Posit AI Weblog: Wavelet Rework


Notice: Like a number of prior ones, this submit 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 laborious 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 Rework

What are wavelets? Just like the Fourier foundation, they’re capabilities; however they don’t prolong infinitely. As an alternative, 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 totally different scales, they seem squished or stretched. Squished, they’ll do higher at detecting excessive frequencies; the converse applies once they’re stretched out in time.

The essential operation concerned within the Wavelet Rework is convolution – have the (flipped) wavelet slide over the information, computing a sequence of dot merchandise. This manner, the wavelet is principally on the lookout for similarity.

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

The subject of wavelets may be very totally 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, also called 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 information, the sorts of knowledge we work with in observe. Thus, (t_k) and (t_n) designate cut-off dates, or equivalently, particular person time-series samples.

This equation seems daunting at first, however we are able to “tame” it a bit by analyzing its construction, and pointing to the principle actors. For concreteness, although, we first have 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, Okay, t_k, and t). It is because the torch code is vectorized: On the one hand, omega, Okay, and t_k, which, within the components, correspond to (omega_{a}), (Okay), and (t_k) , are scalars. (Within the equation, they’re assumed to be fastened.) t, alternatively, is a vector; it can maintain the measurement instances of the collection to be analyzed.

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

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

create_wavelet_plot <- operate(omega, Okay, t_k, sample_time) {
  morlet <- morlet(omega, Okay, t_k, sample_time)
  df <- information.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 , Okay = ", Okay)
    ) +
    theme_minimal()
}

create_wavelet_plot(omega, Okay, t_k, sample_time)
A Morlet wavelet.

What we see here’s a advanced sine curve – notice the true and imaginary components, separated by a part shift of (pi/2) – that decays on each side of the middle. Wanting again on the equation, we are able to establish the components liable 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 concerning the second time period, (e^{-Okay^2}): For given (Okay), it’s only a fixed.)

The third time period really is a Gaussian, with location parameter (t_k) and scale (Okay). We’ll discuss (Okay) 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 (Okay) and (omega_a)

Now, we already mentioned that (Okay) is the size of the Gaussian; it thus determines how far the curve spreads out in time. However there’s additionally (omega_a). Wanting again on the Gaussian time period, it, too, will influence 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 (Okay).

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)
Morlet wavelet: Effects of varying scale and analysis frequency.

Within the left column, we hold (omega_a) fixed, and range (Okay). On the appropriate, (omega_a) modifications, and (Okay) stays the identical.

Firstly, we observe that the upper (Okay), the extra the curve will get unfold out. In a wavelet evaluation, which means that extra cut-off dates 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 influence is twofold. On the one hand, within the Gaussian time period, it counteracts – precisely, even – the size parameter, (Okay). On the opposite, it determines the frequency, or equivalently, the interval, of the wave. To see this, check out the appropriate column. Similar to the totally different frequencies, we’ve got, within the interval between 4 and 6, 4, six, or eight peaks, respectively.

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

This state of issues sounds sophisticated, however is much less problematic than it may appear. In observe, understanding the function of (Okay) is necessary, since we have to choose smart (Okay) values to attempt. As to the (omega_a), alternatively, there might be a mess of them, comparable to the vary of frequencies we analyze.

So we are able to perceive the influence of (Okay) in additional element, we have to take a primary have a look at the Wavelet Rework.

Wavelet Rework: A simple 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 understand. It’s a sequence of native convolutions between wavelet and sign. Right here is the components for particular scale parameter (Okay), 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 rather a lot, as you’ll see quickly.)

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

wavelet_transform <- operate(x, omega, Okay) {
  n_samples <- dim(x)[1]
  W <- torch_complex(
    torch_zeros(n_samples), torch_zeros(n_samples)
  )
  for (i in 1:n_samples) {
    # transfer heart of wavelet
    t_k <- x[i, 1]
    m <- morlet(omega, Okay, t_k, x[, 1])
    # compute native dot product
    # notice 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, length) {
  x <- torch_arange(0, length, 1 / fs)[1:-2]$unsqueeze(2)
  y <- amp * torch_cos(2 * pi * freq * x + part)
  torch_cat(listing(x, y), dim = 2)
}

# sampling frequency
fs <- 8000

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

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

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

df <- information.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()
An example signal, consisting of a low-frequency and a high-frequency half.

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

Okay <- 2
omega <- 2 * pi * f1

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

ggplot(df, aes(x = x, y = y)) +
  geom_line() +
  xlab("time") +
  ylab("Wavelet Rework") +
  theme_minimal()
Wavelet Transform of the above two-part signal. Analysis frequency is 100 Hertz.

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

Now, in actuality we are going to need to run this evaluation not for a single frequency, however a spread of frequencies we’re excited about. And we are going to need to attempt totally different scales Okay. Now, for those who executed the code above, you may be apprehensive 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, totally different scales to be explored. And secondly, spectrograms function on entire home windows (with configurable overlap); a wavelet, alternatively, slides over the sign in unit steps.

Nonetheless, the state of affairs just isn’t as grave because it sounds. The Wavelet Rework being a convolution, we are able to 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 Okay.

Decision in time versus in frequency

We already noticed that the upper Okay, the extra spread-out the wavelet. We will use our first, maximally easy, instance, to analyze one quick consequence. What, for instance, occurs for Okay set to twenty?

Okay <- 20

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

ggplot(df, aes(x = x, y = y)) +
  geom_line() +
  xlab("time") +
  ylab("Wavelet Rework") +
  theme_minimal()
Wavelet Transform of the above two-part signal, with K set to twenty instead of two.

The Wavelet Rework nonetheless picks out the right area of the sign – however now, as a substitute of a rectangle-like outcome, 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 is 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 Okay that properly captures the sign’s frequency. Then every other Okay, be it bigger or smaller, will lead to much less point-wise overlap.

Performing the Wavelet Rework within the Fourier area

Quickly, we are going to run the Wavelet Rework on an extended sign. Thus, it’s time to pace 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 must run the FFT: Its Fourier-domain illustration will be acknowledged in closed kind. We’ll simply make use of that formulation from the outset. Right here it’s:

morlet_fourier <- operate(Okay, omega_a, omega) {
  2 * (torch_exp(-torch_square(
    Okay * (omega - omega_a) / omega_a
  )) -
    torch_exp(-torch_square(Okay)) *
      torch_exp(-torch_square(Okay * 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, instantly is dependent upon sampling frequency). Our wavelet, alternatively, works with frequencies in Hertz (properly, from a person’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 comparable 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 outcome:

Okay <- 3

m <- morlet_fourier(Okay, 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’ve got the next. (Notice tips on how to wavelet_transform_fourier, we now, conveniently, go within the frequency worth in Hertz.)

wavelet_transform_fourier <- operate(x, omega_a, Okay, fs) {
  N <- dim(x)[1]
  omega_bin <- omega_a / fs * N
  m <- morlet_fourier(Okay, 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 can lead to a three-dimensional illustration, the wavelet diagram.

Creating the wavelet diagram

Within the Fourier Rework, the variety of coefficients we get hold of is dependent upon 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 investigate.

Firstly, the vary of frequencies of curiosity will be decided working the DFT. The following 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, Okay.

Iteration over frequencies is then applied as a loop:

wavelet_grid <- operate(x, Okay, 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 * Okay))
  freqs <- seq(f_start, f_end, size.out = ground(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], Okay, fs)
    remodeled[i, ] <- w
  }
  listing(remodeled, freqs)
}

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

Subsequent, we create a utility operate that visualizes the outcome. By default, plot_wavelet_diagram() shows the magnitude of the wavelet-transformed collection; it could actually, nevertheless, plot the squared magnitudes, too, in addition to their sq. root, a way a lot advisable by Vistnes whose effectiveness we are going to 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’s not really current. The components, once more, is taken from Vistnes’ e-book.

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

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

plot_wavelet_diagram <- operate(x,
                                 freqs,
                                 grid,
                                 Okay,
                                 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(Okay / 24 * fs / f_end, 1)
  new_x_length <- ground(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 = "Mild grays")
  )
  foremost <- paste0("Wavelet Rework, Okay = ", Okay)
  sub <- change(kind,
    magnitude = "Magnitude",
    magnitude_squared = "Magnitude squared",
    magnitude_sqrt = "Magnitude (sq. root)"
  )

  mtext(facet = 3, line = 2, at = 0, adj = 0, cex = 1.3, foremost)
  mtext(facet = 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 examine, I’ve chosen what, to me, was essentially 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 sort of evaluation we’re doing, there isn’t any level in protecting two channels round.)

library(torchaudio)
library(tuneR)

wav <- tuneR_loader("/tmp/chaffinch.wav")
wav <- mono(wav, "each")
wav
Wave Object
    Variety of Samples:      1864548
    Length (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 suggestion as to which vary of samples to investigate.

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 really hear to it, in your laptop computer.)

df <- information.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()
Chaffinch’s song.

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 <- information.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()
Chaffinch’s song, Fourier spectrum (excerpt).

Based mostly on this distribution, we are able to safely prohibit the vary of study frequencies to between, roughly, 1800 and 8500 Hertz. (That is additionally the vary advisable by Vistnes.)

First, although, let’s anchor expectations by making a spectrogram for this sign. Appropriate values for FFT measurement and window measurement had been discovered experimentally. And although, in spectrograms, you don’t see this completed typically, I discovered that displaying sq. roots of coefficient magnitudes yielded essentially 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 = "Mild grays")
)
foremost <- paste0("Spectrogram, window measurement = ", window_size)
sub <- "Magnitude (sq. root)"
mtext(facet = 3, line = 2, at = 0, adj = 0, cex = 1.3, foremost)
mtext(facet = 3, line = 1, at = 0, adj = 0, cex = 1, sub)
Chaffinch’s song, spectrogram.

The spectrogram already exhibits a particular sample. Let’s see what will be completed with wavelet evaluation. Having experimented with a couple of totally different Okay, I agree with Vistnes that Okay = 48 makes for a wonderful alternative:

f_start <- 1800
f_end <- 8500

Okay <- 48
c(grid, freqs) %<-% wavelet_grid(x, Okay, f_start, f_end, fs)
plot_wavelet_diagram(
  torch_tensor(1:dim(grid)[2]),
  freqs, grid, Okay, fs, f_end,
  kind = "magnitude_sqrt"
)
Chaffinch’s song, wavelet diagram.

The achieve in decision, on each the time and the frequency axis, is totally spectacular.

Thanks for studying!

Picture by Vlad Panov on Unsplash

Vistnes, Arnt Inge. 2018. Physics of Oscillations and Waves. With Use of Matlab and Python. Springer.

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