noisy speech received by a single microphone is a widely - explored problem .a plethora of approaches can be found in the literature  . although many current devices are equipped with multiple microphones , there are still many applications for which only a single microphone is available .one such application involves systems .it is well - known that such systems are sensitive to mismatch between the train and test environments . enhancing the noisy speech signal prior to the application of the system ,might alleviate the performance degradation caused by the environment .nonstationary noise environments are usually more challenging , since the speech enhancement algorithm should adapt to the changing statistics of the additive noise .the celebrated and  are widely - used model - based algorithms .the estimator and in particular the noise estimator are specifically tailored to nonstationary noise environments  . however , fast changes in noise statistics often yields the _ musical noise _ phenomenon .recently , techniques gained a lot of popularity due to theoretical and algorithmic progress , and the availability of more data and more processing power . unlike past learning algorithms for nn , it is now possible to infer the parameters of the with many layers , and hence the name _deep learning_. deep learning methods were mainly applied to speech recognition and lately , for speech enhancement as well . and a were used as a nonlinear filters in  and  , respectively .the networks are trained on stereo ( noisy and clean ) audio features , to infer the complex mapping from noisy to clean speech . an experimental study with this approachis shown in  .the reduces the noise level significantly , yet , the enhanced signals still suffer from noticeable _ speech distortion_.  other methods attempt to train an to find a mask , which classifies the time - frequency bins into speech / noise classes . given the binary mask , the noisy bins are decreased . in  for instance ,a is used to estimate the for speech separation from non - speech background interference .an is trained to find the input features for the .a simpler approach is to train the itself to find the .different targets for the are presented in  .the has shown advantageous in terms of intelligibility  . yet, the binary mask is known to introduce artifacts such as _ musical noises_. for intelligibility tasks , this might not be problematic , though for speech enhancement the is not sufficient . to circumvent this phenomenon , in  is trained to find the , which is a soft mask .a comparison between the and the is presented in  . the soft mask is better than the binary mask in terms of speech quality .these approaches do not use models nor assumptions for their speech enhancement .however , they are trained with specific noise types , resulting in poor enhancement in an untrained noise environment . to cope with this problem , in  was trained with more than 100 different types of noise .nevertheless , in real - life where the number of noise types are not limited , this approach may not be satisfactory .training - based algorithms , such as mixmax  , were also developed .these algorithms are performed in two phases , the training phase and the test phase . in the trainingphase the parameters of the model are found , usually with an unsupervised machine learning algorithms , such as the algorithm in  . in the test phase ,the enhancement is carried out using the learned model parameters .one weakness of the algorithm is that the speech parameters are found in an unsupervised manner that ignores the phoneme - based structure of speech .another drawback of the mixmax algorithm is that the noise parameters are estimated once at the beginning of the utterance and then are kept fixed during the entire utterance .this enhancement approach is not always sufficient for real - life noises .  in this paper , we apply a hybrid algorithm , which integrates the generative model - based approach with the discriminative tool . as in  , we use a two phase algorithm . in the training phase, the clean speech is modeled with a phoneme - based that is built using phoneme labeled database .a is then trained to classify clean time - frame features as one of the phonemes from the phoneme - based .once the training phase is over , the training does not recur .with the estimated phonemes , an is calculated in the test phase using the generative model .soft spectral subtraction is then carried out using the spp , while , simultaneously , the noise estimation is updated .the continuity of the speech is maintained using the that uses context frames in addition to the current frame .in addition , the assists the calculation of the spp .furthermore , the phoneme - based and the soft preserve the spectral structure of the speech thus alleviating the musical noise phenomenon .this approach utilizes the benefits of both the generative and the discriminative methods to alleviate the drawbacks of the mentioned above algorithms .the rest of the paper is organized as follows . in section [ sec :model ] , a generative model is presented .section [ sec : algorithm ] presents the proposed enhancement algorithm and describes its implementation in details .a comprehensive experimental results using speech databases in various noise types are presented in section [ sec : results ] . in section [ sec : analysis ] the building blocks of the algorithm are analyzed .finally , some conclusions are drawn and the paper is summarized in section [ sec : summer ] .in this section , a generative model of the noisy speech signal is presented .we follow the model proposed by ndas et al .  that was utilized in  .the following notation is used throughout the paper .uppercase letters are used for random variables , lower case for a given value and a boldface symbols denotes vectors .let and denote the speech and noise signals , respectively .the observed noisy signal is given by applying the with frame length set to samples and overlap between successive frames set to samples to yields with the frame index and the frequency index .the frame index is henceforth omitted for brevity , whenever applicable .let denote the dimensional log - spectral vector , defined by note that the other frequency bins can be obtained by the symmetry of the .similarly , we define and to be the log - spectral vectors of the speech and noise signals , respectively .it is assumed that the noise is statistically independent of the speech signal .furthermore , it is assumed that both the speech and noise are zero - mean stochastic processes . due to these assumptionsthe following approximation can be justified : hence   following ndas et al . , the noisy log - spectral can be further approximated : where the maximization is component - wise over the elements of and .this approximation was found useful for speech recognition  , speech enhancement  and speech separation tasks  . in a speech enhancement task ,only the noisy signal is observed , and the aim is to estimate the clean speech .it is well - known that a speech utterance can be described as a time - series of phonemes , i.e. speech is uttered by pronouncing a series of phonemes  . in our approach , we give this observation a probabilistic description , namely the log - spectral vector of the clean speech signal , , is modelled by a distribution , where each mixture component is associated with a specific phoneme .unlike  , that uses unsupervised clustering of the speech frames , we use here a supervised clustering , explicitly utilizing the labels of the phonemes of the training speech signals . based on the mog model ,the probability density function of the clean speech , can be written as where is the number of mixture components and let be the phoneme indicator associated with the , i.e. . the term is the gaussian of given that .the scalar is the probability of the -th mixture and and are the mean and the standard deviation of the -th entry of the -th mixture gaussian , respectively . due to the fouriertransform properties , we neglect any residual correlation between the frequency bins .since for each class the is gaussian , the frequency bins are also statistically independent .consequently , the covariance matrix of each mixture component is diagonal . to set the mog parameters we used the phoneme - labeled timit database  as described in sec .[ subsec : trainnn ] .let define the log - spectral vector of the noise signal , and let denote the of .as with the log - spectral vector of the speech signal , it is assumed that the components of are statistically independent . for simplicity , is modeled as a single gaussian , with diagonal covariance i.e. , where initial estimation and adaptation the noise parameters will be explained in sec .[ subsec : noiseadapt ] .  using the maximum assumption in the log - spectral vector of the noisy speech ,as explained above , it can be verified  that the of is given by the following mixture model : where such that and are the cumulative distribution functions of the gaussian densities and , respectively .the term is the of given that .the generative modeling described above was nicknamed mixmax  , since it is based on the maximum assumption and on the modelling of the clean speech as a ( gaussian ) mixture and the noisy speech is modeled as the maximum of the clean speech and the noise signal .originally , the mixture components were not associated with phonemes , but rather learned in an unsupervised manner .in this section , we describe the proposed enhancement algorithm . in sec .[ subsec : mmse ] we remind the estimator based on the mixmax model  .we then propose in sec .[ subsec : soft ] a new variant of the estimator that utilizes the same model but allows for better noise reduction . in sec .[ subsec : nn ] an approach is introduced as a tool for accurate phoneme classification .issues regarding the training of the are discussed in sec .[ subsec : trainnn ] .finally , test - phase noise adaption is discussed in sec .[ subsec : noiseadapt ] .an of the clean speech from measurement is obtained by the conditional expectation .note , that since the of both and is non - gaussian , this estimator is not expected to be linear . utilizing the generative model described in the previous section we can obtain a closed - form solution for the mmse estimator as follows . the posterior probability can be easily obtained from  by applying the bayes rule : since the gaussian covariance matrices of both the speech and the noise models are diagonal , we can separately compute for each frequency bin . for the -th frequency binwe obtain : such that and for the second term in : the closed - form expression for the mmse estimator of the clean speech  is obtained from , , , .these expressions are the core of the mixmax speech enhancement algorithm proposed by burshtein and gannot  . in their approachthe mog parameters of the clean speech are inferred from a database of speech utterances utilizing the in an unsupervised manner .   assuming the maximization model in  is valid , was obtained in .summing over all the possible mixture components , we obtain : the term can be interpreted as the probability that given the noisy speech vector , the -th frequency bin of the current log - spectral vector is originated from the clean speech and not from the noise .the probability can thus be viewed as a training - based detector , namely the probability that the designated time - frequency bin is dominated by speech .consequently , can be interpreted as the posterior probability that the -th bin is dominated by noise .  using and ,the -th frequency bin of the mmse estimator can be recast as follows : hence , given the generative model , the enhancement procedure in , substitutes the frequency bins identified as noise with the a priori value drawn from the model and using .the structure of voiced speech consists of dominant spectral lines which recur at multiples of the fundamental frequency ( known as _ pitch _ ) .the of different speakers pronouncing the same phoneme share similar properties , but are never identical .hence , the parameters inferred from multiple speakers , is never individualized to the current speaker and therefore can not represent the specific periodicity .the phoneme - based parameters are only capable of preserving the general structure of an averaged phoneme .this phenomenon might lead to _ residual noise _ even when the algorithm identifies the noise correctly .  to circumvent this phenomenon, we propose to substitute the optimal estimator that uses the parameters with a simpler estimate based on the spectral substraction paradigm , namely : is substituted by : where is a noise reduction level .it is well - known that the basic spectral subtraction method is prone to _ musical noise _  . in our proposed method, the estimator also incorporates the soft mask deduced from the , thus potentially alleviating the musical noise phenomenon .  substituting in  weobtain the following simplified expression for the estimated clean speech : or , equivalently which can be interpreted as -driven ( soft ) spectral subtraction algorithm .the gist of our approach is the calculation of the  .this calculation necessitates two terms , which is given by  and the posterior phoneme probability . utilizing the generative model defined in section [ sec : model ] , is obtained from  by applying the bayes rule : this approach exhibits some major shortcomings . estimating the required noise statistics is a cumbersome task , especially in time - varying scenarios .furthermore , as the calculation in  is carried out independently for each frame , continuous and smooth speech output can not be guaranteed .  in our approach ,( unlike  ) we adopt a _ supervised _ learning approach in which each mixture component of the clean speech is associated with a specific phoneme .hence the computation of the mixture index posterior probability becomes a phoneme classification task ( based on the noisy speech ) . to implement thissupervised classification task , we substitute  with an that is known to be significantly better than models for phoneme classification tasks ( see e.g.  ) . the is trained on a phoneme - labeled clean speech . for each log - spectral vector , , we calculate the corresponding features ( and their respective delta and delta - delta features ) . to preserve the continuity of the speech signal, 9 mfcc vectors are concatenated ( the current feature vector , 4 past vectors and 4 future vectors ) to form the feature vector , denoted , which is a standard feature set for phoneme classification .this feature vector is used as the input to the , and the phoneme label as the corresponding target .the phoneme - classification is trained on clean signals . however , as part of the speech enhancement procedure , we apply it to noisy signals . to alleviate the mismatch problem between train and test conditions , we use a standard preprocessing stage for robust phoneme classification , namely  .the is calculated using , which requires both and .while is calculated from the generative model using , we propose to replace  for calculating by a better phoneme - classification method .it is therefore proposed , to infer the posterior phoneme probability by utilizing the _ discriminative _ , rather than resorting to the generative mog model : note , that the compound feature vector is used instead of the original log - spectrum .finally , the spp is obtained using  and : the proposed calculation is based on a _ hybrid _ method , utilizing both the generative model and a discriminative approach to infer the posterior probability . for the latter we harness the known capabilities of the .we used the phoneme - labeled clean speech timit database  to train the phoneme classifier and the phoneme - based generative model .we next describe the training procedure .we used the 462 speaker from the training set of the database excluding all sa sentences , since they consist of identical sentences to all speakers in the database , and hence can bias the results .in training the phoneme - based we set the number of gaussians to ( see  ) , where each gaussian corresponds to one phoneme .all frames labeled by the -th phoneme were grouped , and for each frequency bin the mean and variance were computed using  and , respectively .first , the log - spectrum of the segments of clean speech utterances is calculated . since the database is labeled each segmentis associated with a phoneme .we can then calculate the following first- and second - moment with phone label : where is -th bin of the -th log - spectra vector with phoneme label .the term , is the total number of vectors associated with phoneme labeled .the mixture coefficients are set to be the relative frequency of each phoneme in the training dataset : note that since the data is already labeled , no iterative clustering procedure , such as the algorithm , is required .  for training the as a discriminative phoneme classifier , we used the feature vectors powered by the delta and delta - delta coefficients . in total , coefficients per time frame were used .context frames ( 4 from the future and 4 from the past ) were added to the current frame as proposed in  .hence , each time frame was represented by 351 features .we used a single hidden layer comprising of 500 neurons .( although adding more hidden layers slightly improves phoneme classification rate , we did nt gain any significant improvement in the overall enhancement procedure . )the network is constructed of sigmoid units as the transfer function for the hidden layer : and a softmax output layer to obtain a vector probabilities associated with the various phonemes : where and are the weights matrices , of the hidden layer and the output layer , respectively .given a sequence of mfcc feature vectors , where is the total number of vectors in the training set , with the corresponding phoneme labels , , the is trained to maximize the log - likelihood function : to train the network we can start with random weights ( or use pre - training methods ( see  ) ) and then , by applying back - propagation algorithm as part of a gradient ascent procedure , the parameter sets of the network , and , are found . in our implementationwe used matlab r2014b pattern recognition toolbox  to train the .the default training function , namely the _ scaled conjugate gradient back - propagation _ was used . to avoid mismatch between train and test conditions each utterance was normalized , such that the utterance samples mean and the variance are zero and one , respectively .  to verify the accuracy of the classifier , the trained was applied to a clean test set ( 24-speaker core test set drawn from timit database ) , obtaining correct phoneme classification results , which is a reasonably high score .  during the test phase of the algorithm , the is applied to speech signals contaminated by additive noisewe have therefore applied the procedure before the classifier to circumvent the noisy test condition  .   to estimate the noise parameters it is assumed that the first part of the utterance ( usually 0.25 sec ) the speech is inactive and it consists of noise - only segments .these first segments can therefore be used for initializing the parameters of the noise gaussian distribution as follows : where is the number of vectors constructed form the noise - only samples .the term denotes the -th bin of the -th noise vector .  in  ,the noise parameters remain fixed for the entire utterance , rendering this estimate incapable of processing non - stationary noise scenarios . to alleviate this problem, we will apply an adaptation procedure ( see  for alternative noise adaptation techniques ) . using the derived in, the following adaptation scheme for the noise model parameters can be stated : where and are the updated parameters and and are the parameters before adaption , and is a smoothing parameter .using this scheme , the noise statistics can be adapted during speech utterances , utilizing the frequency bins that are dominated by noise .this scheme is particularly useful in non - stationary noise scenarios . as a consequence ,the first few segments , assumed to be dominated by noise , are only used for initializing the noise statistics and their influence is fading out as more data is collected .  the proposed algorithm is summarized in algorithm [ alg : nn ] .we dub the proposed algorithm to emphasize its hybrid nature , as a combination of the generative mixmax model and the phoneme - classification .  : * input * : log - spectral vectors , mfcc vectors , and their corresponding phoneme labels . :set the phoneme - based parameters using and .: train a nn for phoneme classification using .: * input * : log - spectral vector of the noisy speech and a corresponding mfcc vector .* output * : estimated log - spectral vector of the clean speech .in this section we present a comparative experimental study .we first describe the experiment setup in sec .[ subsec : expsetup ] .objective quality measure results are then presented in sec .[ subsec : objectmeasurs ] . in sec .[ subsec : asr ] results are compared with different approaches .finally , the algorithm is tested with an untrained database in sec .[ subsec : notimit ] .   to test the proposed algorithm we have contaminated speech signal with several types of noise from noisex-92 database  , namely _ speech - like _ , _ babble _ , _ car _ , _ room _ , _ awgn _ and _factory_. the noise was added to the clean signal drawn from the test set of the timit database ( 24-speaker core test set ) , with 5 levels of at  db ,  db ,  db ,  db and  db in order to represent various real - life scenarios .the algorithm was tested similarly , with the untrained database .we compared the proposed algorithm to the algorithm  with noise estimator  , a state - of - the - art algorithm for single channel enhancement .the default parameters of the were set according to  .  in order to evaluate the performance of the speech enhancement algorithm ,several objective and subjective measures were used , namely the quality measure , which has a high correlation with subjective score  , and a composite measure  , weighting the , the and the  .the composite measure outputs , and results .  as an additional measurewe have examined the performance improvement of an system .we used the pocketsphinx asr system  .the feature set of the system is composed of 39 mfcc features powered by delta and delta - delta features .the acoustic model consists of a hidden markov model with 5000 states .each state is represented by a with 16 mixture components .finally , the 20,000-word vocabulary language model was trained using corpus  . finally , we have carried out informal listening tests .we first evaluate the objective results of the proposed algorithm and compare it with the results obtained by the algorithm . to further examine the upper bound of the proposed method we also replaced the classifier with an ideal classifier that always provides the correct phoneme , denoted _ideal-_. the test set was the core set of the timit database .[ fig : known_phonemes ] depicts the results of all examined algorithm for the speech - like , room , factory and babble noise types as a function of the input . in fig . [fig : covl ] we show the covl results for factory and room noises .the results behave in a similar way , with other noise types .it can be clearly deduced that the proposed algorithm outperform the algorithm in the two designated objective measures .the ideal- outperforms the , but the difference is rather marginal .still , there is a room for improvement , would a better phoneme classifier be available .  0.5   0.5   +  0.5   0.5   0.5   0.5   to gain further insight ,we have also compared the enhancement capabilities of the proposed algorithm and the state - of - the - art algorithm in the challenging factory noise environment .it is clearly depicted in fig .[ fig : factory_spectrum ] ( obtained in snr=5 db ) that the proposed is less prone to musical noise , while maintaining comparable noise level at the output.  0.5   0.5   +  0.5   0.5    speech enhancement algorithms can also serve as a preprocessing stage on front of systems . in order to test the performance of the enhancement algorithm we added four types of noise in four different levels to a database comprising five female and five male speakers ,each uttering approximately 150 english sentences .the utterances were taken from different speech databases ( samples from the were not included ) .overall , the test database consists of 1497 sentences , 24 sec long ( 28 words each ) .  as before, we have the proposed algorithm with the original mixmax and the algorithms .the results are depicted in table [ table : asr ] .the algorithm significantly outperforms both competing algorithms for the factory and babble noise , and most the speech - like for most values ( besides 5 db ) . in the white noise case , the original mixmax exhibits slightly better performance .the superior results of the proposed algorithm can be attributed to the improved phoneme classification , which is one of the main building blocks of an system .   method \ & 5[db ] & 10[db ] & 15[db ] & 20[db ] + noisy signal & 8.8 & 43.0 & 68.8 & 79.7 + mixmax & 18.7 & 53.7 & 72.9 & 81.0 + omlsa & 13.7 & 45.0 & 66.0 & 76.2 + nn - mm & * 28.2 * & * 60.3 * & * 76.2 * & * 81.9 * +   method \ & 5[db ] & 10[db ] & 15[db ] & 20[db ] + noisy signal & 1.1 & 32.7 & 62.2 & 76.1 + mixmax & 9.5 & 44.4 & 68.3 & 78.7 + omlsa & 16.3 & 47.4 & 69.0 & 78.0 + nn - mm & * 19.5 * & * 52.9 * & * 71.9 * & * 80.1 * +  [ table : asr_factory ]   method \ &5[db ] & 10[db ] & 15[db ] & 20[db ] + noisy signal & 7.9 & 44.4 & 68.4 & 77.5 + mixmax & 38.5 & 64.9 & 77.4 & 81.6 + omlsa & * 41.3 * & 65.4 & 75.8 & 81.3 + nn - mm & 40.4 & * 66.6 * & * 78.0 * & * 82.2* +  [ table : asr ]   method \ & 5[db ] & 10[db ] & 15[db ] & 20[db ] + noisy signal & 10.4 & 31.8 & 53.6 & 68.9 + mixmax & * 28.9 * & * 51.7 * & * 67.2 * & * 77.1 * + omlsa & 25.8 & 46.1 & 65.1 & 74.5 + nn - mm & 26.1 & 45.8 & 65.5 & 75.8 +  [ table : asr_white ]   finally , we would like to demonstrate the capabilities of the proposed algorithm when applied to speech signals from other databases . in this workwe have trained the phoneme - based and the using the timit database . in this section we apply the algorithm to clean signals drawn from the database  .the signals were contaminated by the challenging factory and babble noise with several levels .note , that the algorithm does not train with noisy signals .[ fig : no_timit ] depicts the measure of the algorithm in comparison with the algorithm .it is that the performance of proposed algorithm and its advantages are maintained even for sentences from different database than the training database . herewe show the challenging factory and babble noise types .the results in other noise types have the same construction .  0.5   0.5in this section , we analyze the individual contributions of each component of the proposed algorithm to the overall performance .first , in sec .[ subsec : phonemog ] the phoneme - based is analyzed . in sec .[ subsec : spp ] example of the is presented and the phoneme classifier is compared to the generative approach in sec .[ subsec : classification ] . finally , in sec .[ subsec : noiseadaptresults ] the noise adaptation is tested in real - life scenario .[ sec : mog ] one of the major differences between the original mixmax algorithm and the proposed algorithm is the construction of the model . while the former uses unsupervised clustering procedure based on the algorithm , the latter uses supervised clustering using the labeled phonemes .consequently , the clusters in the proposed algorithm consists of different variants of the same phoneme , while the cluster obtained by the algorithm mixtures of various phonemes .we postulate that the supervised clustering is therefore advantageous over the unsupervised clustering .we will examine this claim in the current section , using clean speech signal contaminated by room noise with snr=5 db .first , define the averaged of the speech utterance as the weighted sum of the gaussian centroids , as inferred by the two clustering procedures .the weights give the respective posterior probabilities ( either  or ) .the averaged obtained by the supervised clustering and the discriminative is given by : similarly , the averaged obtained by the unsupervised clustering and the generative model is given by   in figs .[ fig : clean_room_5db ] and [ fig : noisy_room_5db ] we show the clean and noisy , respectively . fig .[ fig : mus_em_room_5db ] and fig .[ fig : mus_nn_room_5db ] illustrates the estimated weighted gaussians and .it evident that is not as successful as successful as in estimating the clean speech .0.5   0.5   0.5   0.5   0.5   0.5   0.5   0.5    the is the probability that the time - frequency bin is dominated by speech . in this sectionwe examine the developed in this work as given in algorithm [ alg : nn ] . to further validate the advantages of the hybrid scheme we compare it with the used in the original mixmax ,namely the posterior probabilities are inferred from the generative model and the is trained in an unsupervised manner .the latter is denoted   we continue the example in sec . [sec : mog ] .both , and , are depicted in figs .[ fig : rho_mm_room_5db ] and [ fig : rho_nn_room_5db ] , respectively .it can be easily observed that has a better resemblance to the clean speech spectrogram shown in fig .[ fig : clean_room_5db ] and suffers from less artifacts .additionally , it is smoother than the in both time and frequency aspects .conversely , vertical narrow spectral lines can be easily observed in .this spectral artifacts may be one of the causes for the differences in the enhancement capabilities of the two algorithms , as depicted in figs .[ fig : rho_mm_room_5db ] and [ fig : rho_nn_room_5db ] .we postulate that the designated advantages of the proposed approach stem from the better classification capabilities as exhibited by the .while the original mixmax algorithm is only utilizing the current frame for inferring the posterior probabilities , the proposed algorithm takes into account the context of the phoneme by augmenting past and future frames to the current frame .this guarantees a smoother and consequently less artifacts at the output of the algorithm .  this context - aware feature vector together withthe phoneme - based may also alleviate the musical noise phenomenon .this observation is also supported by the smoother spectral envelop of the centroid as can be deduced from comparing figs .[ fig : mus_em_room_5db ] and [ fig : mus_nn_room_5db ] .we turn now to the assessment of the proposed phoneme classifier . forthat we compare the classification accuracy of the with that of the generative model in  using the phoneme - labeled .[ fig : classification_results ] depicts the percentage of correct classification results obtained on the test data .two types of noise were added to the clean signals , namely factory and babble noise .the results clearly indicate that the based classifier significantly outperforms the classifier based on the generative model , and hence better suited for the task at hand .0.5   0.5   0.5   0.5   +  0.5   0.5   +  0.5   0.5    in this section we examine the noise adaptation scheme described in .a _ city ambiance _ noise  that consists of a siren and passing cars was chosen , as it is a highly non - stationary noise source with fast changes during the speech utterance .the clean and noisy signals are depicted in figs .[ fig : clean_siren_stft ] and [ fig : noisy_siren_stft ] .the input was set to 5 db ( resulting in input = 2.124 .  in fig .[ fig : real_mun ] the real noise is depicted and in fig .[ fig : est_mun ] its estimate using the proposed adaptation scheme and the inferred by the algorithm .it can be observed that the estimate is quite accurate even when the noise changes very fast. note that during speech dominant time - frequency bins , the noise estimate can not adapt .these adaptation capabilities are also reflected at the output of the algorithms , especially in comparison with the algorithm , as depicted in figs .[ fig : omlsa_siren ] and [ fig : ann_mm_siren ] .we observe that the algorithm outperforms the in reducing this challenging noise .this is also indicated by the measure .while the degrade the speech quality ( pesq=1.847 ) , the proposed hybrid algorithm slightly improves it ( pesq=2.361 ) .the reader is also referred to our website where these sound clips can be found .in this paper a novel speech enhancement scheme , denoted , is presented .the proposed algorithm is based on a hybrid scheme which combines phoneme - based generative model for the clean speech signal with a discriminative , -based estimator . in the proposed algorithmwe try to adopt the advantages of model - based approaches and approaches . while the former usually trade - off noise reduction abilities with residual musical noise , the latter often suffer from speech distortion artifacts .  in the proposed algorithm we take advantage of the _ discriminative _ nature of the that preserves speech smoothness by using context frames. moreover , the phoneme - based model , where each gaussian corresponds to a specific phoneme , preserves the general phoneme structure and reduces musical noise .a comprehensive set of experiments demonstrate the capabilities of the proposed algorithm in both improving scores as well as objective quality measures .the algorithm is shown to outperform state - of - the - art algorithm ( ) for both stationary and non - stationary environmental noises and a variety of levels .

Abstract: in this paper we present a single - microphone speech enhancement algorithm . a hybrid approach is proposed merging the generative model and the discriminative . the proposed algorithm is executed in two phases , the training phase , which does not recur , and the test phase . first , the noise - free speech is modeled as a , representing the phoneme based diversity in the speech signal . an is then trained with phoneme labeled database for phoneme classification with as the input features . given the phoneme classification results , an is obtained using both the generative and discriminative models . soft spectral subtraction is then executed while simultaneously , the noise estimation is updated . the discriminative maintain the continuity of the speech and the generative phoneme - based preserves the speech spectral structure . extensive experimental study using real speech and noise signals is provided . we also compare the proposed algorithm with alternative speech enhancement algorithms . we show that we obtain a significant improvement over previous methods in terms of both speech quality measures and speech recognition results .   speech enhancement , mixmax model , neural - network , phoneme classification


animal tissue cells sense the mechanical and geometrical features of their environment by applying traction forces to the extracellular matrix .various studies over the last decades demonstrated the importance of tissue mechanics for cell behavior , including cell adhesion , migration , proliferation , differentiation and fate .it has been shown early that cells respond sensitively to the rigidity of their environment , with larger spreading area and higher force generation on stiffer substrates . even more dramatically , essential cell processes such as cell differentiation are controlled by substrate stiffness . because environmental stiffness is a passive property of the environment , cells have to actively pull on it to determine its magnitude .cellular traction forces are not only used to sense extracellular stiffness , but also to sense geometrical properties of the environment .this is most evident for adhesive patterns , whose geometry also determines cell responses such as cell survival and differentiation .however , geometry sensing also includes cell sensitivity to topographical features of the environment , which spans several orders of magnitude , ranging from the molecular up to the cellular scale .it has been shown that cell differentiation can be controled also by nano - topography .regarding the cytoskeletal response , it has been found that cells tend to align with grooves on substrates with corresponding nano- and micro - topography .another physiological relevant example is topography - driven polarity guidance and directional growth of neurons .early examples for topography sensing on the _ _ scale are studies that showed the alignment of actin microfilaments in cells adhered to microcylinders .interestingly it was found that cells tend to align either parallel or orthogonal to the direction of highest curvature dependent on the cell type .inspired by these observations mechanical cell models have been proposed that suggest the importance of mechanical stress in the cytoskeleton for the detection and the response to curved structures .recently the combined effect of stiffness and geometry has been studied by engineering topographic features into polyacrylamide substrates , which as hydrogels however tend to swell in medium and therefore change dimensions .interestingly , it was found that cells align with the topographic features independent of stiffness .however , no attempt has been made to measure cellular traction forces in these experiments . in general, there are many physiological situations in which cells are exposed to mechanical and geometrical cues simultaneously , and therefore will use cellular traction forces to sense them both at the same time .an interesting example are podocytes , which are epithelial cells lining the basement membrane of the glomerular capillaries in kidneys , which have a wavy shape .however , a setup that allows to quantitatively measure forces for such situations is still missing , despite recent progress in measuring cellular traction forces on planar substrates .  during the last three decades , the measurement of cellular forces ( traction force microscopy , tfm ) has become a mature research field .the standard setup uses a planar elastic substrate whose deformations are tracked using embedded marker beads .standard choices for substrate material are polyacrylamide ( paa ) or crosslinked polydimethylsiloxane ( pdms ) . for sufficiently thick substrates ,one then can use the green s function of an elastic halfspace to relate these deformations to traction forces ( gf - tfm ) . for a thin substrate , the corresponding green s functionis also known .tfm - procedures can be made quite fast by inverting the elastic equations in fourier space ( fourier transform traction cytometry , fttc ) , which is a special case of gf - tfm .traditionally , cell forces had been reconstructed only in the x - y - plane of a planar substrate ( with the z - direction denoting the normal direction ) , but over the last years , the green s function approach has been extended to also reconstruct the z - forces that cells exert to the substrate .alternatively , one can use the finite element method ( fem ) to reconstruct these z - forces ( fem - tfm ) . within a fem - approach, one does not rely on the analytical form of a green s function , but uses numerical solutions to the mechanical problem interpolated on a suitable chosen grid .another alternative to gf - tfm is the direct method , in which deformations are directly converted into a stress tensor , from which the traction forces are extracted .if one aims at implementing tfm for non - planar substrates , from these three methods ( gf - tfm , fem - tfm and direct tfm ) only the second one seems feasible .first it is notoriously hard to analytically calculate green s functions for non - planar ( e.g. wavy ) surfaces , thus ruling out gf - tfm .as we will see below , wavy substrates lead to rather noisy displacement data , which is hard to deal with in direct tfm , because it relies on constructing derivatives of the measured data .therefore we opted for an approach using fem - tfm , which needs more computer time than traditional gf - tfm , but offers the same level of robustness .we first developed a novel experimental technique to prepare curved micromolded pdms substrates with embedded fluorescent marker beads matching the requirements of tfm application .in contrast to the paa - hydrogels often used for studies of mechanosensing , the micromolded pdms - substrates used here do not suffer from swelling and therefore present time - independent geometrical cues to the cells .however , because the point spread function in such a substrate varies in space , special procedures have to be used for tracking the marker beads .because here we focus on measuring traction forces , we did not vary the mechanical stiffness of the substrates . on the computational side we established a complete workflow to reconstruct cellular traction fields on such non - planar substrates .the core of this technique is a parallelized optimization framework that efficiently implements fem to reconstruct cellular traction forces in 3d .we validated our procedures by reconstructing simulated traction patterns under various experimental conditions .in particular , we show that the use of the l1-norm for the definition of the residue strongly improves our force reconstruction because it better deals with outliers than the l2-norm .we then applied tfm to cardiac myofibroblasts cultured on curved elastic substrates , thus complementing a traditional contact guidance experiment with measurements of cell traction forces . by comparing both polarization of the cytoskeleton and the distribution of cellular tractionwe show that cells not only adjust their morphology , but also their moments of traction force to geometrical properties of their surrounding .elastomeric substrates were made of sylgard silicone elastomer kit ( dow corning gmbh , wiesbaden , germany ) as described previously . in brief , both components ( base and cross - linker ) of the elastomer kit were mixed at various ratios to generate after cross - linking either stamps from vinyl disc masters ( 10:1 ) or cell culture substrates of stiffness 15 kpa ( 50:1 ) . for stamp manufacturing round polypropylene rings ( diameter 10 mm ) were placed on top of the vinyl disc and crosslinked at for 4 h. stamps were peeled off and silanized with tricholorosilane . for force analysis cell culture substrateswere equipped with red fluorescent beads ( 0.1 m diameter , non - modified beads , magspheres , ca , usa ) resulting in a dense microstructured volume ( typical bead distance 4 ) .base oil was mixed with a 1:50 dilution of beads in methanol .the modified base oil was incubated at for 3 h to evaporate methanol .the desired amount of cross - linker was added to the modified base oil and mixed intensively .the stamp was placed onto a 80 coverslip with a drop of silicone oil mixture .layer thickness was adjusted to 80 using glass slices of same defined thickness ( menzel gmbh , braunschweig , germany ) as spacers .cross - linking of silicone oil was performed at for 16 h. after curing stamps were carefully peeled off and structured elastomeric substrates were glued to the bottom of 3.5 cm petri - dishes to cover predrilled holes .mechanical properties of all elastomeric mixing ratios were characterized as described resulting in elasticities as given above and a poisson s ratio of .cardiac fibroblasts were isolated from -day - old wistar rat embryos as described previously .primary cardiac fibroblasts were cultured for additional days at and co in a humidified incubator on standard polystyrene cell culture dishes to induce their differentiation to myofibroblasts , which have more prominent focal adhesions and stress fibers .cells were trypsinized and subsequently transfected using nucleofector technology ( lonza - amaxa systems , cologne , germany ) . cells were resuspended in liposome solution containing purified plasmid - dna ( gfp - vasp or gfp - vinculin ) .after transfection , resuspended myofibroblasts were seeded on fibronectin coated (  ) ( bd bioscience , new jersey , usa ) silicone rubber wave substrates at a density of per sample for traction force microscopy .the cells were seeded for at least 24 hours before the measurements were taken .live cell analyses were performed at and co using an inverse confocal laser scanning microscope ( lsm 710 , zeiss , germany , software zen 2011 ) equipped with live cell imaging accessories and a planapochromat oil immersion objective ( ph3 , na , zeiss ) .confocal 3d micrographs were taken using appropriate laser and filter settings for detection of green and red fluorescent light .z - stacks of approximately with optimized overlap were taken .as reference value , z - stacks without cells ( peeled off with glass micro needle ) were performed with the same parameters .acquisition of a single slice took 8 seconds and we typically acquired 100 slices with a distance around 0.4 .thus acquisition time for one stack was around 15 minutes .bead tracking for wavy substrates is challenging because of the long stack acquisition time leading to considerable drift not only between the deformed and the reference states , but even within one stack .we figured that this drift did not result from cell activity , because 24 hours after spreading , they were quiescent. moreover the observed drift was similar at any depth in the substrate .as another special feature of wavy substrates , we observed that in contrast to planar substrates , the point spread function varied in space .our bead tracking routine consists of three steps : determination of image drift , single bead localization , and feature vector based tracking of bead movement . to reduce noise ,the stacks were first binomially filtered with a filter kernel , voxel sizes m or m lateral and m or m vertical. then spots ( usually ) in the undeformed corners of the image were marked ( in the x - y - plane ) . at these spots a cuboid ( voi , volume of interest ) with of the image size in each dimension was cropped from the first image and cross - correlated with the second image to obtain the drift . because the drift in each slice of the stack can differ , this voi is cropped along each slice and so the drift is calculated for every slice separately .the area in which the voi is cross - correlated with the second stack has to be larger than the maximum shift of the two image stacks . to decrease the calculation time for the cross correlation ( cc ) , the size of the image stack and the voisare initially reduced by two levels of gaussian pyramid .the positions of the maximum cc values are then used to calculate the drift on the full image to get the exact drift .this can be done in a very small search area ( pixel ) around the previously calculated positions . to obtain a subpixel accurate positioning of the vois , parabolas were fitted through three points ( maximum of cc and both neighbors ) in x , y , and z - directions , respectively .the extreme values of the fits are defined as the subpixel accurate voi positions .the cc and parabola fitting is also done in the first image stack to get a subpixel accurate position there as well .the mean difference between the voi positions in the first and in the second stack defines the drift , separately for each slice .  to localize beads in each image stack ,both stacks were first filtered with a and a binomial filter .these filtered images were then subtracted , to get a bandpass - filtered stack , with negative gray values set to zero .then a voi that represents one bead in three dimensions is selected manually and fitted to a 3d gaussian .the fit is used as a reference template to find other beads that match the template .the fitted gaussian is cross - correlated with the first and the second stack .then the cross - correlated dataset is segmented with a threshold ( ) and each segment is labeled . at the maximum cc value in each labeled segmentthe data is again fitted to a gaussian and the center of mass of that fit is defined as bead position .once the beads are localized in both stacks , they have to be correlated to each other . here , we adapted a feature vector based tracking method as published previously .before we apply the tracking procedure , the determined drift is subtracted from the bead positions found in the second stack .then the distance of a bead ( b1 ) from stack 1 to all other beads in its neighborhood ( pixel in x - y- and pixel in z- direction ) is calculated . to the at most beads with the lowest distance , vectors from b1 to those beadsare created .these vectors are added to beads ( b2 ) in stack 2 ( again only in a neighborhood of pixel around the position of b1 ) and a cubic surrounding ( e.g. pixel ) is created at each end of the vector .if there is a bead located within this surrounding , a hit is counted .the more hits are counted , the bigger is the probability that bead b1 corresponds to b2 .the bead b2 with the most hits ( at least out of ) is assigned to bead b1 . if two or more beads b2 have the same amount of hits , the one with the lowest deviation at the vector ends is chosen .as pre - processing step for the traction reconstruction we need to determine the substrate shapes .although the substrate preparation described above leads to reproducible samples with m accuracy , shape variations and material relaxation after mold removal occasionally change the final shape .the latter effect can in principle be predicted theoretically , however , in practice local shape variations occur and make the use of theoretical shape predictions difficult . for that reason we determine the substrate shape for each individual tfm data set by image processing of the relaxed configuration of the marker beads , which are distributed sufficiently homogeneously in the substrate as to carve out the substrate shape .  -shape algorithm is applied to projected bead positions in order to determine the envelope .( d ) from multiple envelopes a 3d hexahedral substrate mesh is reconstructed . meshing with _ gmesh _ and visualization with _scale bar m . ]we developed a custom mesh generation program as illustrated in fig .[ femtfm_alpha_shapes_and_hexmesh ] . in fig .[ femtfm_alpha_shapes_and_hexmesh]a we show an image of the marker beads ; the inset shows the strong anisotropy of the point spread function .first bead locations are partitioned into sections along the long axis of the wavy pattern that are separated by a partition width w. in a second step bead positions are mapped to each section plane ( normal direction pointing towards the partition direction ) , compare fig .[ femtfm_alpha_shapes_and_hexmesh]b . by doing this ,we end up with a set of separated slices associated with 2d bead distributions . after that, we calculate the 2d hull for each bead distribution using the 2d -shape algorithm implemented in the open - source computational geometry algorithm library ( cgal ) , compare fig .[ femtfm_alpha_shapes_and_hexmesh]c .after determination of the hull for each segmented slice we again stitch the determined contours together separated by the partition width w. we then form a 3d hexahedral mesh that approximates the substrate shape . in this particular stepthe program uses the open source mesh generator gmsh . a representative result for the described procedureis depicted in fig .[ femtfm_alpha_shapes_and_hexmesh]d .traction reconstruction is established by solving an inverse elastic boundary volume problem ( bvp ) .we first describe the corresponding direct bvp .because green s functions are not known for the wavy shapes considered here , we use the finite element methods ( fem ) . in principle , this allows us to use also non - linear formulations for large deformations and non - linear materials .however , here we deal with linear material ( pdms ) and small strains , thus the linear formulation is sufficient .we therefore have to solve the navier - lam equation for the displacement field in the computational domain : traction exerted by cells is applied to the top surface of the substrate volume in the reference state . accordinglythe traction field enters as stress boundary condition , , where is the normal vector of the unit surface element and the substrate cauchy stress tensor . as illustrated in fig .[ femtfm_fig_bvp_and_reconstruction]a , we choose appropriate mixed bcs at the remaining parts of the boundary arriving at a well - defined bvp . at the bottom surfacewe require zero displacement , , due to rigid coupling between the soft elastomeric substrate and the underlying rigid glass coverslip . since the used mesh represents a cutout of the substrate , which is largely extended in lateral directions , proper boundary conditions at the side surfaces are applied . in case of an infinite half - space , an appropriate boundary condition forthe side surfaces of the cutout would be a counter stress of the same magnitude .this however would lead to an undesired recursive problem . herewe use vanishing stress boundary conditions , , for sufficient large cutouts .based on the knowledge that the displacement field monotonically decays at least as , the influence of boundary conditions can be neglected for sufficient large cutouts .therefore , we occasionally extend the substrate mesh in lateral directions to ensure the repression of boundary effects .   and poisson s ratio .boundary conditions : bottom - zero displacement ( ) , ( sides ) - zero - stress ( ) , top - traction field ( ) .( b ) work flow of traction reconstruction ( inverse bvp ) .starting from 3d image stack raw data several steps are processed : 3d bead tracking , substrate mesh generation , fem - calculations and numeric traction optimization.,scaledwidth=80.0% ]  for a given traction pattern , the direct bvp formulated in eq .[ direc_bvp_first ] is now solved by means of the fem as typically applied to elastic problems . for this purpose , we transform the elastic equations into the weak form and use linear shape functions on a hexahedral lattice to arrive at an algebraic problem ( the mathematical details are provided as supplementary material ) . the algebraic equations are solved with the conjugated gradient ( cg ) method . alternatively , it is also possible to directly invert them by means of e.g. gauss - elimination .the achieved solution then can be used to interpolate the displacement to any position within the domain . for the implementation of our fem - approach, we used the fem c++ library _ deal.ii _ .it provides the essential set of features to achieve a fem - calculation , for instances , managing local and global dof indices , matrix manipulation feature , gauss - quadrature , coordinate transformations and solvers for the linear algebraic system .the inversion of the direct bvp is in general ill - posed , which implies that no unique or sufficiently stable solution exists .therefore regularization methods have to be used to render the inverse problem stable . according to tikhonov and arsenin ,the regularized inverse problems can be formulated as a minimization problem and in tfm the corresponding tikhonov functional  ] represents an error estimate that assigns a value to differences between calculated and measured displacements ( the _ residue _ ) .the larger the deviation , the larger is the scalar value the estimate returns .the second term  ] .we next need to define the form of the residue and the regularization term in eq .[ eq : general_target_functional ] .standard tfm uses the least square estimate to measure deviations between measured and computed displacements .additionally , most methods use order tikhonov regularization = \lambda \int_{\partial \omega_{top } } \|\mathbf{t}(\mathbf{x } ' ) \|^2~da ] and the subsequent line minimization .parallel computing was realized by using the message passing interface ( mpi ) , which is suitable for large scale distributed computing on computer clusters or sheared memory systems . although the cg method is a common tool in the field of inverse problems , e.g. , we tested other optimization methods as well , namely gradient - less downhill - simplex optimization , simple steepest descent and heuristic monte - carlo optimization with simulated annealing ( all described in ) .we found that the cg method led to the shortest computation times and excellent convergence .[ femtfm_fig_bvp_and_reconstruction]b shows a schematic representation of our complete workflow .kpa pdms substrate . substrate displacement field and 2d traction forces obtained from piv and regularized fourier transform traction cytometry ( reg - fttc ) .the cell shows a typical dipole pattern with strong inward directed forces at the cell periphery .( b ) illustration of traction pattern simulation .traction patches are distributed within a ring - shaped area close to the cell edge with normal components that are balanced by a central patch that is consequently pushed into the substrate .traction magnitude in lateral direction is set to a constant value of kpa ., scaledwidth=100.0% ]  as reported earlier , for 2d tfm our implementation of fem - tfm gives equivalent results as the standard approach with fttc .in this case , fttc is much more efficient and faster .here however we want to address 3d tfm for wavy substrates in which a green s function is not known , making fem - tfm the most reasonable approach . in order to validate our method for a 3d situation , we first reconstructed simulated data for planar substrates with both tangential and normal components .we generated these distributions by a simple set of rules that mimic typical cellular force distributions .[ femtfm_fig_cell_like_simulation]a shows a typical fttc traction force pattern for a cardiac myofibroblast cultured on a 15 kpa pdms substrate .this cell shows the typical dipole pattern that has been suggested as a minimal model for a contractile cell .[ femtfm_fig_cell_like_simulation]b illustrates the rules we use to generate artificial traction patterns .a cell is modeled as a circle with traction patches located only in a peripheral annulus of width .each of these patches carries a tangential traction stress of kpa which is a typical value found for cells . however , previous studies also reported appreciable cellular traction stress in normal directions .they found a typical pull - push pattern , such that the cell is pulling up at the periphery and pushing down with the cell body .the upward forces might arise from actin fibers being anchored at the dorsal side of the cell or at the upper side of the nucleus , while the downward force might be the reaction force localized by the large and relatively stiff nucleus . herewe include these normal forces in our simulations by adding force in positive z - direction to our adhesion patches , that are counterbalanced by an extra traction patch located at the cell center .together , these rules allow us to generate realistic traction distributions that satisfy basic properties of a typical cell induced 3d traction pattern .the displacement fields resulting from these simulated traction patterns were calculated using the direct bvp introduced above ( eq . [ direc_bvp_first ] ) with our fem - implementation . from the resulting displacement field, we sampled random displacements . to account for uncertainties due to the contribution of experimental noise , we introduced additive random displacement errors that modify each displacement component by a random value .subsequently a simulated bead displacement is expressed by .the simulated data allows us to validate and characterize features and limits of our method under well - controlled conditions .beads / , which corresponds to a bead number of .( d ) traction reconstructed with fem on regular mesh ( 3.267 degrees of freedom , dofs ) .( e ) reconstruction on locally h - refined mesh ( 1.848 dofs ) works equally well .( f ) comparison of simulation and reconstruction along a traction profile through a force patch ( white line in ( a)).,scaledwidth=100.0% ]  in fig .[ femtfm_fig_adaptive_traction_reconstruction]a - c we show the results of a typical simulation without noise ( ) . figs .[ femtfm_fig_adaptive_traction_reconstruction]d and e depict a comparison of reconstruction results on a homogeneous and a adaptively refined mesh , respectively . for the adaptive reconstruction , we started with a two times larger mesh size compared to the homogeneous mesh .after one refinement step , we subsequently achieve the same mesh size at refined regions , which makes results comparable .both results show an excellent agreement and reconstruct correctly the original traction pattern .the depicted mesh topology shows that the adaptive refinement automatically adjusts the local mesh size to regions of accumulated traction stress , where we accordingly establish higher local accuracy . by doing this we saved of degrees of freedom ( dofs ) which has a positive effect on the computation time .[ femtfm_fig_adaptive_traction_reconstruction]f shows the traction profile through a reconstructed traction patch .both reconstructions lead to a slightly smoothed profile compared to the original .this reduced resolution of a sharp edge can be explained by a limiting bead density ( in our simulation ) according to the sampling theorem by nyquist and shannon . from the simulation results we conclude that adaptive local mesh refinement has no detrimental effect on the traction reconstruction accuracy .however , it clearly needs much less dofs and therefore is much more computer time efficient .  when solving an inverse problem the stability of the solution strongly depends on the uncertainties in the provided data .in particular , the problem might be ill - posed due to the effect of noise .this has been explicitly shown for the case of traction reconstruction .as mentioned in the introduction , experimental conditions limit the resolution measured displacement fields .main reasons here are limited optical resolution of the microscope and errors in the bead tracking procedures .the first issue can be treated as gaussian shaped errors that limits the spatial localization of beads .the latter is less relevant in the planar 2d case since state of the art 2d tracking methods are very accurate with eventually negligible error rates . however , bead localization and tracking in 3d is by far more challenging , even with modern microscopic setups .this is due to anisotropic optical resolution that produce a notably inferior accuracy in z - direction compared to the corresponding lateral directions .there already exist improved setups like dual objective super resolution microscopy techniques , however , these are not standard and rarely available and because of this no application in tfm has been shown so far . therefore , we aimed at improving bead tracking in 3d and on adapting the traction reconstruction method to data with anisotropic optical resolution .in fact we realized that due to the substrate topography we have to deal with a locally varying point spread function .that directly effected the tracking of bead movements when using image cross correlation techniques .hence , we applied a more robust single bead tracking method than done in standard tfm , using feature vectors introduced earlier for 3d tfm for the same reasons . by using this techniquewe obtained a significantly improved displacement field , however , the data showed strong and inhomogeneous drift which could not be corrected by a single drift vector ( evaluated in one focal plane ) .this effect can be explained by relatively long image stack acquisition times of approximately min when recording images per stack . due to this long acquisition timewe observed non - monotonic drift when comparing two image stacks . in order to solve this problem, we applied drift correction for each individual slice , which successfully canceled out most of the drift .nevertheless , the derived displacement data showed anisotropic deviations and occasionally higher densities of outliers .   with gaussian noise added ( noise level ) .( b ) reconstruction without regularization does not give a clear traction pattern .( c ) reconstruction with a regularization parameter , selected by generalized cross validation ( gcv ) , reconstructs the original traction pattern , however , with a reduced maximal traction magnitude of compared to the simulated pattern.,scaledwidth=100.0% ]  fig .[ femtfm_fig_gaussian_noise_reconstruction]a shows the displacement field used in fig .[ femtfm_fig_adaptive_traction_reconstruction ] with gaussian noise of strength ( measured with respect to maximal displacement ) . the force reconstruction without regularizationis shown as fig .[ femtfm_fig_gaussian_noise_reconstruction]b and clearly is not very good .therefore we next used order tikhonov regularization ( ) to find an approximated reconstruction solution . here , we determine an optimal regularization parameter by using generalized cross validation ( gcv ) .the optimal value of depends on the noise level and for the given example , we determined .the regularization improves the result significantly , as shown in fig .[ femtfm_fig_gaussian_noise_reconstruction]c .the penalization of total force induced by tikhonov regularization , however , led to traction underestimation and edge smoothing .next , we simulated the presence of outliers .we drew the additive random displacement error for each bead either from a gaussian distribution or from a box - shaped distribution limited by the box width .we decided from which distribution an error contribution is drawn by calculating an equally distributed random variable over the interval $ ] .if , we draw from the boxed - shaped distribution , corresponding to an outlier . in case of , we calculated a gaussian distribution .here , corresponds to the fraction of outliers .in general , there are two possible approaches to diminish the effect of outliers for the reconstruction .one is to filter out outliers based on predefined criteria .this demands to set up appropriate filter limits .moreover , such a procedure might overlook valuable information in the data . as a second approach one can use robust estimates for the optimization process .we will show in the following that reconstruction can be improved , when the least square estimate in the tikhonov functional gets replaced by a robust measure more insensitive against data outliers .we tested different robust maximum likelihood estimates known from optimization theory .the simplest robust estimate uses the l1-norm , . as the l2-estimate described above, it can be derived analytically from the maximum likelihood assumption , in this case based on a laplace distribution .the essential advantage of this estimate is its weighting of deviations , which compared to the l2 is less sensitive to outlier contributions .as alternative estimates , we implemented also huber functions or biweighting functions .they require an additional cutoff parameter to characterize transition feature of the effective deviation weighting .this indeed requires a good guess about expected outlier strength .   and for two different estimates , l2 ( blue ) and l1 ( red ) .( b ) l2-based reconstruction for outlier fraction .( c ) l1-based reconstruction for outlier fraction . for all simulations the gaussian noise level was .,scaledwidth=100.0% ]  fig .[ femtfm_fig_reconstruction_outliers ] depicts the influence of the used estimate on the quality of the reconstructed traction field .the simulation study considered an isotropic gaussian based noise of and a isotropic box shaped distribution with .we investigated the convergence behavior for the optimization achieved by l1- and l2-estimates with respect the outlier fractions .to compare the results , we introduced the relative displacement deviation . for ( no outliers ) , both estimates converge to a similar result , however , the l1 converges slower .if we chose a non - zero fraction of outliers , the convergence dynamics of the optimization procedure starts to differ between l2 and l1 .the l1 shows still a monotonic decreasing , while in case of the l2 the curve starts to increase again saturating into a different solution .corresponding traction field solutions for are depicted in fig .[ femtfm_fig_reconstruction_outliers ] . in case of using the l2 , the reconstruction is strongly influenced by the fraction of outliers .in contrast , the l1-reconstruction leads to satisfying results comparable to the target field .simulated displacement field with bead volume density beads / ( ) , ( c ) projected to the xy - plane , and ( d ) projected to the xz - plane .( e ) the achieved reconstruction is very close to the original traction pattern.,scaledwidth=60.0% ]  after having established a successful approach for planar substrates with both tangential and normal forces , we finally simulated 3d fem - tfm with wavy substrates as shown schematically in fig .[ femtfm_fig_cellb_reconstruction]a .we first distributed random traction patches at both flanks of the sinusoidal shape , while the traction vectors point at a common virtual point . in a second step, all traction directions got reoriented due to movement of the virtual point in 3d , until a balance of overall force was achieved .[ femtfm_fig_cellb_reconstruction]b and c depict an exemplary simulation result .it shows a traction field with four distinct traction patches with a force density of kpa .the corresponding simulated bead displacements are depicted in c and d as of projections to the xy - plane and xz - planes , respectively .compared to the planar case , the displacements in z - direction became more prominent , see fig .[ femtfm_fig_cellb_reconstruction]d . the resulting reconstruction shown in fig .[ femtfm_fig_cellb_reconstruction]e shows similar good agreement as for the planar case , thereby validating our method also for the experimental setup to be studied .    between cell orientation ( red arrow ) and substrate orientation ( blue arrow ) .( c ) cell shape plotted versus the angular difference .,scaledwidth=100.0% ]  we finally investigated myofibroblasts adhered to elastic wavy pdms - substrates with a focus on their morphology and traction forces . for our context , the choice of pdms has several advantages over the paa - system often employed for planar 2d tfm .in particular , it does not swell due to water uptake , it has superior optical properties , and it can be reproducibly molded into wave structures on a microscopic scale of several tens of micrometers .we first quantified the effect of substrate topography on cell morphology and cytoskeletal polarization .for this purpose we evaluated a data set of stack images of cells adhered to waves of width and height .a representative phase contrast image is shown in fig .[ femtfm_fig_cell_morphology]a .such images were used to determine the orientation of the topographic features .the cells were fluorescently stained for cell adhesion - related molecules to measure their internal organization .we determined the direction and degree of cell polarization by fitting a 3d ellipse to the image and subsequently evaluated the orientation and eccentricity of the ellipse , compare fig .[ femtfm_fig_cell_morphology]b .we finally correlated the two measures .[ femtfm_fig_cell_morphology]c shows the eccentricity as a function of the angle between substrate and cell orientations .the plot confirms that cells tend to align with the long axis of the substrate .2/3 of the cells show a ratio ( semimajor over semiminor ) larger than 2 and are aligned with the wave within the angular range of to .this indicates that even the cells that are close to round have sensed the topographic features of the wavy substrate . from our ellipse analysiswe further found that they tend to adhere most often to the wave flanks and less to the valleys or hill tops of the height - modulated structures .m.,scaledwidth=100.0% ]  in addition to the morphological study we applied the fem traction reconstruction to an exemplary set of cells . here , we wanted to investigate the influence of environmental topography on cellular contractility . the long acquisition time for displacement datahowever made it very difficult to conduct a full statistical analysis and therefore in fig .[ femtfm_fig_result_reconstruction ] we only show a few typical examples of cardiac myofibroblasts cultured on wavy substrates with a young s modulus of kpa . in fig .[ femtfm_fig_result_reconstruction ] two different topographies are used , micro - grooves and wave shapes . for the substrates in cases a / b / c we used 70/142/91 slices with z - distance between two slices of 0.55/0.39/0.30 .we tracked 6205/9614/13613 beads , with an average distance between beads of 5.40/5.91/2.90 .for the reconstruction , we used the adaptive mesh refinement approach and started to calculate a traction solution on a coarser mesh ( optimization steps ) . afterwards the algorithm refined of the computational grid cells at the surface based on obtained traction magnitudes .we subsequently achieved local mesh sizes of for the corresponding cells a / b / c ( see fig .[ femtfm_fig_result_reconstruction ] ) .we further utilized an l1-estimate for the residue and a 0th order tikhonov regularization with l2-norm ( same regularization parameter for all data ) .as explained above , the -direction is weightened only with a factor of due to its poor resolution compared to the two lateral directions .cell a was adhered on top a slightly curved surface between two grooves ( dimensions : , ) .compared to the others it shows the largest spread area and a roughly homogeneous distribution of adhesion sites ( observed in vasodilator - stimulated phosphoprotein ( vasp ) fluorescence image ) .for this cell , we obtained the largest maximal traction as pa .cell b occupied the edge region of two opposite groove flanks ( groove dimensions : , ) .fas were concentrated to limited region at the edge and the cell spanned the groove with actin sfs bridging free space .the maximal traction was pa . strikingly the traction vectors revealed that most of the forces were balanced along the edges and not to the opposite groove flanks .cell c spanned the gap between two wave shaped hills ( wave shape : , with separation distance ) .fas were more concentrated at vertical parts of the edge region with a similar total area compared to cell b. we derived the lowest maximal traction pa for this cell .as for cell b , however , most of the forces seemed to be balanced along the right wave instead of across the gap .therefore we conclude that even if cells span two neighboring hills , they still tend to balance their forces along the ridges , in agreement with the earlier observation that their cytoskeleton is in average organized in this direction .here we have described experimental and theoretical methods to reconstruct cellular traction patterns of cells adhering to non - planar ( wavy ) substrates .this involves novel micromolding techniques to prepare curved elastic substrates with embedded fluorescent beads , improved image processing procedures for bead tracking and a completely novel 3d tfm workflow to achieve traction reconstruction utilizing fem and optimization procedures .we successfully checked the validability of our method by first reconstructing simulated data considering different experimental conditions .as shown in the method validation part , the presence of outliers plays a crucial role in proper traction force reconstruction for our 3d data on topographic substrates . by replacing the typically used l2- by an l1-error estimate, we demonstrated a way to prevent problems in traction reconstruction originated from outliers .the l1-estimate is a much more robust measure in the presence of outliers in noisy data .this can be essentially traced back to a reduced weighting of outliers ( linear weighting ) during the reconstruction process .compared to other known robust estimates , the l1-norm needs no additionally cutoff parameters and converges to the l2-solution for outlier - free data with gaussian error distributions .we note that this approach aims at improving traction field solutions for non - smooth and noisy 3d data .it has no essential influence on the regularization scheme .here we stay with the standard l2-norm , which seems appropriate for our case of dense marker beads .recently , it has been suggested that the l1-norm is favorable for the regularization term in the case of high resolution 2d tfm .although we expect that this approach is not required in our case of rather dense bead positioning ( typical bead difference of 4 and cell size around 100 ) , for future tfm applications it seems very interesting to explore in which situation which combination of norms for the residue and the regularization term are most favorable .we also emphasize the need for adaptations in the image processing procedures on wavy substrates .because the point spread function for the marker beads can vary locally in this case , we can not use the standard cross correlation approach for bead tracking , but resort to a template matching procedure based on feature vectors .we also take into account the anisotropy in the point spread function by using a reduced weight for the z - component in the residue . together with the use of the l1-norm for the residue, this ensures a very good quality of our traction force reconstruction as validated by computer simulations .our analysis is facilitated by the use of topographic substrates with cylinder symmetry , which allows us to use the slicing and meshing procedures shown in fig .[ femtfm_alpha_shapes_and_hexmesh ] .we next investigated the degree and orientation of cellular polarity for cardiac myofibroblasts , which have been exposed to a wavy substrate geometry .we found that these cells strongly align and polarize perpendicular to the direction of maximal curvature , which has been reported before for other types of fibroblasts on rigid microcylinders .we note that even round cells tend to align with the substrates features , presumabely because the feature dimensions are such that every cell is affected by the topographical cues .in addition to this observation we determined traction force maps for an exemplary set of cells and demonstrated the successful reconstruction of the 3d forces .we found that cells also balance their forces along the orientation of the pattern , even if at the same time forming a bridge between neighboring hills . because our reconstruction is strongly regularized due to the noise issues with wavy substrates , the absolute values of the measured traction stresses are expected to be lower than the actual values , as shown earlier with model - based traction force microscopy .in fact , we expect the real traction stress to be closer to the kpa - range as used for the simulated data .  in the future , further advances in the experimental techniquesas described here ( including faster acquisition of the image data , for example with a light sheet microscope ) should make a full statistical analysis feasible. it would be also interesting to conduct these experiments with other cell types whose physiological function depends on curved or corrugated environment , such as podocytes filtering the blood in kidney glomeruli .this work was supported by mechanosys ( bmbf grant no . ) and the ectop programme of the cluster of excellence cellnetworks .bh , rm and uss would like to thank the isaac newton institute for mathematical sciences for its hospitality during the programme `` coupling geometric pdes with physics for cell morphology , motility and pattern formation '' ( cgpw02 ) supported by epsrc grant number ep / k032208/1 .uss is a member of the interdisciplinary center for scientific computing ( iwr ) at heidelberg .for a given traction pattern , the direct boundary value problem ( bvp ) formulated in eq. 1 is now solved by means of the finite element method ( fem ) as typically applied to elastic problems . the first step in sucha calculation is the transformation of the given equations into the weak form .therefore we multiply the equation with an arbitrary field and integrate over the substrate volume :  represents the constant elasticity matrix for isotropic elastic materials , . in the following we impose a discretization scheme , by partitioning the integral over into smaller elements corresponding the generated substrate mesh .we further apply a local interpolation scheme .this has the objective to reduce the infinite dimensional space of displacement solutions to a finite dimensional subspace of nodal displacement values .the weak form for a single element reads : in our calculations , we consider hexahedral elements with eight nodal points at the element boundary . as an advantage of hexahedral elements ,the volume and surface integration can be mapped to an integration over the unit cube parametrized by cartesian coordinates , , while the coordinate transformation is determined by an individual jacoby matrix calculated for each element .thus the integration reads :  represents the jacobian and the tilde marks quantities with respect to the new coordinate . for subsequent interpolation of the displacement field we use interpolation functions , which are based on an elementary set of linear shape functions . by interpolation of nodal valueswe can rewrite the displacement field , which now only dependents on a discrete number of degrees of freedom ( dofs ) ( in the hexahedral case used here , this is the number of nodes multiplied by the dimension , ) :  is a vector of nodal displacement dof values and is the interpolation matrix , which interpolates the element displacement for a given configuration of nodal values .the same can be achieved for the virtual displacement . applying this to eq .[ element_virtual_work ] , we obtain }_{k^e } \mathbf{u}^e_{node } = \delta \mathbf{u}^e_{node } \underbrace{\left[\int_{\partial \omega^e } n(\boldsymbol{\xi})^{t } \mathbf{t }  j_s ~ds \right]}_{\mathbf{r}^e}\ ] ] is called the element stiffness matrix and the element load vector .since is a vector of arbitrary values , we can reduce the problem to solving a linear algebraic system : the components of the element stiffness matrix and the load vector are calculated numerically . here, the integrals have been solved by means of gauss quadrature . in this waywe calculate the stiffness matrix and load vector for each element .in a subsequent step , we assemble a global system that forms the domain :  , , and are the global stiffness matrix , the global load vector , and the global dof vector .nodal dofs are only shared by neighboring elements and thus is a sparse matrix , while most of the matrix entries are zero . is further singular and hence not invertible , since we have still not introduced constraints to avoid rigid body motions .therefore , the system must be further restrained by incorporating appropriate boundary conditions ( bcs ) . for our direct bvpwe consider the bcs illustrated in fig . 2 .the traction bc enters the system through the surface integral in the load vector . the zero stress bc leads to no constraints of the system .only the remaining zero displacement bc applied to the bottom surfaces constrains eq .[ femtfm_fem_global_system ] and by forcing the displacement condition , we can reduce the system to where is the reduced non - 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Abstract: animal cells use traction forces to sense the mechanics and geometry of their environment . measuring these traction forces requires a workflow combining cell experiments , image processing and force reconstruction based on elasticity theory . such procedures have been established before mainly for planar substrates , in which case one can use the green s function formalism . here we introduce a worksflow to measure traction forces of cardiac myofibroblasts on non - planar elastic substrates . soft elastic substrates with a wave - like topology were micromolded from polydimethylsiloxane ( pdms ) and fluorescent marker beads were distributed homogeneously in the substrate . using feature vector based tracking of these marker beads , we first constructed a hexahedral mesh for the substrate . we then solved the direct elastic boundary volume problem on this mesh using the finite element method ( fem ) . using data simulations , we show that the traction forces can be reconstructed from the substrate deformations by solving the corresponding inverse problem with a l1-norm for the residue and a l2-norm for 0th order tikhonov regularization . applying this procedure to the experimental data , we find that cardiac myofibroblast cells tend to align both their shapes and their forces with the long axis of the deformable wavy substrate .