Trying nearer on the non-deep studying components

About half a yr in the past, this weblog featured a publish, written by Daniel Falbel, on use Keras to categorise items of spoken language. The article acquired a variety of consideration and never surprisingly, questions arose apply that code to completely different datasets. We’ll take this as a motivation to discover in additional depth the preprocessing executed in that publish: If we all know why the enter to the community seems to be the best way it seems to be, we will modify the mannequin specification appropriately if want be.

In case you have got a background in speech recognition, and even common sign processing, for you the introductory a part of this publish will most likely not comprise a lot information. Nonetheless, you may nonetheless have an interest within the code half, which exhibits do issues like creating spectrograms with present variations of TensorFlow.
Should you don’t have that background, we’re inviting you on a (hopefully) fascinating journey, barely pertaining to one of many larger mysteries of this universe.

We’ll use the identical dataset as Daniel did in his publish, that’s, model 1 of the Google speech instructions dataset(Warden 2018)
The dataset consists of ~ 65,000 WAV information, of size one second or much less. Every file is a recording of one in every of thirty phrases, uttered by completely different audio system.

The aim then is to coach a community to discriminate between spoken phrases. How ought to the enter to the community look? The WAV information comprise amplitudes of sound waves over time. Listed here are a number of examples, equivalent to the phrases chook, down, sheila, and visible:

A sound wave is a sign extending in time, analogously to how what enters our visible system extends in house.
At every cut-off date, the present sign depends on its previous. The plain structure to make use of in modeling it thus appears to be a recurrent neural community.

Nonetheless, the knowledge contained within the sound wave may be represented in an alternate manner: specifically, utilizing the frequencies that make up the sign.

Right here we see a sound wave (high) and its frequency illustration (backside).

Within the time illustration (known as the time area), the sign consists of consecutive amplitudes over time. Within the frequency area, it’s represented as magnitudes of various frequencies. It could seem as one of many best mysteries on this world that you may convert between these two with out lack of data, that’s: Each representations are primarily equal!

Conversion from the time area to the frequency area is finished utilizing the Fourier remodel; to transform again, the Inverse Fourier Remodel is used. There exist several types of Fourier transforms relying on whether or not time is seen as steady or discrete, and whether or not the sign itself is steady or discrete. Within the “actual world,” the place normally for us, actual means digital as we’re working with digitized alerts, the time area in addition to the sign are represented as discrete and so, the Discrete Fourier Remodel (DFT) is used. The DFT itself is computed utilizing the FFT (Quick Fourier Remodel) algorithm, leading to important speedup over a naive implementation.

Trying again on the above instance sound wave, it’s a compound of 4 sine waves, of frequencies 8Hz, 16Hz, 32Hz, and 64Hz, whose amplitudes are added and displayed over time. The compound wave right here is assumed to increase infinitely in time. Not like speech, which modifications over time, it may be characterised by a single enumeration of the magnitudes of the frequencies it’s composed of. So right here the spectrogram, the characterization of a sign by magnitudes of constituent frequencies various over time, seems to be primarily one-dimensional.

Nonetheless, after we ask Praat to create a spectrogram of one in every of our instance sounds (a seven), it may appear like this:

Right here we see a two-dimensional picture of frequency magnitudes over time (greater magnitudes indicated by darker coloring). This two-dimensional illustration could also be fed to a community, rather than the one-dimensional amplitudes. Accordingly, if we resolve to take action we’ll use a convnet as a substitute of an RNN.

Spectrograms will look completely different relying on how we create them. We’ll check out the important choices in a minute. First although, let’s see what we can’t all the time do: ask for all frequencies that have been contained within the analog sign.

Above, we mentioned that each representations, time area and frequency area, have been primarily equal. In our digital actual world, that is solely true if the sign we’re working with has been digitized appropriately, or as that is generally phrased, if it has been “correctly sampled.”

Take speech for instance: As an analog sign, speech per se is steady in time; for us to have the ability to work with it on a pc, it must be transformed to occur in discrete time. This conversion of the unbiased variable (time in our case, house in e.g. picture processing) from steady to discrete is known as sampling.

On this means of discretization, a vital resolution to be made is the sampling fee to make use of. The sampling fee needs to be at the very least double the best frequency within the sign. If it’s not, lack of data will happen. The way in which that is most frequently put is the opposite manner spherical: To protect all data, the analog sign could not comprise frequencies above one-half the sampling fee. This frequency – half the sampling fee – is known as the Nyquist fee.

If the sampling fee is simply too low, aliasing takes place: Increased frequencies alias themselves as decrease frequencies. Because of this not solely can’t we get them, additionally they corrupt the magnitudes of corresponding decrease frequencies they’re being added to.
Right here’s a schematic instance of how a high-frequency sign may alias itself as being lower-frequency. Think about the high-frequency wave being sampled at integer factors (gray circles) solely:

Within the case of the speech instructions dataset, all sound waves have been sampled at 16 kHz. Because of this after we ask Praat for a spectogram, we must always not ask for frequencies greater than 8kHz. Here’s what occurs if we ask for frequencies as much as 16kHz as a substitute – we simply don’t get them:

Now let’s see what choices we do have when creating spectrograms.

Within the above easy sine wave instance, the sign stayed fixed over time. Nonetheless in speech utterances, the magnitudes of constituent frequencies change over time. Ideally thus, we’d have a precise frequency illustration for each cut-off date. As an approximation to this preferrred, the sign is split into overlapping home windows, and the Fourier remodel is computed for every time slice individually. That is known as the Quick Time Fourier Remodel (STFT).

After we compute the spectrogram by way of the STFT, we have to inform it what measurement home windows to make use of, and the way large to make the overlap. The longer the home windows we use, the higher the decision we get within the frequency area. Nonetheless, what we achieve in decision there, we lose within the time area, as we’ll have fewer home windows representing the sign. It is a common precept in sign processing: Decision within the time and frequency domains are inversely associated.

To make this extra concrete, let’s once more have a look at a easy instance. Right here is the spectrogram of an artificial sine wave, composed of two parts at 1000 Hz and 1200 Hz. The window size was left at its (Praat) default, 5 milliseconds:

We see that with a brief window like that, the 2 completely different frequencies are mangled into one within the spectrogram.
Now enlarge the window to 30 milliseconds, and they’re clearly differentiated:

The above spectrogram of the phrase “seven” was produced utilizing Praats default of 5 milliseconds. What occurs if we use 30 milliseconds as a substitute?

We get higher frequency decision, however on the value of decrease decision within the time area. The window size used throughout preprocessing is a parameter we’d wish to experiment with later, when coaching a community.

One other enter to the STFT to play with is the kind of window used to weight the samples in a time slice. Right here once more are three spectrograms of the above recording of seven, utilizing, respectively, a Hamming, a Hann, and a Gaussian window:

Whereas the spectrograms utilizing the Hann and Gaussian home windows don’t look a lot completely different, the Hamming window appears to have launched some artifacts.

Preprocessing choices don’t finish with the spectrogram. A well-liked transformation utilized to the spectrogram is conversion to mel scale, a scale based mostly on how people truly understand variations in pitch. We don’t elaborate additional on this right here, however we do briefly touch upon the respective TensorFlow code under, in case you’d prefer to experiment with this.
Prior to now, coefficients remodeled to Mel scale have typically been additional processed to acquire the so-called Mel-Frequency Cepstral Coefficients (MFCCs). Once more, we simply present the code. For wonderful studying on Mel scale conversion and MFCCs (together with the explanation why MFCCs are much less usually used these days) see this publish by Haytham Fayek.

Again to our authentic activity of speech classification. Now that we’ve gained a little bit of perception in what’s concerned, let’s see carry out these transformations in TensorFlow.

Code will likely be represented in snippets in line with the performance it offers, so we could straight map it to what was defined conceptually above.
An entire instance is out there right here. The whole instance builds on Daniel’s authentic code as a lot as attainable, with two exceptions:

• The code runs in keen in addition to in static graph mode. Should you resolve you solely ever want keen mode, there are a number of locations that may be simplified. That is partly associated to the truth that in keen mode, TensorFlow operations rather than tensors return values, which we are able to straight go on to TensorFlow capabilities anticipating values, not tensors. As well as, much less conversion code is required when manipulating intermediate values in R.

• With TensorFlow 1.13 being launched any day, and preparations for TF 2.0 working at full velocity, we wish the code to necessitate as few modifications as attainable to run on the following main model of TF. One large distinction is that there’ll now not be a contrib module. Within the authentic publish, contrib was used to learn within the .wav information in addition to compute the spectrograms. Right here, we are going to use performance from tf.audio and tf.sign as a substitute.

All operations proven under will run inside tf.dataset code, which on the R facet is achieved utilizing the tfdatasets package deal.
To elucidate the person operations, we have a look at a single file, however later we’ll additionally show the information generator as an entire.

For stepping via particular person strains, it’s all the time useful to have keen mode enabled, independently of whether or not in the end we’ll execute in keen or graph mode:

We decide a random .wav file and decode it utilizing tf$audio$decode_wav.This can give us entry to 2 tensors: the samples themselves, and the sampling fee.

fname  "information/speech_commands_v0.01/chook/00b01445_nohash_0.wav"
wav  tf$audio$decode_wav(tf$read_file(fname))

wav$sample_rate accommodates the sampling fee. As anticipated, it’s 16000, or 16kHz:

sampling_rate  wav$sample_rate %>% as.numeric()
sampling_rate

16000

The samples themselves are accessible as wav$audio, however their form is (16000, 1), so we’ve to transpose the tensor to get the standard (batch_size, variety of samples) format we’d like for additional processing.

samples  wav$audio
samples  samples %>% tf$transpose(perm = c(1L, 0L))
samples

tf.Tensor(
[[-0.00750732  0.04653931  0.02041626 ... -0.01004028 -0.01300049
  -0.00250244]], form=(1, 16000), dtype=float32)

Computing the spectogram

To compute the spectrogram, we use tf$sign$stft (the place stft stands for Quick Time Fourier Remodel). stft expects three non-default arguments: Apart from the enter sign itself, there are the window measurement, frame_length, and the stride to make use of when figuring out the overlapping home windows, frame_step. Each are expressed in items of variety of samples. So if we resolve on a window size of 30 milliseconds and a stride of 10 milliseconds …

window_size_ms  30
window_stride_ms  10

… we arrive on the following name:

samples_per_window  sampling_rate * window_size_ms/1000 
stride_samples   sampling_rate * window_stride_ms/1000 

stft_out  tf$sign$stft(
  samples,
  frame_length = as.integer(samples_per_window),
  frame_step = as.integer(stride_samples)
)

Inspecting the tensor we acquired again, stft_out, we see, for our single enter wave, a matrix of 98 x 257 complicated values:

tf.Tensor(
[[[ 1.03279948e-04+0.00000000e+00j -1.95371482e-04-6.41121820e-04j
   -1.60833192e-03+4.97534114e-04j ... -3.61620914e-05-1.07343149e-04j
   -2.82576875e-05-5.88812982e-05j  2.66879797e-05+0.00000000e+00j] 
   ... 
   ]],
form=(1, 98, 257), dtype=complex64)

Right here 98 is the variety of intervals, which we are able to compute upfront, based mostly on the variety of samples in a window and the dimensions of the stride:

n_periods  size(seq(samples_per_window/2, sampling_rate - samples_per_window/2, stride_samples))

257 is the variety of frequencies we obtained magnitudes for. By default, stft will apply a Quick Fourier Remodel of measurement smallest energy of two larger or equal to the variety of samples in a window, after which return the fft_length / 2 + 1 distinctive parts of the FFT: the zero-frequency time period and the positive-frequency phrases.

In our case, the variety of samples in a window is 480. The closest enclosing energy of two being 512, we find yourself with 512/2 + 1 = 257 coefficients.
This too we are able to compute upfront:

Again to the output of the STFT. Taking the elementwise magnitude of the complicated values, we receive an power spectrogram:

magnitude_spectrograms  tf$abs(stft_out)

If we cease preprocessing right here, we are going to normally wish to log remodel the values to raised match the sensitivity of the human auditory system:

log_magnitude_spectrograms = tf$log(magnitude_spectrograms + 1e-6)

Mel spectrograms and Mel-Frequency Cepstral Coefficients (MFCCs)

If as a substitute we select to make use of Mel spectrograms, we are able to receive a change matrix that can convert the unique spectrograms to Mel scale:

lower_edge_hertz  0
upper_edge_hertz  2595 * log10(1 + (sampling_rate/2)/700)
num_mel_bins  64L
num_spectrogram_bins  magnitude_spectrograms$form[-1]$worth

linear_to_mel_weight_matrix  tf$sign$linear_to_mel_weight_matrix(
  num_mel_bins,
  num_spectrogram_bins,
  sampling_rate,
  lower_edge_hertz,
  upper_edge_hertz
)

Making use of that matrix, we receive a tensor of measurement (batch_size, variety of intervals, variety of Mel coefficients) which once more, we are able to log-compress if we wish:

mel_spectrograms  tf$tensordot(magnitude_spectrograms, linear_to_mel_weight_matrix, 1L)
log_mel_spectrograms  tf$log(mel_spectrograms + 1e-6)

Only for completeness’ sake, lastly we present the TensorFlow code used to additional compute MFCCs. We don’t embrace this within the full instance as with MFCCs, we would want a unique community structure.

num_mfccs  13
mfccs  tf$sign$mfccs_from_log_mel_spectrograms(log_mel_spectrograms)[, , 1:num_mfccs]

Accommodating different-length inputs

In our full instance, we decide the sampling fee from the primary file learn, thus assuming all recordings have been sampled on the identical fee. We do permit for various lengths although. For instance in our dataset, had we used this file, simply 0.65 seconds lengthy, for demonstration functions:

fname  "information/speech_commands_v0.01/chook/1746d7b6_nohash_0.wav"

we’d have ended up with simply 63 intervals within the spectrogram. As we’ve to outline a set input_size for the primary conv layer, we have to pad the corresponding dimension to the utmost attainable size, which is n_periods computed above.
The padding truly takes place as a part of dataset definition. Let’s rapidly see dataset definition as an entire, leaving out the attainable technology of Mel spectrograms.

data_generator  perform(df,
                           window_size_ms,
                           window_stride_ms) {
  
  # assume sampling fee is similar in all samples
  sampling_rate 
    tf$audio$decode_wav(tf$read_file(tf$reshape(df$fname[[1]], record()))) %>% .$sample_rate
  
  samples_per_window  (sampling_rate * window_size_ms) %/% 1000L  
  stride_samples   (sampling_rate * window_stride_ms) %/% 1000L   
  
  n_periods 
    tf$form(
      tf$vary(
        samples_per_window %/% 2L,
        16000L - samples_per_window %/% 2L,
        stride_samples
      )
    )[1] + 1L
  
  n_fft_coefs 
    (2 ^ tf$ceil(tf$log(
      tf$forged(samples_per_window, tf$float32)
    ) / tf$log(2)) /
      2 + 1L) %>% tf$forged(tf$int32)
  
  ds  tensor_slices_dataset(df) %>%
    dataset_shuffle(buffer_size = buffer_size)
  
  ds  ds %>%
    dataset_map(perform(obs) {
      wav 
        tf$audio$decode_wav(tf$read_file(tf$reshape(obs$fname, record())))
      samples  wav$audio
      samples  samples %>% tf$transpose(perm = c(1L, 0L))
      
      stft_out  tf$sign$stft(samples,
                                 frame_length = samples_per_window,
                                 frame_step = stride_samples)
      
      magnitude_spectrograms  tf$abs(stft_out)
      log_magnitude_spectrograms  tf$log(magnitude_spectrograms + 1e-6)
      
      response  tf$one_hot(obs$class_id, 30L)

      enter  tf$transpose(log_magnitude_spectrograms, perm = c(1L, 2L, 0L))
      record(enter, response)
    })
  
  ds  ds %>%
    dataset_repeat()
  
  ds %>%
    dataset_padded_batch(
      batch_size = batch_size,
      padded_shapes = record(tf$stack(record(
        n_periods, n_fft_coefs,-1L
      )),
      tf$fixed(-1L, form = form(1L))),
      drop_remainder = TRUE
    )
}

The logic is similar as described above, solely the code has been generalized to work in keen in addition to graph mode. The padding is taken care of by dataset_padded_batch(), which must be instructed the utmost variety of intervals and the utmost variety of coefficients.

Time for experimentation

Constructing on the full instance, now could be the time for experimentation: How do completely different window sizes have an effect on classification accuracy? Does transformation to the mel scale yield improved outcomes? You may additionally wish to strive passing a non-default window_fn to stft (the default being the Hann window) and see how that impacts the outcomes. And naturally, the easy definition of the community leaves a variety of room for enchancment.

Talking of the community: Now that we’ve gained extra perception into what’s contained in a spectrogram, we’d begin asking, is a convnet actually an ample answer right here? Usually we use convnets on photos: two-dimensional information the place each dimensions signify the identical form of data. Thus with photos, it’s pure to have sq. filter kernels.
In a spectrogram although, the time axis and the frequency axis signify essentially several types of data, and it isn’t clear in any respect that we must always deal with them equally. Additionally, whereas in photos, the interpretation invariance of convnets is a desired characteristic, this isn’t the case for the frequency axis in a spectrogram.

Closing the circle, we uncover that attributable to deeper information in regards to the topic area, we’re in a greater place to purpose about (hopefully) profitable community architectures. We depart it to the creativity of our readers to proceed the search…