Posit AI Weblog: Wavelet Rework

Observe: Like a number of prior ones, this put up 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 Rework

What are wavelets? Just like the Fourier foundation, they’re features; 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, additionally 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 after they’re stretched out in time.

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

As to the wavelet features themselves, there are various of them. In a sensible utility, we’d need to experiment and decide 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 may 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 mirror 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 closing dates, 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 have a look at an instance wavelet.

We begin by implementing the above equation:

Evaluating code and mathematical formulation, we discover a distinction. The perform 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 components, correspond to (omega_{a}), (Ok), and (t_k) , are scalars. (Within the equation, they’re assumed to be fastened.) t, then again, is a vector; it can maintain the measurement occasions of the collection to be analyzed.

We decide instance values for omega, Ok, and t_k, in addition to a variety of occasions 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 <- perform(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 complicated sine curve – word the true and imaginary elements, separated by a section shift of (pi/2) – that decays on either side of the middle. Wanting again on the equation, we will determine the components accountable 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 really 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 method 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 stated 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). 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 (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 range (Ok). On the correct, (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 extra closing 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 dimensions parameter, (Ok). On the opposite, it determines the frequency, or equivalently, the interval, of the wave. To see this, check out the correct column. Comparable to the completely different frequencies, we now have, 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 (Ok), preserving (omega_a) fixed, or enhance (omega_a), holding (Ok) fastened.

This state of issues sounds difficult, however is much less problematic than it may appear. In apply, understanding the function of (Ok) is essential, since we have to decide wise (Ok) values to attempt. As to the (omega_a), then again, there shall be a large number of them, similar to the vary of frequencies we analyze.

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

Wavelet Rework: A simple implementation

Whereas general, the subject of wavelets is extra multifaceted, and thus, could appear extra enigmatic than Fourier evaluation, the remodel itself is less complicated to know. It’s a sequence of native convolutions between wavelet and sign. Right here is the components 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 complicated conjugation flips the wavelet in time, making this convolution, not correlation – a incontrovertible fact that issues rather a lot, as you’ll see quickly.)

Correspondingly, easy implementation leads to a sequence of dot merchandise, every similar to a distinct alignment of wavelet and sign. Beneath, 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 <- perform(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 <- perform(amp, freq, section, fs, length) {
  x <- torch_arange(0, length, 1 / fs)[1:-2]$unsqueeze(2)
  y <- amp * torch_cos(2 * pi * freq * x + section)
  torch_cat(listing(x, y), dim = 2)
}

# sampling frequency
fs <- 8000

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

s1 <- gencos(1, f1, section, fs, length)
s2 <- gencos(1, f2, section, 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 <- 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 Rework 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 Rework") +
  theme_minimal()

The remodel accurately picks out the a part of the sign that matches the evaluation frequency. In the event 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 variety of frequencies we’re concerned about. And we are going to need to attempt completely different scales Ok. Now, for those who executed the code above, you could 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, completely different scales to be explored. And secondly, spectrograms function on entire home windows (with configurable overlap); a wavelet, then again, slides over the sign in unit steps.

Nonetheless, the state of affairs shouldn’t be as grave because it sounds. The Wavelet Rework being a convolution, we will implement it within the Fourier area as a substitute. We’ll try 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 analyze one speedy 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 Rework") +
  theme_minimal()

The Wavelet Rework nonetheless picks out the proper 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 shall 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, once we compute the dot product at location t_k = 1, only a single pattern of the sign is taken into account.

Aside from presumably 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, ultimately, is similar) 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 some other Ok, be it bigger or smaller, will end in 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 stated 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 will be acknowledged in closed type. We’ll simply make use of that formulation from the outset. Right here it’s:

morlet_fourier <- perform(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, instantly will depend on sampling frequency). Our wavelet, then again, 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 cross 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 elements
omega <- 2 * pi * f1

# want the bin similar 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
reworked <- torch_fft_ifft(prod)

Placing collectively wavelet instantiation and the steps concerned within the evaluation, we now have the next. (Observe find out how to wavelet_transform_fourier, we now, conveniently, cross within the frequency worth in Hertz.)

wavelet_transform_fourier <- perform(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 variety of frequencies of curiosity. This may end in a three-dimensional illustration, the wavelet diagram.

Creating the wavelet diagram

Within the Fourier Rework, the variety of coefficients we get hold of 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 would as nicely determine which frequencies to research.

Firstly, the vary of frequencies of curiosity will 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 is predicated on the relation between present frequency worth and wavelet scale, Ok.

Iteration over frequencies is then applied as a loop:

wavelet_grid <- perform(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 = ground(num_freqs))
  
  reworked <- 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)
    reworked[i, ] <- w
  }
  listing(reworked, 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 perform that visualizes the end result. By default, plot_wavelet_diagram() shows the magnitude of the wavelet-transformed collection; it might probably, nevertheless, plot the squared magnitudes, too, in addition to their sq. root, a technique a lot beneficial by Vistnes whose effectiveness we are going to quickly have alternative to witness.

The perform deserves a number of additional feedback.

Firstly, similar as we did with the evaluation frequencies, we down-sample the sign itself, avoiding to recommend 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 could even be crucial if we maintain the unique grid, since when distances between grid factors are very small, R’s picture() could refuse to just accept axes as evenly spaced.

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

plot_wavelet_diagram <- perform(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 <- 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 = "Gentle grays")
  )
  primary <- paste0("Wavelet Rework, Ok = ", Ok)
  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, primary)
  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 music

For the case research, 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 accessible 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 type of evaluation we’re doing, there isn’t a level in preserving 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 whole sequence. Helpfully, Vistnes additionally revealed a suggestion 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 really 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 affordable vary of research 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 research 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 measurement and window measurement have been discovered experimentally. And although, in spectrograms, you don’t see this accomplished 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 = "Gentle grays")
)
primary <- paste0("Spectrogram, window measurement = ", window_size)
sub <- "Magnitude (sq. root)"
mtext(facet = 3, line = 2, at = 0, adj = 0, cex = 1.3, primary)
mtext(facet = 3, line = 1, at = 0, adj = 0, cex = 1, sub)

The spectrogram already exhibits a particular sample. Let’s see what will be accomplished with wavelet evaluation. Having experimented with a number 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!

Picture by Vlad Panov on Unsplash