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The Laplace transform allows us to examine the relations between the space-time and space-kime representations of longitudinal data. The Fourier transformation is a linear operator that maps complex-valued functions of real variables (e.g., space, time domains) to complex valued functions of other real variables (e.g., frequency domain). The Laplace transform is similar. However, it sends complex-valued functions of positive real variables (e.g., time) to complex-valued functions defined on complex variables (e.g., kime).
In this vignette, we introduce the use of inv_kimesurface_transform() and kimesurface_transform() functions in our package. We demonstrate how to do Laplace transform and kimesurface transform with our functions.
Here we first apply the discrete Laplace Transform (LT) function in our package on the sine function with the domain of \([0 : 2\pi]\) to see whether it has the same 2D surface function as the analytic form of LT of sine. The analytic form is \(1/(1+z^2)\).
# For this part of code, we comment it out and import the output plot already generated before to reduce time
# But this part of code can be run successfully. If you are interested, you can try it on your computer!
# discrete Laplace Transform of sine
range_limit = 2
x2 = seq(from = 0, to = range_limit, length.out = 50)[2:50]
# drop the first row to avoid real part value of 0
y2 = seq(from = 0, to = range_limit, length.out = 50)[2:50]
# drop the first column to avoid imaginary part value of 0
# Recompute the LT(sin) discretized on lower-res grid
z2_grid = array(dim=c(length(x2), length(y2)))# x2 %o% y2
f_sin <- function(t) { sin(t) }
# kime surface transform
# use parallel computing to speed up code
ncors = 2 # please choose the ncors according to the number of cores your PC has
# it is better that you increase the number of cores used for parallel computing if your computer allows
cl <- makeCluster(ncors)
registerDoParallel(cl)
F = list()
for (i in 1:length(x2) ){
F[[i]] =
foreach(j = 1:length(y2),
.export='cubintegrate',
.packages='cubature') %dopar% {
TCIU::LT(FUNCT=f_sin, complex(real=x2[i], imaginary = y2[j]))
}
}
stopCluster(cl)
F_vec = lapply(F, unlist)
z2_grid = unlist(do.call(rbind, F_vec))
# explicit form of Laplace Transform of sine
laplace_sine = function(p) { 1/(p^2 + 1) } # Exact Laplace transform of sin(x), continuous function
XY = expand.grid(X=x2, Y=y2)
complex_xy = mapply(complex, real=XY$X,imaginary=XY$Y)
sine_z =laplace_sine(complex_xy)
dim(sine_z) = c(length(x2), length(y2))# dim(sine_z) # [1] 49 49
# make the two plots in the same plot to compare
lt_ilt_plotly = plot_ly(hoverinfo="none", showscale = FALSE)%>%
add_trace(z=Re(sine_z)-1, type="surface", surfacecolor=Im(sine_z)) %>%
add_trace(z = Re(z2_grid), type="surface", opacity=0.7, surfacecolor=Im(z2_grid) )%>%
layout(title =
"Laplace Transform, LT(sin()), Height=Re(LT(sin())), Color=Re(LT(sin())) \n Contrast Exact (Continuous) vs.
Approximate (Discrete) Laplace Transform", showlegend = FALSE)
lt_ilt_plotlyFrom the plot, we can easily tell that our discrete LT function generate the same 2D surface as the analytic LT function does.
Here we first apply the discrete Laplace Transform (LT) function in our package on the sine function with the domain of \([0, 2\pi]\), and then use the inverse Laplace Transform (ILT) function in our package to prove the Laplace Transformation has been converted back to sine.
# For this part of code, we comment it out and import the output plot already generated before to reduce time
# But this part of code can be run successfully. If you are interested, you can try it on your computer!
# discrete Laplace Transform of sine
f_sin = function(t) { sin(t) }
lt_sine = function(z) TCIU::LT(f_sin, z)
# inverse Laplace Transform on the lt_sine
# using parallel computing to speed up code
tvalsn <- seq(0, pi*2, length.out = 20)
cl <- makeCluster(ncors)
registerDoParallel(cl)
sinvalsn <- foreach(t=1:length(tvalsn),
.export='cubintegrate',
.packages='cubature') %dopar% {
TCIU::ILT(FUNCT=lt_sine, t=tvalsn[t])
}
stopCluster(cl)
sinvalsn = unlist(sinvalsn)
# make the plot of the result from ILT
# to see whether it still looks like sine
sinvalsn_df2 <- as.data.frame(cbind(Re=Re(sinvalsn),Im=Im(sinvalsn),
Sin=sin(tvalsn), time_points=tvalsn))
lt_ilt_sine = ggplot(sinvalsn_df2, aes(x=time_points))+
geom_line(aes(y=Re, color="Real"), linetype=1, lwd=2) +
geom_line(aes(y = Sin, color="Sin"), linetype=2, lwd=1) +
scale_color_manual(name="Index",
values = c("Real"="steelblue", "Sin"="darkred"))+
labs(title = "Original fMRI Time-series f(t)=sin(t) and \n Reconstructed f'(t)=ILT(F)=ILT(discrete LT(f))",
subtitle = bquote("F" ~ "=" ~ "discrete LT(sine)")) +
xlab("Time") + ylab("fMRI Image Intensities (f and f')") +
theme_grey(base_size = 16) +
theme(legend.title = element_text(size=14, color = "black", face="bold"),
panel.grid.minor.y = element_blank(),
panel.grid.major.y = element_blank(),
plot.title = element_text(hjust = 0.5),
plot.subtitle = element_text(hjust = 0.5))
lt_ilt_sine## $data
## Re Im Sin time_points
## 1 -7.404072e-05 -1.888474e-15 0.000000e+00 0.0000000
## 2 3.295033e-01 3.517522e-15 3.246995e-01 0.3306940
## 3 6.149702e-01 -2.954290e-15 6.142127e-01 0.6613879
## 4 8.370462e-01 -2.184423e-15 8.371665e-01 0.9920819
## 5 9.746300e-01 1.059702e-14 9.694003e-01 1.3227759
## 6 9.983078e-01 -2.224456e-14 9.965845e-01 1.6534698
## 7 9.184352e-01 3.337463e-14 9.157733e-01 1.9841638
## 8 7.195434e-01 -3.831961e-14 7.357239e-01 2.3148577
## 9 4.427572e-01 1.611943e-14 4.759474e-01 2.6455517
## 10 1.536455e-01 2.195190e-14 1.645946e-01 2.9762457
## 11 -1.836331e-01 -8.396361e-14 -1.645946e-01 3.3069396
## 12 -4.670591e-01 1.166413e-13 -4.759474e-01 3.6376336
## 13 -7.172075e-01 -1.330107e-13 -7.357239e-01 3.9683276
## 14 -8.618044e-01 1.381377e-13 -9.157733e-01 4.2990215
## 15 -9.920689e-01 -6.506112e-14 -9.965845e-01 4.6297155
## 16 -1.070833e+00 -1.005387e-13 -9.694003e-01 4.9604095
## 17 -8.353279e-01 3.045155e-13 -8.371665e-01 5.2911034
## 18 -6.811159e-01 -5.070069e-13 -6.142127e-01 5.6217974
## 19 -2.926268e-01 5.270047e-13 -3.246995e-01 5.9524913
## 20 -8.648900e-03 -4.600098e-13 -2.449213e-16 6.2831853
##
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## geom_line: na.rm = FALSE, orientation = NA
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## [1] "gg" "ggplot"In this part, we still apply the Kimesurface transform on our sine function, and then check whether the inverse Kimesurface transform can convert it back to our original form of sine. Kimesurface transform mainly does a Laplace Transform and then applies a height and surface transformation to have a better view of the plot.
Below, we apply the Kimesurface transform on the sine function, and make the plot of the 2D function using the magnitude of the complex value as y.
x = seq(0, 2, length.out=50)[2:50]; y = seq(0, 2, length.out=50)[2:50];
# do Kimesurface transform on sine function
z2_grid = kimesurface_transform(FUNCT = function(t) { sin(t) },
real_x = x, img_y = y)
# make the plot after Kimesurface transformation
surf_color <- atan2(Im(z2_grid), Re(z2_grid))
colorscale = cbind(seq(0, 1, by=1/(length(x) - 1)), rainbow(length(x)))
magnitude <- (sqrt( Re(z2_grid)^2+ Im(z2_grid)^2))
p <- plot_ly(hoverinfo="none", showscale = FALSE) %>%
add_trace(z = magnitude,
surfacecolor=surf_color, colorscale=colorscale, #Phase-based color
type = 'surface', opacity=1, visible=T) %>%
layout(title = "fMRI Kime-Surface, F=LT(fMRI) \n Height=Mag(F), Color=Phase(F)", showlegend = FALSE,
scene = list(aspectmode = "manual", aspectratio = list(x=1, y=1, z=1.0) ) ) # 1:1:1 aspect ratio
pAlso, you can try to apply the Kimesurface transformation on our fMRI data for a brain. Each voxel of the brain contains a set of time series data that can be analyzed using our transform. However, due to the number of cores limited for parallel computing in the vignette, we did not run this chuck. You can try it if you are interested.
# load the fMRI data
fMRIURL <- "http://socr.umich.edu/HTML5/BrainViewer/data/fMRI_FilteredData_4D.nii.gz"
fMRIFile <- file.path(tempdir(), "fMRI_FilteredData_4D.nii.gz")
download.file(fMRIURL, dest=fMRIFile, quiet=TRUE)
fMRIVolume <- readNIfTI(fMRIFile, reorient=FALSE)
# dimensions: 64 x 64 x 21 x 180 ; 4mm x 4mm x 6mm x 3 sec
# extract a set of time series data from a voxel of fMRI data
xA_fMRI_1D_x20_y20_z11 <- fMRIVolume[20, 20, 11, ]
# get the smooth function of this time series data
# Instead of using the extremely noisy fMRI data and avoiding integration problems,
# smooth "f" and use the **smooth version, f**
time_points <- seq(0+0.001, 2*pi, length.out = 180)
f <- smooth.spline(ceiling((180*time_points)/(2*pi)),
xA_fMRI_1D_x20_y20_z11, df = 10)$y
# Define the f(t)=smooth(fMRI)(t) signal as a function of real time 0<t<=2*pi
fmri_funct <- function(t) {
if (t < 0+0.001 || t > 2*pi) { return ( 0 ) } else {
return ( f[ceiling((180*t)/(2*pi))] )
}
}
# do the Kimesurface transform
ncors = 8
# please choose the ncors according to the number of cores your PC has
x = seq(0, 2, length.out=50)[2:50]; y = seq(0, 2, length.out=50)[2:50];
# do Kimesurface transform on sine function
z2_grid_fmri = kimesurface_transform(FUNCT = fmri_funct,
glb_para="f",
real_x = x, img_y = y,
parallel_computing = TRUE,
ncor=ncors)
# make the plot of function after Kimesurface transformation
surf_color <- atan2(Im(z2_grid_fmri), Re(z2_grid_fmri))
colorscale = cbind(seq(0, 1, by=1/(length(x) - 1)), rainbow(length(x)))
magnitude <- (sqrt( Re(z2_grid_fmri)^2+ Im(z2_grid_fmri)^2))
p_fmri <- plot_ly(hoverinfo="none", showscale = FALSE) %>%
add_trace(z = magnitude,
surfacecolor=surf_color, colorscale=colorscale, # Phase-based color
type = 'surface', opacity=1, visible=T) %>%
layout(title = "fMRI Kime-Surface, F=LT(fMRI) \n Height=Mag(F), Color=Phase(F)", showlegend = FALSE,
scene = list(aspectmode = "manual", aspectratio = list(x=1, y=1, z=1.0) ) ) # 1:1:1 aspect ratio
p_fmriAfter the Kimesurface transformation, we apply the inverse Kimesurface transformation, and see that we can get a curve that captures the most trend of the sine function.
time_points <- seq(0+0.001, 2*pi, length.out = 180)
inv_data = inv_kimesurface_transform(time_points, z2_grid)
inv_data = inv_kimesurface_transform(time_points, z2_grid,num_length = 23,
m=1, msg=TRUE)
time_Intensities_ILT_df2 <- as.data.frame(cbind(Re=scale(Re(inv_data$Smooth_Reconstruction)),
Im=scale(Re(inv_data$Raw_Reconstruction)),
fMRI=scale(Re(sin(time_points))),
time_points=time_points))
colnames(time_Intensities_ILT_df2) = c("Smooth Reconstruction",
"Raw Reconstruction",
"Original Sin", "time_points")
df = reshape2::melt(time_Intensities_ILT_df2, id.var = "time_points")
pppp<-ggplot(df, aes(x = time_points, y = value, colour = variable)) +
geom_line(linetype=1, lwd=3) +
ylab("Function Intensity") + xlab("Time") +
theme(legend.position="top")+
labs(title= bquote("Comparison between" ~ "f(t)=Smooth(Sin)(t)" ~ "and Smooth(ILT(LT(Sin)))(t); Range [" ~ 0 ~":"~ 2*pi~"]"))## $data
## time_points variable value
## 1 0.00100000 Smooth Reconstruction 2.407143e-01
## 2 0.03609601 Smooth Reconstruction 2.434831e-01
## 3 0.07119201 Smooth Reconstruction 2.462513e-01
## 4 0.10628802 Smooth Reconstruction 2.490181e-01
## 5 0.14138403 Smooth Reconstruction 2.517820e-01
## 6 0.17648004 Smooth Reconstruction 2.545411e-01
## 7 0.21157604 Smooth Reconstruction 2.572931e-01
## 8 0.24667205 Smooth Reconstruction 2.600349e-01
## 9 0.28176806 Smooth Reconstruction 2.627631e-01
## 10 0.31686407 Smooth Reconstruction 2.654735e-01
## 11 0.35196007 Smooth Reconstruction 2.681616e-01
## 12 0.38705608 Smooth Reconstruction 2.708221e-01
## 13 0.42215209 Smooth Reconstruction 2.734493e-01
## 14 0.45724809 Smooth Reconstruction 2.760366e-01
## 15 0.49234410 Smooth Reconstruction 2.785770e-01
## 16 0.52744011 Smooth Reconstruction 2.810627e-01
## 17 0.56253612 Smooth Reconstruction 2.834853e-01
## 18 0.59763212 Smooth Reconstruction 2.858357e-01
## 19 0.63272813 Smooth Reconstruction 2.881043e-01
## 20 0.66782414 Smooth Reconstruction 2.902803e-01
## 21 0.70292015 Smooth Reconstruction 2.923529e-01
## 22 0.73801615 Smooth Reconstruction 2.943098e-01
## 23 0.77311216 Smooth Reconstruction 2.961387e-01
## 24 0.80820817 Smooth Reconstruction 2.978258e-01
## 25 0.84330418 Smooth Reconstruction 2.993573e-01
## 26 0.87840018 Smooth Reconstruction 3.007181e-01
## 27 0.91349619 Smooth Reconstruction 3.018929e-01
## 28 0.94859220 Smooth Reconstruction 3.028656e-01
## 29 0.98368820 Smooth Reconstruction 3.036194e-01
## 30 1.01878421 Smooth Reconstruction 3.041368e-01
## 31 1.05388022 Smooth Reconstruction 3.043998e-01
## 32 1.08897623 Smooth Reconstruction 3.043896e-01
## 33 1.12407223 Smooth Reconstruction 3.040868e-01
## 34 1.15916824 Smooth Reconstruction 3.034711e-01
## 35 1.19426425 Smooth Reconstruction 3.025220e-01
## 36 1.22936026 Smooth Reconstruction 3.012179e-01
## 37 1.26445626 Smooth Reconstruction 2.995366e-01
## 38 1.29955227 Smooth Reconstruction 2.974555e-01
## 39 1.33464828 Smooth Reconstruction 2.949512e-01
## 40 1.36974428 Smooth Reconstruction 2.919995e-01
## 41 1.40484029 Smooth Reconstruction 2.885758e-01
## 42 1.43993630 Smooth Reconstruction 2.846549e-01
## 43 1.47503231 Smooth Reconstruction 2.802109e-01
## 44 1.51012831 Smooth Reconstruction 2.752174e-01
## 45 1.54522432 Smooth Reconstruction 2.696475e-01
## 46 1.58032033 Smooth Reconstruction 2.634736e-01
## 47 1.61541634 Smooth Reconstruction 2.566677e-01
## 48 1.65051234 Smooth Reconstruction 2.492023e-01
## 49 1.68560835 Smooth Reconstruction 2.410499e-01
## 50 1.72070436 Smooth Reconstruction 2.321858e-01
## 51 1.75580037 Smooth Reconstruction 2.225862e-01
## 52 1.79089637 Smooth Reconstruction 2.122296e-01
## 53 1.82599238 Smooth Reconstruction 2.010955e-01
## 54 1.86108839 Smooth Reconstruction 1.891662e-01
## 55 1.89618439 Smooth Reconstruction 1.764249e-01
## 56 1.93128040 Smooth Reconstruction 1.628576e-01
## 57 1.96637641 Smooth Reconstruction 1.484517e-01
## 58 2.00147242 Smooth Reconstruction 1.331971e-01
## 59 2.03656842 Smooth Reconstruction 1.170856e-01
## 60 2.07166443 Smooth Reconstruction 1.001116e-01
## 61 2.10676044 Smooth Reconstruction 8.227138e-02
## 62 2.14185645 Smooth Reconstruction 6.356425e-02
## 63 2.17695245 Smooth Reconstruction 4.399136e-02
## 64 2.21204846 Smooth Reconstruction 2.355736e-02
## 65 2.24714447 Smooth Reconstruction 2.268738e-03
## 66 2.28224047 Smooth Reconstruction -1.986416e-02
## 67 2.31733648 Smooth Reconstruction -4.282901e-02
## 68 2.35243249 Smooth Reconstruction -6.660937e-02
## 69 2.38752850 Smooth Reconstruction -9.118661e-02
## 70 2.42262450 Smooth Reconstruction -1.165374e-01
## 71 2.45772051 Smooth Reconstruction -1.426367e-01
## 72 2.49281652 Smooth Reconstruction -1.694571e-01
## 73 2.52791253 Smooth Reconstruction -1.969701e-01
## 74 2.56300853 Smooth Reconstruction -2.251454e-01
## 75 2.59810454 Smooth Reconstruction -2.539520e-01
## 76 2.63320055 Smooth Reconstruction -2.833560e-01
## 77 2.66829656 Smooth Reconstruction -3.133227e-01
## 78 2.70339256 Smooth Reconstruction -3.438143e-01
## 79 2.73848857 Smooth Reconstruction -3.747916e-01
## 80 2.77358458 Smooth Reconstruction -4.062124e-01
## 81 2.80868058 Smooth Reconstruction -4.380325e-01
## 82 2.84377659 Smooth Reconstruction -4.702041e-01
## 83 2.87887260 Smooth Reconstruction -5.026775e-01
## 84 2.91396861 Smooth Reconstruction -5.353987e-01
## 85 2.94906461 Smooth Reconstruction -5.683117e-01
## 86 2.98416062 Smooth Reconstruction -6.013558e-01
## 87 3.01925663 Smooth Reconstruction -6.344677e-01
## 88 3.05435264 Smooth Reconstruction -6.675791e-01
## 89 3.08944864 Smooth Reconstruction -7.006190e-01
## 90 3.12454465 Smooth Reconstruction -7.335109e-01
## 91 3.15964066 Smooth Reconstruction -7.661753e-01
## 92 3.19473666 Smooth Reconstruction -7.985260e-01
## 93 3.22983267 Smooth Reconstruction -8.304741e-01
## 94 3.26492868 Smooth Reconstruction -8.619272e-01
## 95 3.30002469 Smooth Reconstruction -8.927932e-01
## 96 3.33512069 Smooth Reconstruction -9.229853e-01
## 97 3.37021670 Smooth Reconstruction -9.524191e-01
## 98 3.40531271 Smooth Reconstruction -9.810143e-01
## 99 3.44040872 Smooth Reconstruction -1.008692e+00
## 100 3.47550472 Smooth Reconstruction -1.035379e+00
## 101 3.51060073 Smooth Reconstruction -1.061001e+00
## 102 3.54569674 Smooth Reconstruction -1.085490e+00
## 103 3.58079274 Smooth Reconstruction -1.108777e+00
## 104 3.61588875 Smooth Reconstruction -1.130798e+00
## 105 3.65098476 Smooth Reconstruction -1.151490e+00
## 106 3.68608077 Smooth Reconstruction -1.170793e+00
## 107 3.72117677 Smooth Reconstruction -1.188646e+00
## 108 3.75627278 Smooth Reconstruction -1.204993e+00
## 109 3.79136879 Smooth Reconstruction -1.219778e+00
## 110 3.82646480 Smooth Reconstruction -1.232948e+00
## 111 3.86156080 Smooth Reconstruction -1.244450e+00
## 112 3.89665681 Smooth Reconstruction -1.254234e+00
## 113 3.93175282 Smooth Reconstruction -1.262249e+00
## 114 3.96684883 Smooth Reconstruction -1.268448e+00
## 115 4.00194483 Smooth Reconstruction -1.272783e+00
## 116 4.03704084 Smooth Reconstruction -1.275210e+00
## 117 4.07213685 Smooth Reconstruction -1.275683e+00
## 118 4.10723285 Smooth Reconstruction -1.274159e+00
## 119 4.14232886 Smooth Reconstruction -1.270593e+00
## 120 4.17742487 Smooth Reconstruction -1.264943e+00
## 121 4.21252088 Smooth Reconstruction -1.257164e+00
## 122 4.24761688 Smooth Reconstruction -1.247214e+00
## 123 4.28271289 Smooth Reconstruction -1.235050e+00
## 124 4.31780890 Smooth Reconstruction -1.220630e+00
## 125 4.35290491 Smooth Reconstruction -1.203913e+00
## 126 4.38800091 Smooth Reconstruction -1.184856e+00
## 127 4.42309692 Smooth Reconstruction -1.163421e+00
## 128 4.45819293 Smooth Reconstruction -1.139567e+00
## 129 4.49328893 Smooth Reconstruction -1.113255e+00
## 130 4.52838494 Smooth Reconstruction -1.084448e+00
## 131 4.56348095 Smooth Reconstruction -1.053110e+00
## 132 4.59857696 Smooth Reconstruction -1.019204e+00
## 133 4.63367296 Smooth Reconstruction -9.826972e-01
## 134 4.66876897 Smooth Reconstruction -9.435580e-01
## 135 4.70386498 Smooth Reconstruction -9.017557e-01
## 136 4.73896099 Smooth Reconstruction -8.572627e-01
## 137 4.77405699 Smooth Reconstruction -8.100530e-01
## 138 4.80915300 Smooth Reconstruction -7.601040e-01
## 139 4.84424901 Smooth Reconstruction -7.073949e-01
## 140 4.87934502 Smooth Reconstruction -6.519095e-01
## 141 4.91444102 Smooth Reconstruction -5.936340e-01
## 142 4.94953703 Smooth Reconstruction -5.325606e-01
## 143 4.98463304 Smooth Reconstruction -4.686845e-01
## 144 5.01972904 Smooth Reconstruction -4.020077e-01
## 145 5.05482505 Smooth Reconstruction -3.325356e-01
## 146 5.08992106 Smooth Reconstruction -2.602816e-01
## 147 5.12501707 Smooth Reconstruction -1.852628e-01
## 148 5.16011307 Smooth Reconstruction -1.075052e-01
## 149 5.19520908 Smooth Reconstruction -2.703960e-02
## 150 5.23030509 Smooth Reconstruction 5.609364e-02
## 151 5.26540110 Smooth Reconstruction 1.418487e-01
## 152 5.30049710 Smooth Reconstruction 2.301689e-01
## 153 5.33559311 Smooth Reconstruction 3.209915e-01
## 154 5.37068912 Smooth Reconstruction 4.142415e-01
## 155 5.40578512 Smooth Reconstruction 5.098374e-01
## 156 5.44088113 Smooth Reconstruction 6.076838e-01
## 157 5.47597714 Smooth Reconstruction 7.076785e-01
## 158 5.51107315 Smooth Reconstruction 8.097034e-01
## 159 5.54616915 Smooth Reconstruction 9.136330e-01
## 160 5.58126516 Smooth Reconstruction 1.019326e+00
## 161 5.61636117 Smooth Reconstruction 1.126630e+00
## 162 5.65145718 Smooth Reconstruction 1.235376e+00
## 163 5.68655318 Smooth Reconstruction 1.345388e+00
## 164 5.72164919 Smooth Reconstruction 1.456505e+00
## 165 5.75674520 Smooth Reconstruction 1.568570e+00
## 166 5.79184120 Smooth Reconstruction 1.681448e+00
## 167 5.82693721 Smooth Reconstruction 1.795014e+00
## 168 5.86203322 Smooth Reconstruction 1.909158e+00
## 169 5.89712923 Smooth Reconstruction 2.023780e+00
## 170 5.93222523 Smooth Reconstruction 2.138794e+00
## 171 5.96732124 Smooth Reconstruction 2.254121e+00
## 172 6.00241725 Smooth Reconstruction 2.369695e+00
## 173 6.03751326 Smooth Reconstruction 2.485460e+00
## 174 6.07260926 Smooth Reconstruction 2.601367e+00
## 175 6.10770527 Smooth Reconstruction 2.717376e+00
## 176 6.14280128 Smooth Reconstruction 2.833456e+00
## 177 6.17789729 Smooth Reconstruction 2.949580e+00
## 178 6.21299329 Smooth Reconstruction 3.065731e+00
## 179 6.24808930 Smooth Reconstruction 3.181894e+00
## 180 6.28318531 Smooth Reconstruction 3.298062e+00
## 181 0.00100000 Raw Reconstruction 1.009695e-01
## 182 0.03609601 Raw Reconstruction 1.009695e-01
## 183 0.07119201 Raw Reconstruction 1.009695e-01
## 184 0.10628802 Raw Reconstruction 1.009695e-01
## 185 0.14138403 Raw Reconstruction 1.009695e-01
## 186 0.17648004 Raw Reconstruction 1.009695e-01
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##
## $layers
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##
## $labels
## $labels$title
## "Comparison between" ~ "f(t)=Smooth(Sin)(t)" ~ "and Smooth(ILT(LT(Sin)))(t); Range [" ~
## 0 ~ ":" ~ 2 * pi ~ "]"
##
## $labels$x
## [1] "Time"
##
## $labels$y
## [1] "Function Intensity"
##
## $labels$colour
## [1] "variable"
##
##
## attr(,"class")
## [1] "gg" "ggplot"