATY (COOKED, GOOD) MEDICINAL METALLIC MINTY, PEPPERMINT MOLASSES MOTHBALLS MOUSE MUSHROOM MUSK MUSTY, EARTHY, MOLDY NAIL POLISH REMOVER NEW RUBBER NUTTY (WALNUT ETC) OAK WOOD, COGNAC OILY, FATTY ORANGE PAINT PEACH (FRUIT) PEANUT BUTTER PEAR PERFUMERY PINEAPPLE POPCORN PUTRID, FOUL, DECAYED RAISINS RANCID RAW CUCUMBER RAW POTATO ROPE ROSE SAUERKRAUT SEASONING (FOR MEAT) SEMINAL, SPERM SEWER SHARP, PUNGENT, ACID SICKENING SOAPY SOOTY SOUPY SOUR MILK SOUR, VINEGAR SPICY STALE STALE TOBACCO SMOKE STRAWBERRY SULFIDIC SWEATY SWEET TAR TEA LEAVES TURPENTINE (PINE OIL) URINE VANILLA VARNISH VIOLETS WARM WET PAPER WET WOOL, WET DOG WOODY, RESINOUS YEASTY Durian Katsuobushi Sauerkraut Onion −5 0 5 10 Figure 7: Percept ual representat i on of residual odor after cancellation of duri an, k atsuobushi, sauerkraut, and onion with µ = 0 . 25. k atsuobushi 0.12, sauerkraut 2.29, and onion 0.24 . The µ = 1 solution do es provide a cer tain level o f o dor c a ncellation, but just by dec reasing the sparsity a little bit, we are able to get very g o o d ca nc e lla tion. Only the residual of sauerkr a ut is non-neg ligible in the µ = 0 . 2 5 so lution, and even that is nearing negligibilit y . W e no te that cer tain parts of the v arious o dor sig natures are easier to cancel tha n others . F or example, the descriptor ‘medicinal’ is mostly remov ed from the s auerkra ut s olution with µ = 1 but ‘eucalyptus’ is not. With a limited budget on their num b er, comp ounds tha t affect all four o dor s are a t a premium. Thirteen compo unds (out of 22 and 38 , resp ectively) ar e common to the tw o solutions: ‘(+)-cyclo sativene,’ ‘(E,E,Z)- 1,3,5,8- undecatetraene,’ ‘(R)-3-hydroxy-2-p en tano ne,’ ‘1,3,5 ,8 -undecatetraene,’ ‘10-methyl-2-undecenal,’ ‘cis-pip eritol oxide,’ ‘cub enene,’ ‘cy c lo o c ta tetraene,’ ‘dehydrocurdio ne,’ ‘ethylp yrro le (unkn.str.),’ ‘heptatriacontane,’ ‘junip er cam- phor,’ and ‘methane.’ 18 first perceptual principal component second perceptual principal component Figure 8: Principal comp onent pro jection of p erceptual v ectors of dictionary and four o dors. The blue squares are the four o dors to b e canceled, the red triangles are compounds selected only in the µ = 1 sol ution, the magen ta diamonds are comp ounds selected only in the µ = 0 . 25 solution, the maro on circles are the comp ounds selected in b oth the µ = 1 and µ = 0 . 25 solutions, and the black p oin ts are all other compounds in the dictionary . As discuss ed in Section 1, our formulation of active odo r cancellatio n is as- so ciated w ith the concept of olfactor y white, whic h emerges with ar ound thirty (but not with fewer) comp ounds o f equal intensit y cov ering the space of com- po unds fairly evenly . W e visualize the space of co mpounds using the first tw o principal compo nen ts of the p erceptual vectors of the comp ounds in the dictio- nary a nd the four o dors under co nsideration in Fig. 8. The comp ounds with non-zero co efficient v alues do span the space as b est as they can to pro duce something akin to olfactor y white. It is in ter esting to note that the mo dest in- crease from 2 2 to 38 comp ounds yields such a lar ge improv ement in cance llation quality where these tw o v alues ar e on either side of the num be r r equired for olfactory white. In the vis ualization, we also s ee that the dictionary we hav e used do es no t well-cov er the full space; this is partly b ecause the only com- po unds we have used are present in fo o d pro ducts, s uggesting that for improv ed cancellation, we s ho uld co ns ider a more diverse dictionary that covers the spa ce of olfactory per ception b etter. 19 −15 −10 −5 0 5 10 15 20 SICKENING GARLIC, ONION SHARP, PUNGENT, ACID PUTRID, FOUL, DECAYED SULFIDIC HEAVY RANCID HOUSEHOLD GAS SOUR, VINEGAR SEWER MINTY, PEPPERMINT CAMPHOR EUCALIPTUS SPICY COOL, COOLING BLACK PEPPER CADAVEROUS (DEAD ANIMAL) SWEATY CEDARWOOD OILY, FATTY BURNT RUBBER FECAL (LIKE MANURE) SEASONING (FOR MEAT) SAUERKRAUT CINNAMON COOKED VEGETABLES POPCORN HERBAL, GREEN, CUT GRASS STALE BLOOD, RAW MEAT MEATY (COOKED, GOOD) SOUR MILK METALLIC PEANUT BUTTER BURNT, SMOKY TURPENTINE (PINE OIL) GASOLINE, SOLVENT WET WOOL, WET DOG FERMENTED (ROTTEN) FRUIT DIRTY LINEN RAW CUCUMBER CRUSHED WEEDS MOUSE YEASTY GRAPEFRUIT ROPE CAT URINE BITTER CANTALOUPE, HONEY DEW MELON DILL BURNT CANDLE FRIED CHICKEN APPLE (FRUIT) SOUPY BEANY FRESH GREEN VEGETABLES PINEAPPLE MUSTY, EARTHY, MOLDY ANIMAL GREEN PEPPER AMMONIA NEW RUBBER MEDICINAL BUTTERY, FRESH BUTTER TAR LAUREL LEAVES CRUSHED GRASS CARDBOARD SOOTY STALE TOBACCO SMOKE BURNT MILK FRESH TOBACCO SMOKE CORK PEAR WET PAPER RAW POTATO EGGY (FRESH EGGS) HAY BANANA KEROSENE URINE CARAWAY CHEESY PEACH (FRUIT) GRAINY (AS GRAIN) BURNT PAPER CHALKY MUSHROOM COCONUT TEA LEAVES GRAPE JUICE NUTTY (WALNUT ETC) SEMINAL, SPERM ALMOND GERANIUM LEAVES CLOVE BAKERY (FRESH BREAD) SOAPY WARM CHOCOLATE CHEMICAL STRAWBERRY INCENSE COFFEE OAK WOOD, COGNAC MOLASSES KIPPERY (SMOKED FISH) HONEY NAIL POLISH REMOVER LEMON CELERY DRY, POWDERY CHERRY (BERRY) CREOSOTE RAISINS BARK, BIRCH BARK VARNISH CLEANING FLUID CARAMEL WOODY, RESINOUS LEATHER MALTY BEERY ORANGE VANILLA ETHERISH, ANAESTHETIC FRUITY, CITRUS MUSK DISINFECTANT, CARBOLIC MAPLE SYRUP ALCOHOLIC LAVENDER FRUITY, OTHER THAN CITRUS PAINT VIOLETS MOTHBALLS FISHY ANISE (LICORICE) COLOGNE AROMATIC ROSE LIGHT FLORAL PERFUMERY SWEET FRAGRANT Coefficient Value Figure 9: Perceptual pro jection of the mixture of compounds con tained in bro ccoli. 6.3. F o o d Ste gano gr aphy T o demonstr ate our approa ch to fo o d stegano graphy , we design fo o d addi- tives to act a s stega nographic keys fo r co oked br o c c oli, wher e the cov er medium may be cheese or mango juice. (As discussed in Section 4, the cover medium do es not matter under the linearity assumption.) Similar ly to dur ian, k atsuobus hi, sauerkr a ut, and onion, we firs t characterize bro cco li physicochemically and pe r - ceptually . The 21 comp ounds in co oked bro ccoli from VCF ar e g iv en in T able 1. W e take the concentration v alues as the weigh ts w j and norma lize to unit ℓ 2 norm, obtaining the physicochemical repres en tatio n of the mixture X hid w hid . The r esult o f pro jecting the mixtur e into p erceptual spa ce using A ∗ is s hown in Fig. 9 . The mos t pro minen t predicted o dor descr iptors for co oked bro ccoli are sick ening, g arlic/onio n, a nd sharp/ pungent / acid, which s pea ks to why many peo ple dislike it. The pure comp ounds dictionar y asso ciated with inv er se pr o blem (6) is the same one used in the active o do r cancella tion study o f comp ounds found nat- urally in fo o d with n = 5736 . W e als o construct an n ′ = 297 fo o d ingr edien ts dictionary from V CF data asso cia ted with the in verse pro blem (7). Sp ecifically , we only include foo d pro ducts with at least 15% of their listed comp ounds hav- ing b oth a match in PubChem a nd having a concentration v alue listed. If a range of concentrations is listed in VCF we use the midp oint of the rang e; if the v a lue is listed as ‘trace,’ we use the v alue 10 − 6 parts per million. All foo d ingredient concentrations are norma lized to hav e unit ℓ 2 norms. The unio n of comp ounds found in the 297 fo o d ingredients is a subset of the 5736 comp ounds in the pure comp ound dictionary . 20 Conc. Comp ound Name 0.0065 benz a ldehyde 0.0324 1-o ctanol 0.0162 4-methylacetophenone 0.0811 phen yla cetaldehyde (=b enzeneacetaldehyde) 0.2596 nonanal (=pela rgonaldehyde) 0.0162 limonene 0.0973 pheneth yl isothio cyanate 0.0162 (E,E)-2,4- decadienal 0.0649 dimeth yl trisulfide (=2,3,4-trithiap entane, methyltrithiomethane) 0.0162 2-p ent ylfuran 0.0162 2,3,5-tr ithia hexane 0.0162 (E,Z)-2,4-he pta dienal 0.0973 (E,E)-2,4- heptadienal 0.4867 4-(methylthio)but yl isothio cyanate 0.0162 2-hexenal 0.6489 5-(methylthio)pentanenitrile 0.0162 dimeth yl disulfide (=methyldith io methane) 0.4867 3-phenylpropanenitrile (=phenethyl cyanide, b enzenepropa nenitrile) 0.0227 1,2-dimethoxybenze ne (=veratrole) 0.0649 (Z)-3-hexen-1 -ol (=leaf alcoho l) 0.0162 benz o thiazole T able 1: Comp ounds in co oke d br occoli with concen trations. 21 Conc. Comp ound 10.452 0 methane 5.6617 2,5-hexa ne dio ne (=acetonylacetone) 4.6890 cyclotetraco sane 3.1862 cube ne ne 1.7275 1,1’-dioxybis(1-decano l) 0.6456 2,4-diphenylp yrr ole 0.5931 propanamide 0.5685 cyclo o ctatetraene 0.5044 heptatriacontene (unkn.str.) 0.3386 p-1,5-menthadien-7-ol 0.3376 2-ethyl-5-pe n tanoylthiophene 0.1209 eth ylpyrrole (unkn.str.) 0.1106 do cosahexa e no ic acid (unkn.str.) 0.0224 10-methyl-2-undecenal 0.0055 α -maaliene 0.0041 2-(2-methylbutanoyl)f ura n T able 2: Additive mixture comp osed of pure compounds for fo o d steganograph y with co ok ed broccoli as the hidden data. W e use a s parsity-promoting p enalty for J as a demonstr ation. The result based on the pure comp ound dictionary is shown in T able 2 and the result based on the fo o d ingredient dictionar y is s hown in T able 3 . It is difficult to in terpr et the pure comp ounds solutio n. The fo o d ingre dient solution is easier to inter- pret. Angelica seeds, which are the main co mpo nen t of the fo o d pro duct-based additive, have a very unique pleasant smell en tir e ly unlike similar plants such a s fennel, parsley , anise, and c araw ay , and are used as a flav o r ing in Scandinavian cuisine. 22 Conc. F oo d P ro duct Name 13.299 9 ANGELICA SEED OIL 7.5619 CUMIN SEED (Cuminum cyminum L.) 7.5328 MUSSEL 4.3985 BARLEY (unpro cessed) 2.8275 LOBSTER 2.7808 BLACKBERR Y BRANDY 2.5717 R O SE WINE 2.3048 OTHER VITIS SPECIES 1.4727 TURNIP 1.3033 LAMB and MUTTON F A T (heated) 0.8432 INDIAN DILL ROOT (Anethum sow a Roxb.) 0.6520 LOGANBERR Y (Rubus ursinus v ar . log anobaccus) 0.4794 ELDERBERR Y FRU IT 0.1626 PEANUT (raw) 0.0989 MICROCITR US SPECIES OIL 0.0285 PRA W N T able 3: Addi tive mixture comp osed of f oo d ingredien ts for fo o d steganograph y with co ok ed broccoli as the hidden data. 23 7. Conclusion This pa per represents a firs t foray for statistical sig na l pro cessing into the new multimedia domain o f human o lfaction, building o n new development s in the science of smell. T he general fra mew o rk was demonstra ted through t wo sp ecific applica tions in a ctive o dor cancellation a nd in foo d steganogr aphy , and metho ds for so lving a broader class of problems were also indica ted. Empiri- cal results fro m the des ign pro cedures r equired bringing toge ther da ta o n the flav or comp osition of ingr edient s (from g as ch r omatogra ph y– mass sp ectro me- try), the molecular pro per ties of o dor co mp ounds (from chemoinformatics), and the human p erception of flav or s (from hedonic psy chophysics) with algorithmic techn ique s for function le a rning and in verse problem solution. By addressing o ne of the fundamental pr oblems of signal pro cessing , nois e cancellation, this work op ens up a new categor y of techniques for dea ling with bad o dors b eyond masking, absorbing, eliminating, and oxidizing; the most im- po rtant a pplication is to indo or air quality . F urthermor e, s ince hu ma n foo d av ersio n and foo d intake behavior can hav e significant consequence for health, well-being, and happiness, wa ys to steganogr a phically hide one fo o d inside an- other can be quite p owerful. Although the sig nal pro cessing r esults are pr omis- ing, it remains to v alida te the efficacy of these methods with exp erimental tests using hu man sub jects. References [1] K. R. V ar shney , L. R. V arshney , Activ e odo r cancella tio n, in: Pro c. IEEE W orkshop Stat. Signal Pr o cess. (SSP 2014 ), Gold Co a st, Australia , 2014 , pp. 25–2 8. doi:1 0.1109 /SSP.2014.6884566 . [2] K. R. V arshney , L. R. 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