Posit AI Weblog: Wavelet Remodel

Be aware: Like a number of prior ones, this publish is an excerpt from the forthcoming e book, Deep Studying and Scientific Computing with R torch. And like many excerpts, it’s a product of onerous trade-offs. For added depth and extra examples, I’ve to ask you to please seek the advice of the e book.

Wavelets and the Wavelet Remodel

What are wavelets? Just like the Fourier foundation, they’re capabilities; however they don’t prolong infinitely. As a substitute, they’re localized in time: Away from the middle, they rapidly decay to zero. Along with a location parameter, in addition they have a scale: At completely different scales, they seem squished or stretched. Squished, they are going to do higher at detecting excessive frequencies; the converse applies once they’re stretched out in time.

The essential operation concerned within the Wavelet Remodel is convolution – have the (flipped) wavelet slide over the info, computing a sequence of dot merchandise. This manner, the wavelet is mainly searching for similarity.

As to the wavelet capabilities themselves, there are various of them. In a sensible software, we’d wish to experiment and choose the one which works finest for the given knowledge. In comparison with the DFT and spectrograms, extra experimentation tends to be concerned in wavelet evaluation.

The subject of wavelets could be very completely different from that of Fourier transforms in different respects, as nicely. Notably, there’s a lot much less standardization in terminology, use of symbols, and precise practices. On this introduction, I’m leaning closely on one particular exposition, the one in Arnt Vistnes’ very good e book on waves (Vistnes 2018). In different phrases, each terminology and examples replicate the alternatives made in that e book.

Introducing the Morlet wavelet

The Morlet, often known as Gabor, wavelet is outlined like so:

[
Psi_{omega_{a},K,t_{k}}(t_n) = (e^{-i omega_{a} (t_n – t_k)} – e^{-K^2}) e^{- omega_a^2 (t_n – t_k )^2 /(2K )^2}
]

This formulation pertains to discretized knowledge, the sorts of information we work with in apply. Thus, (t_k) and (t_n) designate deadlines, or equivalently, particular person time-series samples.

This equation appears to be like daunting at first, however we will “tame” it a bit by analyzing its construction, and pointing to the primary actors. For concreteness, although, we first take a look at an instance wavelet.

We begin by implementing the above equation:

Evaluating code and mathematical formulation, we discover a distinction. The operate itself takes one argument, (t_n); its realization, 4 (omega, Ok, t_k, and t). It’s because the torch code is vectorized: On the one hand, omega, Ok, and t_k, which, within the formulation, correspond to (omega_{a}), (Ok), and (t_k) , are scalars. (Within the equation, they’re assumed to be mounted.) t, alternatively, is a vector; it can maintain the measurement instances of the collection to be analyzed.

We choose instance values for omega, Ok, and t_k, in addition to a spread of instances to guage the wavelet on, and plot its values:

omega <- 6 * pi
Ok <- 6
t_k <- 5
 
sample_time <- torch_arange(3, 7, 0.0001)

create_wavelet_plot <- operate(omega, Ok, t_k, sample_time) {
  morlet <- morlet(omega, Ok, t_k, sample_time)
  df <- knowledge.body(
    x = as.numeric(sample_time),
    actual = as.numeric(morlet$actual),
    imag = as.numeric(morlet$imag)
  ) %>%
    pivot_longer(-x, names_to = "half", values_to = "worth")
  ggplot(df, aes(x = x, y = worth, coloration = half)) +
    geom_line() +
    scale_colour_grey(begin = 0.8, finish = 0.4) +
    xlab("time") +
    ylab("wavelet worth") +
    ggtitle("Morlet wavelet",
      subtitle = paste0("ω_a = ", omega / pi, "π , Ok = ", Ok)
    ) +
    theme_minimal()
}

create_wavelet_plot(omega, Ok, t_k, sample_time)

What we see here’s a advanced sine curve – word the actual and imaginary components, separated by a part shift of (pi/2) – that decays on either side of the middle. Trying again on the equation, we will determine the elements answerable for each options. The primary time period within the equation, (e^{-i omega_{a} (t_n – t_k)}), generates the oscillation; the third, (e^{- omega_a^2 (t_n – t_k )^2 /(2K )^2}), causes the exponential decay away from the middle. (In case you’re questioning in regards to the second time period, (e^{-Ok^2}): For given (Ok), it’s only a fixed.)

The third time period truly is a Gaussian, with location parameter (t_k) and scale (Ok). We’ll discuss (Ok) in nice element quickly, however what’s with (t_k)? (t_k) is the middle of the wavelet; for the Morlet wavelet, that is additionally the situation of most amplitude. As distance from the middle will increase, values rapidly strategy zero. That is what is supposed by wavelets being localized: They’re “energetic” solely on a brief vary of time.

The roles of (Ok) and (omega_a)

Now, we already mentioned that (Ok) is the dimensions of the Gaussian; it thus determines how far the curve spreads out in time. However there’s additionally (omega_a). Trying again on the Gaussian time period, it, too, will affect the unfold.

First although, what’s (omega_a)? The subscript (a) stands for “evaluation”; thus, (omega_a) denotes a single frequency being probed.

Now, let’s first examine visually the respective impacts of (omega_a) and (Ok).

p1 <- create_wavelet_plot(6 * pi, 4, 5, sample_time)
p2 <- create_wavelet_plot(6 * pi, 6, 5, sample_time)
p3 <- create_wavelet_plot(6 * pi, 8, 5, sample_time)
p4 <- create_wavelet_plot(4 * pi, 6, 5, sample_time)
p5 <- create_wavelet_plot(6 * pi, 6, 5, sample_time)
p6 <- create_wavelet_plot(8 * pi, 6, 5, sample_time)

(p1 | p4) /
  (p2 | p5) /
  (p3 | p6)

Within the left column, we maintain (omega_a) fixed, and differ (Ok). On the suitable, (omega_a) adjustments, and (Ok) stays the identical.

Firstly, we observe that the upper (Ok), the extra the curve will get unfold out. In a wavelet evaluation, which means that extra deadlines will contribute to the remodel’s output, leading to excessive precision as to frequency content material, however lack of decision in time. (We’ll return to this – central – trade-off quickly.)

As to (omega_a), its affect is twofold. On the one hand, within the Gaussian time period, it counteracts – precisely, even – the dimensions parameter, (Ok). On the opposite, it determines the frequency, or equivalently, the interval, of the wave. To see this, check out the suitable column. Akin to the completely different frequencies, we have now, within the interval between 4 and 6, 4, six, or eight peaks, respectively.

This double position of (omega_a) is the explanation why, all-in-all, it does make a distinction whether or not we shrink (Ok), holding (omega_a) fixed, or improve (omega_a), holding (Ok) mounted.

This state of issues sounds difficult, however is much less problematic than it might sound. In apply, understanding the position of (Ok) is essential, since we have to choose smart (Ok) values to attempt. As to the (omega_a), alternatively, there might be a large number of them, equivalent to the vary of frequencies we analyze.

So we will perceive the affect of (Ok) in additional element, we have to take a primary take a look at the Wavelet Remodel.

Wavelet Remodel: An easy implementation

Whereas total, the subject of wavelets is extra multifaceted, and thus, could seem extra enigmatic than Fourier evaluation, the remodel itself is simpler to know. It’s a sequence of native convolutions between wavelet and sign. Right here is the formulation for particular scale parameter (Ok), evaluation frequency (omega_a), and wavelet location (t_k):

[
W_{K, omega_a, t_k} = sum_n x_n Psi_{omega_{a},K,t_{k}}^*(t_n)
]

That is only a dot product, computed between sign and complex-conjugated wavelet. (Right here advanced conjugation flips the wavelet in time, making this convolution, not correlation – a indisputable fact that issues quite a bit, as you’ll see quickly.)

Correspondingly, easy implementation ends in a sequence of dot merchandise, every equivalent to a distinct alignment of wavelet and sign. Under, in wavelet_transform(), arguments omega and Ok are scalars, whereas x, the sign, is a vector. The result’s the wavelet-transformed sign, for some particular Ok and omega of curiosity.

wavelet_transform <- operate(x, omega, Ok) {
  n_samples <- dim(x)[1]
  W <- torch_complex(
    torch_zeros(n_samples), torch_zeros(n_samples)
  )
  for (i in 1:n_samples) {
    # transfer middle of wavelet
    t_k <- x[i, 1]
    m <- morlet(omega, Ok, t_k, x[, 1])
    # compute native dot product
    # word wavelet is conjugated
    dot <- torch_matmul(
      m$conj()$unsqueeze(1),
      x[, 2]$to(dtype = torch_cfloat())
    )
    W[i] <- dot
  }
  W
}

To check this, we generate a easy sine wave that has a frequency of 100 Hertz in its first half, and double that within the second.

gencos <- operate(amp, freq, part, fs, period) {
  x <- torch_arange(0, period, 1 / fs)[1:-2]$unsqueeze(2)
  y <- amp * torch_cos(2 * pi * freq * x + part)
  torch_cat(record(x, y), dim = 2)
}

# sampling frequency
fs <- 8000

f1 <- 100
f2 <- 200
part <- 0
period <- 0.25

s1 <- gencos(1, f1, part, fs, period)
s2 <- gencos(1, f2, part, fs, period)

s3 <- torch_cat(record(s1, s2), dim = 1)
s3[(dim(s1)[1] + 1):(dim(s1)[1] * 2), 1] <-
  s3[(dim(s1)[1] + 1):(dim(s1)[1] * 2), 1] + period

df <- knowledge.body(
  x = as.numeric(s3[, 1]),
  y = as.numeric(s3[, 2])
)
ggplot(df, aes(x = x, y = y)) +
  geom_line() +
  xlab("time") +
  ylab("amplitude") +
  theme_minimal()

Now, we run the Wavelet Remodel on this sign, for an evaluation frequency of 100 Hertz, and with a Ok parameter of two, discovered by means of fast experimentation:

Ok <- 2
omega <- 2 * pi * f1

res <- wavelet_transform(x = s3, omega, Ok)
df <- knowledge.body(
  x = as.numeric(s3[, 1]),
  y = as.numeric(res$abs())
)

ggplot(df, aes(x = x, y = y)) +
  geom_line() +
  xlab("time") +
  ylab("Wavelet Remodel") +
  theme_minimal()

The remodel accurately picks out the a part of the sign that matches the evaluation frequency. Should you really feel like, you may wish to double-check what occurs for an evaluation frequency of 200 Hertz.

Now, in actuality we’ll wish to run this evaluation not for a single frequency, however a spread of frequencies we’re all in favour of. And we’ll wish to attempt completely different scales Ok. Now, in the event you executed the code above, you is perhaps anxious that this might take a lot of time.

Nicely, it by necessity takes longer to compute than its Fourier analogue, the spectrogram. For one, that’s as a result of with spectrograms, the evaluation is “simply” two-dimensional, the axes being time and frequency. With wavelets there are, as well as, completely different scales to be explored. And secondly, spectrograms function on complete home windows (with configurable overlap); a wavelet, alternatively, slides over the sign in unit steps.

Nonetheless, the scenario will not be as grave because it sounds. The Wavelet Remodel being a convolution, we will implement it within the Fourier area as a substitute. We’ll do this very quickly, however first, as promised, let’s revisit the subject of various Ok.

Decision in time versus in frequency

We already noticed that the upper Ok, the extra spread-out the wavelet. We are able to use our first, maximally easy, instance, to research one instant consequence. What, for instance, occurs for Ok set to twenty?

Ok <- 20

res <- wavelet_transform(x = s3, omega, Ok)
df <- knowledge.body(
  x = as.numeric(s3[, 1]),
  y = as.numeric(res$abs())
)

ggplot(df, aes(x = x, y = y)) +
  geom_line() +
  xlab("time") +
  ylab("Wavelet Remodel") +
  theme_minimal()

The Wavelet Remodel nonetheless picks out the right area of the sign – however now, as a substitute of a rectangle-like end result, we get a considerably smoothed model that doesn’t sharply separate the 2 areas.

Notably, the primary 0.05 seconds, too, present appreciable smoothing. The bigger a wavelet, the extra element-wise merchandise might be misplaced on the finish and the start. It’s because transforms are computed aligning the wavelet in any respect sign positions, from the very first to the final. Concretely, after we compute the dot product at location t_k = 1, only a single pattern of the sign is taken into account.

Aside from probably introducing unreliability on the boundaries, how does wavelet scale have an effect on the evaluation? Nicely, since we’re correlating (convolving, technically; however on this case, the impact, in the long run, is identical) the wavelet with the sign, point-wise similarity is what issues. Concretely, assume the sign is a pure sine wave, the wavelet we’re utilizing is a windowed sinusoid just like the Morlet, and that we’ve discovered an optimum Ok that properly captures the sign’s frequency. Then every other Ok, be it bigger or smaller, will end in much less point-wise overlap.

Performing the Wavelet Remodel within the Fourier area

Quickly, we’ll run the Wavelet Remodel on an extended sign. Thus, it’s time to velocity up computation. We already mentioned that right here, we profit from time-domain convolution being equal to multiplication within the Fourier area. The general course of then is that this: First, compute the DFT of each sign and wavelet; second, multiply the outcomes; third, inverse-transform again to the time area.

The DFT of the sign is rapidly computed:

F <- torch_fft_fft(s3[ , 2])

With the Morlet wavelet, we don’t even need to run the FFT: Its Fourier-domain illustration might be said in closed type. We’ll simply make use of that formulation from the outset. Right here it’s:

morlet_fourier <- operate(Ok, omega_a, omega) {
  2 * (torch_exp(-torch_square(
    Ok * (omega - omega_a) / omega_a
  )) -
    torch_exp(-torch_square(Ok)) *
      torch_exp(-torch_square(Ok * omega / omega_a)))
}

Evaluating this assertion of the wavelet to the time-domain one, we see that – as anticipated – as a substitute of parameters t and t_k it now takes omega and omega_a. The latter, omega_a, is the evaluation frequency, the one we’re probing for, a scalar; the previous, omega, the vary of frequencies that seem within the DFT of the sign.

In instantiating the wavelet, there’s one factor we have to pay particular consideration to. In FFT-think, the frequencies are bins; their quantity is decided by the size of the sign (a size that, for its half, immediately will depend on sampling frequency). Our wavelet, alternatively, works with frequencies in Hertz (properly, from a consumer’s perspective; since this unit is significant to us). What this implies is that to morlet_fourier, as omega_a we have to go not the worth in Hertz, however the corresponding FFT bin. Conversion is finished relating the variety of bins, dim(x)[1], to the sampling frequency of the sign, fs:

# once more search for 100Hz components
omega <- 2 * pi * f1

# want the bin equivalent to some frequency in Hz
omega_bin <- f1/fs * dim(s3)[1]

We instantiate the wavelet, carry out the Fourier-domain multiplication, and inverse-transform the end result:

Ok <- 3

m <- morlet_fourier(Ok, omega_bin, 1:dim(s3)[1])
prod <- F * m
remodeled <- torch_fft_ifft(prod)

Placing collectively wavelet instantiation and the steps concerned within the evaluation, we have now the next. (Be aware how one can wavelet_transform_fourier, we now, conveniently, go within the frequency worth in Hertz.)

wavelet_transform_fourier <- operate(x, omega_a, Ok, fs) {
  N <- dim(x)[1]
  omega_bin <- omega_a / fs * N
  m <- morlet_fourier(Ok, omega_bin, 1:N)
  x_fft <- torch_fft_fft(x)
  prod <- x_fft * m
  w <- torch_fft_ifft(prod)
  w
}

We’ve already made important progress. We’re prepared for the ultimate step: automating evaluation over a spread of frequencies of curiosity. This may end in a three-dimensional illustration, the wavelet diagram.

Creating the wavelet diagram

Within the Fourier Remodel, the variety of coefficients we acquire will depend on sign size, and successfully reduces to half the sampling frequency. With its wavelet analogue, since anyway we’re doing a loop over frequencies, we’d as nicely resolve which frequencies to research.

Firstly, the vary of frequencies of curiosity might be decided operating the DFT. The subsequent query, then, is about granularity. Right here, I’ll be following the advice given in Vistnes’ e book, which relies on the relation between present frequency worth and wavelet scale, Ok.

Iteration over frequencies is then carried out as a loop:

wavelet_grid <- operate(x, Ok, f_start, f_end, fs) {
  # downsample evaluation frequency vary
  # as per Vistnes, eq. 14.17
  num_freqs <- 1 + log(f_end / f_start)/ log(1 + 1/(8 * Ok))
  freqs <- seq(f_start, f_end, size.out = flooring(num_freqs))
  
  remodeled <- torch_zeros(
    num_freqs, dim(x)[1],
    dtype = torch_cfloat()
    )
  for(i in 1:num_freqs) {
    w <- wavelet_transform_fourier(x, freqs[i], Ok, fs)
    remodeled[i, ] <- w
  }
  record(remodeled, freqs)
}

Calling wavelet_grid() will give us the evaluation frequencies used, along with the respective outputs from the Wavelet Remodel.

Subsequent, we create a utility operate that visualizes the end result. By default, plot_wavelet_diagram() shows the magnitude of the wavelet-transformed collection; it will probably, nevertheless, plot the squared magnitudes, too, in addition to their sq. root, a technique a lot beneficial by Vistnes whose effectiveness we’ll quickly have alternative to witness.

The operate deserves a couple of additional feedback.

Firstly, identical as we did with the evaluation frequencies, we down-sample the sign itself, avoiding to counsel a decision that isn’t truly current. The formulation, once more, is taken from Vistnes’ e book.

Then, we use interpolation to acquire a brand new time-frequency grid. This step could even be vital if we maintain the unique grid, since when distances between grid factors are very small, R’s picture() could refuse to simply accept axes as evenly spaced.

Lastly, word how frequencies are organized on a log scale. This results in way more helpful visualizations.

plot_wavelet_diagram <- operate(x,
                                 freqs,
                                 grid,
                                 Ok,
                                 fs,
                                 f_end,
                                 kind = "magnitude") {
  grid <- change(kind,
    magnitude = grid$abs(),
    magnitude_squared = torch_square(grid$abs()),
    magnitude_sqrt = torch_sqrt(grid$abs())
  )

  # downsample time collection
  # as per Vistnes, eq. 14.9
  new_x_take_every <- max(Ok / 24 * fs / f_end, 1)
  new_x_length <- flooring(dim(grid)[2] / new_x_take_every)
  new_x <- torch_arange(
    x[1],
    x[dim(x)[1]],
    step = x[dim(x)[1]] / new_x_length
  )
  
  # interpolate grid
  new_grid <- nnf_interpolate(
    grid$view(c(1, 1, dim(grid)[1], dim(grid)[2])),
    c(dim(grid)[1], new_x_length)
  )$squeeze()
  out <- as.matrix(new_grid)

  # plot log frequencies
  freqs <- log10(freqs)
  
  picture(
    x = as.numeric(new_x),
    y = freqs,
    z = t(out),
    ylab = "log frequency [Hz]",
    xlab = "time [s]",
    col = hcl.colours(12, palette = "Gentle grays")
  )
  essential <- paste0("Wavelet Remodel, Ok = ", Ok)
  sub <- change(kind,
    magnitude = "Magnitude",
    magnitude_squared = "Magnitude squared",
    magnitude_sqrt = "Magnitude (sq. root)"
  )

  mtext(aspect = 3, line = 2, at = 0, adj = 0, cex = 1.3, essential)
  mtext(aspect = 3, line = 1, at = 0, adj = 0, cex = 1, sub)
}

Let’s use this on a real-world instance.

An actual-world instance: Chaffinch’s tune

For the case research, I’ve chosen what, to me, was probably the most spectacular wavelet evaluation proven in Vistnes’ e book. It’s a pattern of a chaffinch’s singing, and it’s obtainable on Vistnes’ web site.

url <- "http://www.physics.uio.no/pow/wavbirds/chaffinch.wav"

obtain.file(
 file.path(url),
 destfile = "/tmp/chaffinch.wav"
)

We use torchaudio to load the file, and convert from stereo to mono utilizing tuneR’s appropriately named mono(). (For the form of evaluation we’re doing, there isn’t a level in holding two channels round.)

library(torchaudio)
library(tuneR)

wav <- tuneR_loader("/tmp/chaffinch.wav")
wav <- mono(wav, "each")
wav

Wave Object
    Variety of Samples:      1864548
    Period (seconds):     42.28
    Samplingrate (Hertz):   44100
    Channels (Mono/Stereo): Mono
    PCM (integer format):   TRUE
    Bit (8/16/24/32/64):    16 

For evaluation, we don’t want the entire sequence. Helpfully, Vistnes additionally revealed a advice as to which vary of samples to research.

waveform_and_sample_rate <- transform_to_tensor(wav)
x <- waveform_and_sample_rate[[1]]$squeeze()
fs <- waveform_and_sample_rate[[2]]

# http://www.physics.uio.no/pow/wavbirds/chaffinchInfo.txt
begin <- 34000
N <- 1024 * 128
finish <- begin + N - 1
x <- x[start:end]

dim(x)

[1] 131072

How does this look within the time area? (Don’t miss out on the event to truly hear to it, in your laptop computer.)

df <- knowledge.body(x = 1:dim(x)[1], y = as.numeric(x))
ggplot(df, aes(x = x, y = y)) +
  geom_line() +
  xlab("pattern") +
  ylab("amplitude") +
  theme_minimal()

Now, we have to decide an inexpensive vary of study frequencies. To that finish, we run the FFT:

On the x-axis, we plot frequencies, not pattern numbers, and for higher visibility, we zoom in a bit.

bins <- 1:dim(F)[1]
freqs <- bins / N * fs

# the bin, not the frequency
cutoff <- N/4

df <- knowledge.body(
  x = freqs[1:cutoff],
  y = as.numeric(F$abs())[1:cutoff]
)
ggplot(df, aes(x = x, y = y)) +
  geom_col() +
  xlab("frequency (Hz)") +
  ylab("magnitude") +
  theme_minimal()

Primarily based on this distribution, we will safely limit the vary of study frequencies to between, roughly, 1800 and 8500 Hertz. (That is additionally the vary beneficial by Vistnes.)

First, although, let’s anchor expectations by making a spectrogram for this sign. Appropriate values for FFT dimension and window dimension had been discovered experimentally. And although, in spectrograms, you don’t see this executed usually, I discovered that displaying sq. roots of coefficient magnitudes yielded probably the most informative output.

fft_size <- 1024
window_size <- 1024
energy <- 0.5

spectrogram <- transform_spectrogram(
  n_fft = fft_size,
  win_length = window_size,
  normalized = TRUE,
  energy = energy
)

spec <- spectrogram(x)
dim(spec)

[1] 513 257

Like we do with wavelet diagrams, we plot frequencies on a log scale.

bins <- 1:dim(spec)[1]
freqs <- bins * fs / fft_size
log_freqs <- log10(freqs)

frames <- 1:(dim(spec)[2])
seconds <- (frames / dim(spec)[2])  * (dim(x)[1] / fs)

picture(x = seconds,
      y = log_freqs,
      z = t(as.matrix(spec)),
      ylab = 'log frequency [Hz]',
      xlab = 'time [s]',
      col = hcl.colours(12, palette = "Gentle grays")
)
essential <- paste0("Spectrogram, window dimension = ", window_size)
sub <- "Magnitude (sq. root)"
mtext(aspect = 3, line = 2, at = 0, adj = 0, cex = 1.3, essential)
mtext(aspect = 3, line = 1, at = 0, adj = 0, cex = 1, sub)

The spectrogram already exhibits a particular sample. Let’s see what might be executed with wavelet evaluation. Having experimented with a couple of completely different Ok, I agree with Vistnes that Ok = 48 makes for a superb selection:

f_start <- 1800
f_end <- 8500

Ok <- 48
c(grid, freqs) %<-% wavelet_grid(x, Ok, f_start, f_end, fs)
plot_wavelet_diagram(
  torch_tensor(1:dim(grid)[2]),
  freqs, grid, Ok, fs, f_end,
  kind = "magnitude_sqrt"
)

The acquire in decision, on each the time and the frequency axis, is completely spectacular.

Thanks for studying!

Photograph by Vlad Panov on Unsplash