Appendix A Smoothing Techniques for Imaging Problems

This appendix intends to present issues in nonparametric regression and image denoising that are relevant for the analyses of neuroimaging experiments presented in this book. It aims at a deeper level of understanding for the methods presented in this book and does not aim for a comprehensive overview on available methods. For a broad overview on nonparametric regression and smoothing techniques, we refer, e.g., to the books of Fan and Gijbels (1996), Simonoff (1996), Wand and Jones (1995), Bowman and Azzalini (1997), Gu (2013), Mallat (2009), Nason (2008), and Katkovnik et al. (2006). A more detailed text on adaptive smoothing methods available in R is Polzehl et al. (2019).

A.1 Nonparametric Regression

Regression is commonly used to describe and analyze the relation between explana- tory input variables X ∈ X ⊂ Rd and responses Y ∈ Y ⊂ Rq . Usually, this relation is given in terms of the mean response as

E(Y |X = x) = f (x) (A.1) or as a model with varying coefficients

Y |(X = x) ∼ Pf (x) assuming that the conditional probability law P of Y depends on x only through a function f . Images naturally fit into this scenario, with X classically being a two-dimensional grid and the response variable Y denoting random intensity values at the grid loca- tions. Moreover, in neuroimaging, see Chaps. 4–6, images are generally considered to be defined on a three-dimensional grid. In a more general setting, the response

© Springer Nature Switzerland AG 2019 171 J. Polzehl and K. Tabelow, Magnetic Resonance Brain Imaging, Use R!, https://doi.org/10.1007/978-3-030-29184-6 172 Appendix A: Smoothing Techniques for Imaging Problems variable Y can also be a vector of observations, see Chaps. 4 and 5 for examples. In all these cases, the design variable X usually is fixed. We therefore denote the design points by xi . The probability law Pf (x) generally depends on the physical process that generates the image. Examples of Pf (x) considered in this book are Gaussian distributions with expectation f (x) and known variance σ 2 or χ-distributions with f (x) = μ(ζ(x), σ, L) depending on an unknown noncentrality parameter and known scale σ and degrees of freedom L. We will see later how the assumption of known σ can be relaxed. Regression analysis then infers on the parameter function f (x) and its spatial structure. Parametric regression assumes that the regression function f = f (x,θ)is known up to a parameter θ, i.e.,

f : X × Θ → Y , with some parameter space Θ ⊂ Rp and X ⊂ Rd . In the simplest case, f may be ( ) = θ + θ T ,θ= (θ ,θ ) ∈ R(d+1) ( ) = assumed linear, i.e., f x 0 1 x 0 1 or quadratic, f x θ + θ T + T Θ θ = (θ ,θ ,Θ ) ∈ R(d2+d+1) 0 1 x x 2x 0 1 2 . The inference problem for the model (A.1) then simplifies to the estimation of a global parameter θ rather than a whole function f . For a given data sample (xi , Yi )i=1,...,n, this can then be done by either least squares n ˆ 2 θ = argmin |Yi − f (xi ,θ)| (A.2) θ i=1 or maximum-likelihood

n θˆ = ( ), argmax log pθ(xi ) Yi (A.3) θ i=1 where pθ(x) denotes the density associated to the probability law Pθ(x) = Pf (x,θ). Both estimates coincide in case of an independent Gaussian response variables and homoskedastic errors. In many applications like image processing, the assumption of a parametric regression function f (x,θ) is by far too restrictive. A common way to increase the flexibility of parametric regression models is to assume that a parametric model g(ξ − x,θ(x)) for some function g provides a good local approximation for f in a neighborhood U(x) ={ξ :||ξ − x|| < h} of x ∈ X . The parameter h > 0isusu- ally referred to as bandwidth and obviously controls the size of the neighborhood U(x). The regression function f (x) is then characterized by the parameter function θ(x) and the function g. To estimate θ(x) at location x, we define a local model by assigning a weight wi (x) with 0 ≤ wi (x) ≤ 1 to each pair of observations (xi , Yi ). We call W(x) = (w1(x),...,wn(x)) a weighting scheme. Local estimates of θ(x) are then obtained by weighted least squares Appendix A: Smoothing Techniques for Imaging Problems 173

n θ(ˆ ) = (( )n , ( ); θ) = ( )| − ( − ,θ)|2 x argmin R Yi i=1 W x argmin wi x Yi g xi x θ θ i=1 or weighted maximum-likelihood

n θ(ˆ ) = (( )n , ( ); θ) = ( ) ( ) x argmax l Yi i=1 W x argmax wi x log pg(xi −x,θ) Yi θ θ i=1 providing local estimates of the regression function as

fˆ(x) = g(0, θ(ˆ x)).

The weighting scheme W(x) determines the influence of the observation pairs on the estimate. To determine a local parametric estimate, we need to specify the local parametric function g(., θ) and a method for generating the local weighting scheme W(x).

A.1.1 Kernel Smoothing

The Nadaraya–Watson kernel estimator (Nadaraya 1964;Watson1964) is a special local parametric regression method which is obtained from the general framework above by specifying g(., θ) ≡ θ, i.e., as a constant function, and defining weights

wi (x) = K (δ(xi , x)/h).

Here, K is a monotone nonincreasing function K : R+ →[0, 1] with K (0) = 1. K is called kernel.1 h > 0 denotes a bandwidth and δ : X × X → R+ defines a distance in X . Common choices for the kernel K include

2 γ K (t) = ((1 − t )+) γ ∈ (0, 1, 2, 3) corresponding to rescaled versions of the uniform (γ = 0), Epanechnikov (γ = 1), bi-weight (γ = 2), or tri-weight (γ = 3) kernel. Another popular choice is the Gaussian kernel   2 γ 1 2 1 t K (t) = √ et /2 = √ lim 1 − . 2π 2π γ →∞ 2γ

+ + 1This definition slightly differs from the standard assumptions, i.e., K : R → R , K monotone ∞ ( ) = nonincreasing and 0 K t dt 1 but by a scale factor only. The requirements on the kernel can be relaxed, changing the properties of the resulting estimates, see, e.g., Fan and Gijbels (1996). 174 Appendix A: Smoothing Techniques for Imaging Problems

Using weighted least squares, the kernel estimate at design point x is then explicitly given as   n K (δ(x , x)/h)Y n w (x)Y fˆ(x) = i i i = i i i . n (δ( , )/ ) n ( ) (A.4) i K xi x h i wi x

Equation (A.4) also provides the weighted likelihood estimate for one-parameter exponential families parametrized such that EYi = θ(xi ). The regression function can be estimated at any point x with wi (x)>0 for at least one i, i.e., not only at points xi with observations Yi . ˜ Kernel estimates can, in case of δ(xi , x) = δ(||xi − x||), be efficiently computed as convolutions of the image data with the kernel function using fast Fourier trans- form. This approach is for 1D, 2D, and 3D images implemented in function kernsm from package aws. We use the first T1 weighted image from the quantitative MRI experiment con- sidered in Chap. 6 to illustrate the properties of kernel smoothers. The 3D image is read using library(oro.nifti) t1Name <- file.path("data", "MPM", "t1w_mfc_3dflash_v1i_R4_0015", "anon_s2018-02-28_18-26-190921-00001-00224-1.nii") T1 <- as.array(readNIfTI(t1Name, reorient = FALSE))

We now calculate a kernel estimate of the intensity function. library(aws) T1sm <- kernsm(T1, h = 2., kern = "Epanechnikov")@yhat

For visualization of the result, we use the function rimage from the package adimpro, with function rimage.options used to modify the default behavior. library(adimpro) rimage.options(ylab = "z", zquantile = c(0.001, 0.999))

A comparison between the original T1 image and its smoothed version is provided in Fig. A.1. rimage(T1[,160,], main="Original", zlim= c(0,2600)) rimage(T1sm[,160,], main="Smoothed image", zlim= c(0,2600))

Even with a small bandwidth 3D kernel smoothing leads to significant noise reduction. Unfortunately, this comes at the cost of severe blurring of tissue borders and consequently loss of spatial resolution. Appendix A: Smoothing Techniques for Imaging Problems 175

A.2 Adaptive Weights Smoothing

In this section, we introduce a class of smoothing procedures that do not require global smoothness assumptions. Instead, they rely on a concept of local homogeneity. The design space is assumed to consist of regions where the regression function is smooth or even approximately constant. Discontinuities show up as edges of homogeneous regions. Focused on recovering the homogeneity structure the procedures are edge preserving. The package aws (Polzehl 2019) implements four main classes of edge- preserving smoothing procedures: the propagation-separation approach (Polzehl and Spokoiny 2000, 2006), pointwise adaptive estimates as presented in Katkovnik et al. (2006), the nonlocal means filter (Buades et al. 2005) and total variation (TV) or total generalized variation (TGV) regularization (Rudin et al. 1992; Bredies et al. 2010). Here, we concentrate on the first class. For a comprehensive coverage of the package capabilities, we refer to Polzehl et al. (2019).

A.2.1 Local Constant Likelihood Models

The nonparametric regression method described in Sect.A.1.1 is designed for the estimation of a smooth regression function f . The assumption that f is twice differ- entiable is questionable in many applications. The relationship between explanatory variables and the response may exhibit abrupt changes at some design points or con- tours. A typical example is the image where the intensity Y usually depends on the image content, e.g., tissue type, in a certain location. This is, e.g., the case in the T1 image used in Fig. A.1 or the images analyzed in Chaps. 4–6. In this section, we take a different look at the nonparametric regression prob- lem (A.1) introduced in Sect. A.1.1. Let us assume that there exists a partitioning of the design space

Original Smoothed image z z 50 100 150 200 50 100 150 200

50 100 150 200 250 50 100 150 200 250 x x

Fig. A.1 Original and smoothed T1 image 176 Appendix A: Smoothing Techniques for Imaging Problems

M X = Xm , Xl ∩ Xm if l = m, (A.5) m=1 such that f (xi ) ≈ f (x j ) if xi and x j belong to the same set Xm . We call the sets Xm homogeneity regions of the design. Change points or discontinuities are implicitly given as borders of the homogeneity regions. Modeling the regression function f now requires two tasks, to determine the regions of homogeneity where variation of the regression function is small and to estimate the parameter function θ(x) by local least squares (A.2) or local maximum-likelihood (A.3), using only observations from the region of homogeneity containing x. Let us first consider what happens if we vary the bandwidth h in our example. ix <- 151:200;iy<- 160;iz<- 121:165 for (h in c(1, 1.15, 1.7, 2.7, 4.2, 6.6)){ T1sm <- kernsm(T1, h, kern = "Epanechnikov")@yhat rimage(ix, iz, T1sm[ix, iy, iz], main = paste("bw=", h)) }

In Fig. A.2, we see the effect of the size of the bandwidth. Larger bandwidths lead to more variance reduction and increase the contrast between intensities in spatially extended regions. This is achieved at the cost of spatial resolution, i.e., loss

bw= 1 bw= 1.15 bw= 1.7 z z z 130 140 150 160 130 140 150 160 130 140 150 160

160 170 180 190 200 160 170 180 190 200 160 170 180 190 200 x x x

bw= 2.7 bw= 4.2 bw= 6.6 z z z 130 140 150 160 130 140 150 160 130 140 150 160

160 170 180 190 200 160 170 180 190 200 160 170 180 190 200 x x x

Fig. A.2 Effect of bandwidth for a detail of the T1 map Appendix A: Smoothing Techniques for Imaging Problems 177 in detail. Different types of inhomogeneities are visible at different scales. For small bandwidths, we can detect tissue borders characterized by large intensity differences with high spatial precision. At larger scales, smaller intensity differences can be identified. A second idea that we will use is the sigma filter introduced by Lee (1983) and generalized in the form of the bilateral filter (Tomasi and Manduchi 1998)orthe M-smoother (Chu et al. 1998)  n w j (x)Y j fˆ(x) = j n ( ) (A.6) j w j x with weights w j (x) given by    (Y − fˆ(x)) δ(x , x) w (x) = L j K j , j g h

L being a Gaussian kernel and g a bandwidth. This additionally penalizes deviations from the regression function and makes the smoother edge preserving. In Chu et al. ˆ (1998), the estimate f (xi ) is calculated from a d + 1-dimensional density estimate in Z as the mode of the conditional density of Y |xi that is closest to Yi . Using this idea allows to design an algorithm that sequentially learns the homo- geneity structure of the design space by inspecting scale space from small to large bandwidths. In each step, we characterize the region of homogeneity containing a design point x by a weighting scheme W(x) = (w1(x),...,wn(x)) with adaptive weights    δ(x , x) Δ( fˆ(x), fˆ(x )) w (x) = K j K N(x) j , (A.7) j loc h st λ

ˆ where f (x) is the estimate of f (x) obtained in the previous step. Kloc and Kst denote two kernel functions. The first term in (A.7) defines a local vicinity of the ˆ ˆ point x ∈ X , leading to vanishing weights outside this set. N(x)Δ( f (x), f (x j )) measures our confidence in x j belonging to the same homogeneity region as x and ˆ ˆ λ is a suitable bandwidth. It depends on the estimates f (x j ), f (x) and a value N(x) characterizing the concentration of distribution of fˆ(x) around its mean. Therefore, the second term in (A.7) penalizes observations that, given the information in the estimates fˆ, are not likely to belong to the homogeneity region containing x. A suitable choice for Δ is ˆ ˆ ˆ ˆ Δ( f (x), f (x j )) = KL ( f (x), f (x j )) (A.8) with   ( ) pθ1 y KL (θ ,θ ) = log pθ (y)dy (A.9) 1 2 ( ) 1 Y pθ2 y 178 Appendix A: Smoothing Techniques for Imaging Problems being the Kullback–Leibler divergence (Kullback and Leibler 1951) between the distributions with parameters θ1 and θ2. Possible choices for N(x) are  n [ n ( )]2 i=1 wi x N(x) = wi (x) or N(x) =  , n w2(x) i=1 i=1 i both reflecting the increased precision of the estimate in x due to local averaging. The first is preferable due to its computationally more simple form. The weighting schemes W(xi ) are then, for all i ∈ (1,...,n), employed to obtain ˆ new estimates f (xi ) as local maximum-likelihood (A.3) or local least squares (A.2) estimates. The process is started using the observed values as initial estimates and then iterated increasing the bandwidth h with each step. As long as N(xi ) increases with increasing bandwidth h the variance of the esti- ˆ mates f (xi ) decreases and the information about the local homogeneity structure becomes more precise. We provide a formal description of the algorithm, denoted as adaptive weights smoothing (AWS) or propagation-separation (PS) algorithm.

Conventions 1 Propagation-Separation algorithm ( , ) , ∈ X , ∈ Y , ∼ ,θ( ) ∈ Θ Data: xi Yi i=1,n xi Yi Yi Pθ(xi ) xi δ( , ) X Δ(θ( ), θ( )) P Distances: xi x j in , xi x j between Pθ(xi ) and Pθ(x j ) in Θ . Kernels: Kloc :[0, ∞] → R+ monotone non-increasing, square integrable Kst :[0, ∞] → R+ monotone non-increasing, compactly supported. ˆ n d Estimates: θ(xi ) = arg min q(Y, W(xi ), θ) with q : Y ⊗[0, 1] ⊗ Θ → R+ some risk func- tion, e.g. the weighted negative log-likelihood or weighted least squares. ∗ Sequence of bandwidths: 0 < h(0) < h(1) < ···< h(k ) Parameters: Adaptation scale parameter λ, number of iterations k∗.

Algorithm 1 Propagation-Separation

( ) ( ) δ(x ,x )  ( ) • Initialization: k = 1, W (0)(x ) ={w 0 }, w 0 = K j i , N (0)(x ) = w 0 and i ij ij loc h(0) i j ij θˆ(0)( ) = ( , (0),θ) xi arg min q Y Wi . • Adaptation: ∀i, j compute weights    δ( , ) Δ(θˆ(k−1)( ), θˆ(k−1)( )) (k) x j xi (k−1) xi x j w = Kloc Kst N (xi ) ij h(k) λ

ˆ(k) (k) (k) • Estimation: ∀i define θ (xi ) = arg min q(Y, W (xi ), θ). compute N (xi ) =  ( ) max( n w k , N (k−1)(x )) j=1 ij i  • ≥ ∗ (k)( ) = ( n (k), (k−1)( )) Iterate: Stop if k k ,elsesetN xi max j=1 wij N xi , increment k by 1 and continue with adaptation. Appendix A: Smoothing Techniques for Imaging Problems 179

Table A.1 Kullback–Leibler divergences for one-parameter families implemented in package aws. Family refers to the corresponding parameter in function aw ( ) KL ( , ) Model Density/probabilities T Y Pθ1 Pθ2 Family 2 2 (θ −θ )2 Gaussian √ 1 e−(y−θ) /(2σ ) y 1 2 “Gaussian” 2πσ 2σ 2 N(θ, σ 2) 2 θ θ Gaussian √ 1 e−y /(2θ) y2 1 ( 1 − 1 + ln 2 ) “Volatility” 2πθ 2 θ2 θ1 N(0,θ) − x θ θ Exponential 1 e θ y 1 − 1 + ln 2 “Exponential” θ θ2 θ1 p p θ θ p − θ x p−1 1 2 Gamma p e x y p( − 1 + ln ) “Variance” θ (p) θ2 θ1 θ y θ Poisson e−θ , y = 0, 1,... y θ − θ + θ ln 1 “Poisson” y! 2 1 1 θ2 θ −θ Bernoulli θ y (1 − θ)(1−y), y ∈{1, 2} y θ ln 1 + (1 − θ ) ln 1 1 “Bernoulli” 1 θ2 1 1−θ2

For the special case of an exponential family PΘ in canonical form, i.e., with E(T (Yi )|xi ) = θ(xi ) we obtain, using weighted negative log-likelihood,  ( ) ˆ j wijT Y j θ(xi ) =  j wij and specify Δ(θ(x j ), θ(xi )) by Eq. (A.8). TableA.1 provides explicit expressions for several exponential families. The adaptive weights smoothing procedure depends on several parameters. The kernelKloc : R+ →[0, 1] is chosen to be monotonic nondecreasing with Kloc(0) = ∞ ( ) < ∞ 1 and 0 Kloc x dx . Kernel functions with compact support, which excludes the Gaussian kernel, are preferable from a computational point of view. The kernel : R →[ , ] ( ) = ∞ ( ) < ∞ Kst + 0 1 again with Kst 0 1 and 0 Kst x dx need to have com- pact support and ideally exhibits a constant value in an interval [0, a). The default choice for both kernels in package aws is

2 Kloc(x) = max((1 − x ), 0) and Kst(x) = min(1, max(2 − 2x, 0)). (A.10)

d If the design points xi form an equidistant grid in R (for some integer d > 0) we can specify δ(xi , x j ) =||xi − x j ||. The series of bandwidths is preferably selected (0) such that h = mini = j δ(xi , x j ) and   δ( , ) 2  δ( , ) xi x j 2 xi x j j Kloc h(k−1) j Kloc h(k)  = ch.  δ( , )  δ( , ) 2 2 xi x j xi x j j Kloc h(k−1) j Kloc h(k)

The package aws uses ch = 1.25 which, for the nonadaptive estimate, provides a variance reduction by 25% from step to step. 180 Appendix A: Smoothing Techniques for Imaging Problems

The parameter k∗ determines the number of iteration steps, and therefore the maximal bandwidth used. For compactly supported Kloc, the number of weights ∗ k to be computed in iteration step k is approximately n ch, i.e., the computational complexity of the algorithm increases exponentially with k∗. To avoid excessive ∗ computations, k∗ should be chosen to either provide a hk in the range of the diameter of the largest homogeneous regions or to provide a sufficient variance reduction for ∗ the nonadaptive estimate with bandwidth h(k ). To achieve a variance reduction by a ∗ ∗ factor CVR k needs to be specified as k ≥ log CVR/ log ch. The properties of the algorithm are mainly determined by the parameter λ.Fortu- nately, in many cases, this parameter only depends on the probability law Pθ(x) but not on the value θ(x) itself. This can be shown theoretically for P being a continu- ous exponential family distribution but is observed for a much wider class. We refer to Becker and Mathé (2013) and Polzehl and Spokoiny (2006) for further details. If the probability law PΘ ={Pθ ,θ ∈ Θ} is specified, the parameter λ can, for dimension d of the design space X , be chosen by a propagation condition. This con- dition requires that for a completely homogeneous situation, i.e., θ(x) ≡ θ ∀ x ∈ X , the adaptive estimate θ(ˆ x) behaves, for all bandwidths h, similar to its nonadaptive ¯ counterpart θ(x) employing weights w¯ i (x) = Kloc(δ(xi , x)/h). More precisely we require the following propagation condition in the form pro- posed in Becker and Mathé (2013). We introduce the function

ζ ( ,δ,θ)= { > : ( ¯ (k)KL (θˆ(k)(λ), θ) > ) ≤ δ}∀≤ ∗ ∀ > λ k inf z 0 P Ni i z k k z 0  (A.11) ¯ (k) = n ¯ ( ) λ with Ni j=1 w j xi .Wesay to be chosen in accordance with the propagation condition at level ε>0 and parameter θ if ζλ(k,δ,θ) is nonincreasing in k for all δ ∈ (ε, 1). For details, we refer to Becker and Mathé (2013). The propagation condition can be checked using the function awstestprop. require(aws) setCores(3) z1 <- awstestprop(c(200, 200, 200), hmax = 6) z2 <- awstestprop(c(200, 200, 200), hmax = 6, ladjust = 1.5)

To test the propagation condition, a 3D image with dimension 2003 and inde- pendent standard Gaussian intensities is generated. The two calls evaluate the prop- agation condition for two values, the default λ = 8.8usedinfunctionaws for a 3D image with Gaussian intensities and a value of λ = 13.2 adjusted by the factor ladjust=1.5. The results are illustrated in the two plots of Fig. A.3 that show contour lines of ( ¯ (k)KL (θˆ(k)(λ), θ) estimates of P Ni i . The dashed red lines correspond to the nonadap- θ¯ = θˆ(k)(∞) tive estimate i while the black solid lines refer to the adaptive estimate θˆ(k)(λ) δ>ε= − δ>ε= − i . We see that the condition is fulfilled for 1e 2 and 5e 4 for the two values of λ, respectively. Appendix A: Smoothing Techniques for Imaging Problems 181

Exceedence probabilities ladjust=1 Exceedence probabilities ladjust=1.5

0.005

0.002 1e−04 1e−04

0 0 1e−05 1e−05

. .

0 0

0 0

1

5 2

2 2 0

0 1 0 0

. . . .

0 0 0 0 0.5 0.2 0.01 0.2 step step

0.005 0.005 2e−04 0.05 0.002 0.05

0.001 5e−04 0.002 1e−04

5e−05 5e−05 5e−04

2

2e−04 0

5e−05 .

0 5e−05 1e−05 0.1 5e−04 5e−06 1e−06 0.2 0.1 5e−04 5e−06 1e−06 2e−05

0.2 0.01 2e−05 0.01 0.02 1e−04 5e−06 2e−05 2e−05 5e−06 1e−06 0.01

0.001 2e−04 2e−06 1e−06 0.001 2e−04 2e−06 0.05 0.05

0.5 0.1

00

..

55 1e−05 2e−06 2e−06 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 z z

Fig. A.3 Contour plots to evaluate the propagation condition step <- 1:length(z1$h) contour(z1$z, step, z1$prob, levels =z1$levels, xlab = "z", ylab = "step", lwd = 2) title("Exceedence probabilities ladjust=1") contour(z1$z, step, z1$probna, levels =z1$levels, add = TRUE, col = 2, lty = 2) contour(z2$z, step, z2$prob, levels =z2$levels, xlab = "z", ylab = "step", lwd = 2) title("Exceedence probabilities ladjust=1.5") contour(z2$z, step, z2$probna, levels =z2$levels, add = TRUE, col = 2, lty = 2)

Due to the specific choice of parameter λ by a propagation condition and its separation (between homogeneous regions) properties, see Polzehl and Spokoiny (2006) and Becker and Mathé (2013), the approach is also referred to as propagation- separation approach. For calculating the Kullback–Leibler divergence, we need, in case of Gaussian intensities, to specify the variance σ 2 of the intensity distribution. This can be achieved under the assumption of a homogeneous variance using a slightly mod- ified AWS algorithm that starts with a nonadaptive kernel estimates employing a small bandwidth and in the adaptation step k uses a global estimate σˆ 2 obtained from the residuals with respect to the reconstruction obtained in step k − 1. For structured images, the error variance is overestimated in the beginning, with esti- mates stabilizing for large bandwidths. This approach is implemented in function aws.gaussian and employed by 182 Appendix A: Smoothing Techniques for Imaging Problems require(aws) sigma2 <- mean(aws.gaussian( T1[121:220, 121:200, 121:200], 10)@sigma2) for a sub-cube. We are now ready to perform adaptive weights smoothing on the T1 image. To illustrate the properties of AWS, we calculate image reconstructions for different values of hmax corresponding to 6, 12, 18, and 24 steps of the algorithm.

T1aws6 <- aws(T1, hmax = 1.15, sigma2 = sigma2)@theta T1aws12 <- aws(T1, hmax = 1.7, sigma2 = sigma2)@theta T1aws18 <- aws(T1, hmax = 2.7, sigma2 = sigma2)@theta T1aws24 <- aws(T1, hmax = 4.2, sigma2 = sigma2)@theta T1aws30 <- aws(T1, hmax = 6.6, sigma2 = sigma2)@theta

The results are presented in Fig. A.4. rimage(ix, iz, T1[ix, iy, iz], main = "original") rimage(ix, iz, T1aws6[ix, iy, iz], main = "steps = 6") rimage(ix, iz, T1aws12[ix, iy, iz], main = "steps = 12") rimage(ix, iz, T1aws18[ix, iy, iz], main = "steps = 18") rimage(ix, iz, T1aws24[ix, iy, iz], main = "steps = 24") rimage(ix, iz, T1aws30[ix, iy, iz], main = "steps = 30")

While the tissue structure visible in the original is preserved and the reconstruc- tion stabilizes with increasing bandwidth Fig.A.4 also reveals the shortcomings of the adaptive weights algorithm. It enforces the structural assumption of a locally constant image, and therefore leads to a cartoon-like reconstruction. Additionally, the algorithm has no information about the smoothness of discontinuities resulting in rather rough tissue borders.

A.2.2 Patchwise Adaptive Weights Smoothing (PAWS)

In Polzehl et al. (2019), a refinement of AWS was introduced that overcomes these shortcomings using local spatial configurations in patches Vi instead of single voxel k ={ ( )||| ( ) − || ≤ } to define a statistical penalty sij. A patch Vi vl i vl i i 1 s is the local vicinity of voxel i that contains all voxel within L1-distance less or equal patch Appendix A: Smoothing Techniques for Imaging Problems 183

original steps = 6 steps = 12 z z z 130 140 150 160 130 140 150 160 130 140 150 160

160 170 180 190 200 160 170 180 190 200 160 170 180 190 200 x x x

steps = 18 steps = 24 steps = 30 z z z 130 140 150 160 130 140 150 160 130 140 150 160

160 170 180 190 200 160 170 180 190 200 160 170 180 190 200 x x x

Fig. A.4 Smoothing using AWS: results for a detail of the T1 map size s from voxel i. This patchwise adaptive weights smoothing (PAWS) procedure (k) employs a new form of the statistical penalty sij based on patches Vi . The variability θˆ (k) of the estimates i at iteration step k depends on the (local) weighting schemes Wi . (k) This is taken into account in the definition of the statistical penalty sij by the use (k) X of the sum of weights Ni . Depending on the unknown underlying structure n the θˆ ( ) ∈ variability of the estimates vl (i) may vary considerably over grid points vl i Vi . (k) Thus, when we extend the definition sij to comparisons between patches, it should consider the accuracy of the parameter estimates reflected by Nvl (i) as achieved in former iteration steps. We thus define a suitable statistical penalty for PAWS by

( ) ( − )  ( − ) ( − ) s˜ k = max N k 1 · η θˆ k 1 , θˆ k 1 /λ. ij vl (i) vl (i) vl ( j) l=1,...,ns

Taking the maximum over all locations l = 1,...,ns in the patch enables to balance spatial differences in the variance of the estimates. λ (k) As for AWS, the adaptation bandwidth in sij depends only on the parametric ∈ P family Pθi Θ , the dimension of the design space d, and, additionally, on the patch size s. We again choose it by a propagation condition, see Becker and Mathé (2013). The algorithm is summarized in Algorithm 2. 184 Appendix A: Smoothing Techniques for Imaging Problems

Algorithm 2 Patch wise adaptive weights smoothing (PAWS)

( ) ( ) δ(x ,x )  ( ) • Initialization: k = 1, W (0)(x ) ={w 0 }, w 0 = K j i , N (0)(x ) = w 0 , i ij ij loc h(0) i j ij θˆ(0)( ) = ( , (0),θ) ={ ( )||| ( ) − || ≤ } xi arg min q Y Wi and define Vi vl i vl i i 1 s . • Adaptation: ∀i, j compute weights     (k) δ(x j , xi ) w = Kloc Kst maxl= ,...,n s ( ) ( ) ij h(k) 1 s vl i vl j

with Δ(θˆ(k−1)( ), θˆ(k−1)( )) (k−1) xvl (i) xvl ( j) s ( ) ( ) = N (x ( )) . vl i vl j vl i λ ˆ(k) (k) (k) • Estimation: ∀i define θ (xi ) = arg min q(Y, W (xi ), θ). compute N (xi ) =  ( ) max( n w k , N (k−1)(x )) j=1 ij i  • ≥ ∗ (k)( ) = ( n (k), (k−1)( )) Iterate: Stop if k k ,elsesetN xi max j=1 wij N xi , increment k by 1 and continue with adaptation.

Patchwise adaptive weights reconstructions employing the same maximal band- widths as before are computed using function paws.

T1paws6 <- paws(T1, hmax = 1.15, sigma2 = sigma2)@theta T1paws12 <- paws(T1, hmax = 1.7, sigma2 = sigma2)@theta T1paws18 <- paws(T1, hmax = 2.7, sigma2 = sigma2)@theta T1paws24 <- paws(T1, hmax = 4.2, sigma2 = sigma2)@theta T1paws30 <- paws(T1, hmax = 6.6, sigma2 = sigma2)@theta

FigureA.5 provides the results for the same region as shown in Figs. A.2 and A.4 rimage(ix, iz, T1[ix, iy, iz], main = "original") rimage(ix, iz, T1paws6[ix, iy, iz], main = "steps = 6") rimage(ix, iz, T1paws12[ix, iy, iz], main = "steps = 12") rimage(ix, iz, T1paws18[ix, iy, iz], main = "steps = 18") rimage(ix, iz, T1paws24[ix, iy, iz], main = "steps = 24") rimage(ix, iz, T1paws30[ix, iy, iz], main = "steps = 30")

The PAWS reconstruction stabilizes with increasing bandwidth to a locally smooth image with smooth tissue borders. It also reveals, using 3D information, anisotropic structures that are hardly visible in the original (2D) slice, see also the resulting quantitative maps in Fig. 6.10 in Chap. 6 for a comparison. Appendix A: Smoothing Techniques for Imaging Problems 185

original steps = 6 steps = 12 z z z 130 140 150 160 130 140 150 160 130 140 150 160

160 170 180 190 200 160 170 180 190 200 160 170 180 190 200 x x x

steps = 18 steps = 24 steps = 30 z z z 130 140 150 160 130 140 150 160 130 140 150 160

160 170 180 190 200 160 170 180 190 200 160 170 180 190 200 x x x

Fig. A.5 Smoothing using patchwise AWS: results for a detail of the T1 map

A.3 Special Settings in Neuroimaging Experiments

Several refinements of the AWS and PAWS algorithms for special needs in the anal- ysis of neuroimaging data are implemented in the R packages fmri, dti, and qMRI. We here only shortly summarize the differences to the original approaches.

A.3.1 Simultaneous Mean and Variance Estimation

The propagation-separation algorithms can be modified to simultaneously estimate additional parameters that are either global, slowly varying in space or functions of a parameter θ. The most interesting case is problems where probability distribu- tion P of the image intensities depends on both mean θ and slowly varying scale parameter σ. Let h pre > h0 be some small bandwidth, hmed a medium, or large bandwidth and σ (0) some reasonable pre-estimate of the scale parameter. The latter may, e.g., in case of Gaussian errors, be obtained using residuals from a nonadaptive kernel smoother employing bandwidth h pre.LetU(xi , h) ={x j |δ(x j , xi ) ≤ h}. 186 Appendix A: Smoothing Techniques for Imaging Problems

Algorithm 3 PS algorithm for simultaneous mean and scale estimation

( ) ( ) δ(x ,x )  ( ) • Initialization: k = 1, W (0)(x ) ={w 0 }, w 0 = K j i , N (0)(x ) = w 0 , i ij ij loc h(0) i j ij σˆ (0)( ) ≡ σ (0) θˆ(0)( ) = ( , (0),θ,σ(0)) xi and xi arg min q Y Wi . • Adaptation: ∀i, j compute weights    δ( , ) Δ(θˆ(k−1)( ), θˆ(k−1)( ), σˆ (k−1)( )) (k) x j xi (k−1) xi x j xi w = Kloc Kst N (xi ) ij h(k) λ

(k) • Estimation: ∀i if h ≤ h pre then set

( ) ( ) ( ) ( ) ( ) θˆ k (x ) = arg min q(Y, W k (x ), θ, σ 0 ) and σˆ k (x ) ≡ σ 0 i θ i i

else define

ˆ(k) (k) (k) (k) (k) (θ (xi ), σ˜ (xi ))= arg min q(Y, W (xi ), θ, σ) and, σˆ (xi )= median σ˜ (x j ) (θ,σ) j∈U(xi ,hmed)

 ( ) Compute N (k)(x ) = max( n w k , N (k−1)(x )). i j=1 ij i • ≥ ∗ (k)( ) = ( n (k), (k−1)( )) Iterate: Stop if k k ,elsesetN xi max j=1 wij N xi , increment k by 1 and continue with adaptation.

The algorithm provides a spatially adaptive estimate of θ(xi ) and simultane- ously a smooth estimate of the scale σ(xi ). This is implemented, e.g., in function awslsigma in package dti. A similar idea is employed in function aws.gaussian in package aws where (k) instead of computing σˆ (xi ) as a median over some spatial neighborhood it is defined by a parametric fit of squared residuals as a polynomial of θˆ(k).

A.3.2 Vector-Valued Data

The R-packages aws and qMRI contain functions for adaptive weights smooth- ing and patchwise adaptive weights smoothing for vector-valued image data. The probabilistic model used is a multivariate Gaussian distribution

Y ∼ N p(θ, Σ) with mean θ ∈ Rp and known covariance matrix Σ. Functions vaws and vpaws employ a distance

Δ(θ( ), θ( )) = σ −2||θ( ) − θ( )||2 xi x j xi x j 2 assuming independent identically distributed Gaussian components and are intended for spatial smoothing of registered time series or collections of images using either Appendix A: Smoothing Techniques for Imaging Problems 187

AWS or PAWS. Functions vawscov and vpawscov in aws are designed for smoothing of vector-valued parameter maps resulting from fitting regression models to a sample of images. They use, for each voxel, an estimated inverse covariance −1 matrix Σ (xi ) and define

−1 T −1 Δ(θ(xi ), θ(x j ), Σ (xi )) = (θ(xi ) − θ(x j )) Σ (xi )(θ(xi ) − θ(x j )).

The last two functions are integrated into function smoothESTATICS in qMRI to enable adaptive smoothing of the parametric map objects created by function estimateESTATICS. The argument patchsize is used to specify either AWS or PAWS. The function smoothInvCov is designed to stabilize the estimated inverse covariance matrices provided in objects of class "ESTATICSModel" cre- ated by estimateESTATICS.

A.3.3 Diffusion Data

Data in diffusion-weighted MRI live on a grid in orientation space S2 × R3, with coordinates on the sphere S2 approximately uniformly sampled and voxel coordinates in R3 forming a regular grid. In case of multi-shell data, the image intensities form vector-valued observations in each design point with components corresponding to the individual b-values. If identical gradients are applied on all shells, the intensity vectors are completely observed, if not, unobserved (missing) values are imputed by interpolation. Position-orientation adaptive smoothing (POAS) (Becker and Mathé 2013; Becker et al. 2014) is an adaptive weights smoothing algorithm that employs a location penalty

δ ( , ) =|| − || + κ−1 (|T  |), κ m n vm vn k acos gm gn (A.12)

3 where ||.|| denotes the L2-norm in R and κk steers the influence of the geodesic distance on the sphere S2 (Hagmann et al. 2006). For multi-shell acquisitions, the procedure uses a statistical penalty for vector-valued data. The approach is imple- mented in function dwi.smooth in dti. Especially on the sphere the assumption of local homogeneous (constant) image intensities is questionable. Therefore, the implementation of POAS reduces/limits potential smoothing on the sphere by decreasing κ with increasing h. Patchwise adaptive weights smoothing would require an appropriate definition of corresponding patches on the sphere and is currently not available.

A.3.4 Tensor-Valued Data

Euclidean operations are inappropriate for tensor-valued data. They, e.g., do not preserve positive definiteness and therefore may lead out of the space of tensors. Smoothing of such data is best carried out in a tensor space equipped with an appro- 188 Appendix A: Smoothing Techniques for Imaging Problems priate Riemannian metric. Arsigny et al. (2006), Fillard et al. (2007), and Fletcher and Joshi (2007) introduced an affine-invariant Riemannian metric based on matrix log- arithms and exponentials. For a tensor D, its exponential and logarithm are obtained via singular value transformation as D = U T Diag(d)U, with orthogonal U,as exp(D) = U T Diag(exp d)U and log(D) = U T Diag(log d)U. The log-Euclidean metric is then defined as

Δ(D1, D2) =||log(D1) − log(D2)||2.

This induces arithmetic operations in tensor space as

D1  D2 = exp(log(D1) + log(D2))

λ  D1 = exp(log(λD1)) and enables averaging, i.e., smoothing and interpolation, the definition of variances and the construction of, e.g., principal components. A similar approach for high angular diffusion data for ODFs is described in Goh et al. (2011). Tensor-valued estimates have been considered in Chap. 5. Estimated diffusion tensors can be smoothed using function dti.smooth from package dti, see, e.g., Tabelow et al. (2008a). The algorithm employs a statistical penalty

N      s = i (Dˆ − Dˆ )T Cov(Dˆ )(Dˆ − Dˆ ) ij λ i j i i j and a location parameter

= δ2( , , Dˆ ) = Dˆ ( − )T Dˆ −1( − ). lij xi x j i det i xi x j i xi x j

To avoid the application of Euclidean operations on the diffusion tensors, the original diffusion data are smoothed using the obtained weighting schemes and the diffusion tensors are then re-estimated from the smoothed data after each iteration step.

A.3.5 Model-Driven Smoothing of Observed Images

The idea of model-based smoothing of the experimental data or of residuals from modeling is employed in both the analysis of fMRI and qMRI data. For example, in function fmri.smooth, the weighting schemes from the final step of the smoothing algorithm are used to smooth residuals and then estimate the spatial correlation of the smoothed parametric map in order to correctly handle the multiple test problem. In qMRI, adaptive smoothing of the observed images using the weighting schemes enables to re-evaluate the quantitative maps. Appendix B Resources for Neuroimaging in R

Several large open-source software environments and collections have emerged from collaborative work in the neuroscience and imaging communities. These are well established and widely used in the analysis of neuroimaging experiments. Statis- ticians have made contribution to several of these software collections. For most of these software collections, interface packages for easy access from R are under development within the Neuroconductor project (Muschelli et al. 2018). They cur- rently provide a partial coverage of the functionality. We here give a short overview on the commonly used open-source tools for neuroimaging, mainly citing from the respective websites.

• The Analysis of Functional NeuroImages (AFNI) software (Cox 1996, 2001)is a software suite of C, Python, R programs, and shell scripts primarily developed for the analysis and display of anatomical and functional MRI (FMRI) data. An interface to R is provided through package afnir. • The National Library of Medicine Insight Segmentation and Registration Toolkit (ITK) (ITK Community 2019) is an open-source, cross-platform system that pro- vides developers with an extensive suite of software tools for image analysis. ITK employs leading-edge algorithms for registering and segmenting multidi- mensional data. The Advanced Normalization Tools (ANTs) (Avants 2019a)is a state-of-the-art medical image registration and segmentation toolkit built upon ITK. ANTs and ITK can be used from R using packages ANTsR (Avants 2019b), ANTsRCore (Kandel et al. 2019) and ITKR (Avants 2019c). The packages are used in Chaps. 4–6. • FreeSurfer (Laboratory for Computational Neuroimaging at the Athinoula A. Mar- tinos Center for Biomedical Imaging 2019; Fischl 2012) is a software package for the analysis and visualization of structural and functional neuroimaging data from cross-sectional or longitudinal studies. FreeSurfer is the structural MRI analysis software of choice for the Human Connectome Project. Freesurfer can be accessed from R using the interface package freesurfer (Muschelli 2019e). The package is used in Sect. 4.1.6.

© Springer Nature Switzerland AG 2019 189 J. Polzehl and K. Tabelow, Magnetic Resonance Brain Imaging, Use R!, https://doi.org/10.1007/978-3-030-29184-6 190 Appendix B: Resources for Neuroimaging in R

• The FMRIB Software Library (FSL) (Jenkinson et al. 2012; FMRIB Analysis Group, Oxford, UK 2018) is a comprehensive library of analysis tools for FMRI, MRI, and DTI brain imaging data that has been developed at the Wellcome Center for Integrative Neuroimaging at the University of Oxford. The package fslr (Muschelli et al. 2015) provides an interface from R and is used in Chaps. 4 and 5. • Statistical parametric mapping (SPM) refers to the construction and assessment of spatially extended statistical processes used to test hypotheses about functional imaging data. These ideas have been instantiated in a software called SPM. The SPM software package has been designed for the analysis of brain imaging data sequences. The sequences can be series of images from different cohorts or time series from the same subject. The current release is designed for the analysis of fMRI, PET, SPECT, EEG, and MEG. SPM12 (The FIL Methods Group (and honorary members) 2014, 2018)isthe current version of the software that has been developed at the Wellcome Centre for Human Neuroimaging. SPM12 is open source but requires MATLAB (Math- works 2019) as backend. An interface to SPM12 is provided by the package spm12r (Muschelli 2019j).

B.1 An Overview on Selected R Packages for Neuroimaging

The CRAN Task View Medical Image Analysis (Whitcher 2019a) provides a struc- tured overview on packages for different imaging tasks that are available from the Comprehensive R Archive Network (CRAN) (CRAN Team 2019). A coordinated effort to promote the use of R in neuroimaging is led by the Neuroconductor (Muschelli et al. 2018) project. Neuroconductor (Muschelli et al. 2019) provides a GitHub-based open-source platform of R packages generally related to neuroimaging analyses and processing. TableB.1 provides an overview on packages for neuroimaging currently (Jan. 2019) available from either CRAN (C) or Neuroconductor (N). The following list provides short information, again mainly from the package description, on the functionality of a selection of these packages that are related to topics addressed in this book.

• adaptsmoFMRI (Hughes 2013) contains R functions for estimating the blood- oxygenation-level-dependent (BOLD) effect by using functional magnetic reso- nance imaging (fMRI) data based on adaptive Gauss Markov random fields. Infer- ence of the underlying models is performed by efficient Markov chain Monte Carlo simulation, with the Metropolis–Hastings algorithm for the non-approximate case and the Gibbs sampler for the approximate case. Appendix B: Resources for Neuroimaging in R 191

Table B.1 Neuroimaging packages available from Neuroconductor (N) and CRAN (C) Package Version Title Maintainer aal 0.1.1 Automated Anatomical Labeling John Muschelli N (“AAL”) Atlas afnir 0.4.6 Wrapper Functions for “AFNI” (Analysis John Muschelli N of Functional “NeuroImages”) adaptsmoFMRI 1.1 Adaptive Smoothing of FMRI Data Max Hughes C adimpro 0.9.0 Adaptive Smoothing of Digital Images Karsten Tabelow C AnalyzeFMRI 1.1-17 Functions for Analysis of fMRI Datasets Pierre Lafaye de C stored in the ANALYZE or NIFTI Format Micheaux ANTsR 0.4.7 ANTs in R: Quantification Tools for Brian B. Avants N Biomedical Images ANTsRCore 0.6.3 Core Software Infrastructure for Brian B. Avants N “ANTsR” arf3DS4 2.5-10 Activated Region Fitting, fMRI data Wouter D. Weeda C analysis (3D) aws 2.2-0 Adaptive Weights Smoothing Jörg Polzehl C bayesImageS 0.6-0 Bayesian Methods for Image Matt Moores C Segmentation using a Potts Model bftools 0.2.0 BioFormats Tools John Muschelli N brainKCCA 0.0.0.9 Region-level Connectivity Network Jian Kang N Construction via Kernel Canonical Correlation Analysis brainR 1.5.2 Helper Functions to “misc3d” and “rgl” John Muschelli NC Packages for Brain Imaging brainwaver 1.6 Basic wavelet analysis of multivariate Sophie Achard C time series with a visualization and parametrization using graph theory cap 1.0 Covariate Assisted Principal (CAP) Yi Zhao N Regression for Covariance Matrix Outcomes cfma 1.0 Causal Functional Mediation Analysis Yi Zhao N cifti 0.4.5.9 Toolbox for Connectivity Informatics John Muschelli N Technology Initiative (“CIFTI”) Files DATforDCEMRI 0.55 Deconvolution Analysis Tool for Gregory Z. Ferl C Dynamic Contrast Enhanced MRI dcemriS4 0.57.1 A Package for Image Analysis of Brandon Whitcher N DCE-MRI (S4 Implementation) dcm2niir 0.6.6 Conversion of “DICOM” to “NIfTI” John Muschelli N Imaging Files Through R dcmsort 0.2.5.1 Sort DICOM Images John Muschelli N dcmtk 0.6.3.1 Wrapper for “DICOM” Toolkit John Muschelli N (“DCMTK”) DensParcorr 1.1 Dens-Based Method for Partial Yikai Wang C Correlation Estimation in Large Scale Brain Networks (continued) 192 Appendix B: Resources for Neuroimaging in R

Table B.1 (continued) Package Version Title Maintainer divest 0.7.2 Get Images Out of DICOM Format Jon Clayden N Quickly dpmixsim 0.0-9 Dirichlet Process Mixture Model Adelino Ferreira da C Simulation for Clustering and Image Silva Segmentation dti 1.3.4 Analysis of Diffusion Weighted Imaging Karsten Tabelow NC (DWI) Data edfReader 1.2.0 Reading EDF(+) and BDF(+) Files Jan Vis C eegkit 1.0-4 Toolkit for Electroencephalography Data Nathaniel E. C Helwig eegkitdata 1.0 Data for package eegkit Nathaniel E. C Helwig eegUtils 0.3.0.9 A collection of utilities for EEG analysis Matt Craddock N EveTemplate 0.99.14 JHU-MNI-ss (Eve) template Jean-Philippe N Fortin extrantsr 3.9.6 Extra Functions to Build on the John Muschelli N “ANTsR” Package flexconn 0.5.2 FLEXCONN Model John Muschelli N fmri 1.8.4 Analysis of fMRI Experiments Karsten Tabelow NC fmriqa 0.3.0 Functional MRI Quality Assurance Martin Wilson N Routines freesurfer 1.6.2.9 Wrapper Functions for “Freesurfer” John Muschelli N fslr 2.22.0 Wrapper Functions for “FSL” (“FMRIB” John Muschelli N Software Library) from Functional MRI of the Brain (“FMRIB”) gdimap 0.1-9 Generalized Diffusion Magnetic Adelino Ferreira da C Resonance Imaging Silva gganatogram 1.1.1 Create Anatograms of Various Species Jesper Maag N ggBrain 0.1 ggplot Brain Images Aaron Fisher N ggneuro 0.5.0 Plotting Functions for Neuroimaging John Muschelli N Data in “ggplot2” ggseg 1.3 Plotting tool for brain atlases Athanasia Mo N Mowinckel gifti 0.7.5.9 Reads in “Neuroimaging” “GIfTI” Files John Muschelli N with Geometry Information I2C2 0.2.3 Image Intraclass Correlation Coefficient Haochang Shou N ichseg 0.13.5 Intracerebral Hemorrhage Segmentation John Muschelli N of X-Ray Computed Tomography (CT) Images ITKR 0.4.17.5 ITK in R Brian B. Avants N itksnapr 2.1 Package of ITK-SNAP John Muschelli N kirby21.asl 1.7.0 Example ASL Data from the John Muschelli N Multi-Modal MRI Reproducibility Resource kirby21.base 1.7.0 Example Data from the Multi-Modal John Muschelli N MRI “Reproducibility” Resource (continued) Appendix B: Resources for Neuroimaging in R 193

Table B.1 (continued) Package Version Title Maintainer kirby21.det2 1.7.0 Example DET2 Structural Data from the John Muschelli N Multi-Modal MRI Reproducibility Resource kirby21.dti 1.7.0 Example DTI Data from the Multi-Modal John Muschelli N MRI Reproducibility Resource kirby21.flair 1.7.0 Example FLAIR Structural Data from John Muschelli N the Multi-Modal MRI Reproducibility Resource kirby21.fmri 1.7.0 Example Functional Imaging Data from John Muschelli N the Multi-Modal MRI ’Reproducibility’ Resource kirby21.mricloud 0.0.0.9 A dataset containing correlation data for Adi Gherman N 20 subjects from Kennedy Krieger kirby21.mt 1.7.0 MT Structural Data from the John Muschelli N Multi-Modal MRI Reproducibility Resource kirby21.smri 1.5 Example Structural Data from the John Muschelli N Multi-Modal MRI Reproducibility Resource

kirby21.t1 1.7.0 Example T1 Structural Data from the John Muschelli N Multi-Modal MRI “Reproducibility” Resource

kirby21.t2 1.7.0 Example T2 Structural Data from the John Muschelli N Multi-Modal MRI Reproducibility Resource kirby21.vaso 1.7.0 Example VASO Data from the John Muschelli N Multi-Modal MRI Reproducibility Resource LESYMAP 0.0.0.92 Lesions to Symptom Mapping in R Dorian Pustina N LINDA 0.5.0 Lesion Identification with Neighborhood Dorian Pustina N Data Analysis lungct 0.7.2 Processing of Lung CT Scans John Muschelli N malf.templates 0.3.1 Template Images for Multi-Atlas Label John Muschelli N Fusion (“MALF”) medals 0.3.0 Performs Memory Efficient Jacob Maronge N Decomposition for Analysis of Local neighborhood moments for Segmentation mimosa 0.5.7 “MIMoSA”: A Method for Inter-Modal Alessandra N Segmentation Analysis Valcarcel mmand 1.5.3 Mathematical Morphology in Any Jon Clayden C Number of Dimensions mni 0.2.0 Human “MNI” (Montreal Neurological John Muschelli N Institute) Adult Templates MNITemplate 0.99.4 MNI152 template Jean-Philippe N Fortin Morpho 2.6 Calculations and Visualizations Related Stefan Schlager C to Geometric Morphometrics MriCloudR 0.9.2 R wrapper for MriCloud API Brian Caffo N (continued) 194 Appendix B: Resources for Neuroimaging in R

Table B.1 (continued) Package Version Title Maintainer

MRIcloudT1v... 0.2.0 MRIcloud Analysis of T1 Volumetric Brian Caffo N Output mritc 0.5-1 MRI Tissue Classification Dai Feng C msmri 0.3.1 Open Multiple Sclerosis Magnetic John Muschelli N Resonance Imaging Data neurobase 1.27.6 “Neuroconductor” Base Package with John Muschelli NC Helper Functions for “nifti” Objects neurocInstall 0.11.0 “Neuroconductor” Installer John Muschelli N neurohcp 0.8.1 Human “Connectome” Project Interface John Muschelli NC neuroim 0.0.6 Data Structures and Handling for Bradley R. C Neuroimaging Data Buchsbaum neuRosim 0.2-12 Functions to Generate fMRI Data Marijke Welvaert C Including Activated Data, Noise Data and Resting State Data neurovault 0.5.5 “Neurovault” Database API Access John Muschelli N nitrcbot 1.2 Download Image Files from the Adi Gherman N “NeuroImaging Tools and Resources Collaboratory” oasis 3.0.1 Multiple Sclerosis Lesion Segmentation Elizabeth M. N using Magnetic Resonance Imaging Sweeney (MRI) occ 1.1 Estimates PET Neuroreceptor Joaquim Radua C Occupancies oro.asl 0.1.1 Rigorous—Aterial Spin Labelling Brandon Whitcher N oro.dicom 0.5.2 Rigorous—DICOM Input/Output Brandon Whitcher NC oro.nifti 0.9.11 Rigorous—“NIfTI” + “ANALYZE” + Brandon Whitcher NC “AFNI” : Input/Output oro.pet 0.2.5 Rigorous—Positron Emission Brandon Whitcher NC Tomography pain21 0.1.0 21 Pain Studies Simon Vandekar N papayar 1.0 View Medical Research Images using the John Muschelli N Papaya JavaScript Library papayaWidget 0.5.4 Rmarkdown papaya embedding John Muschelli N pbj 0.1.2 Parametric Bootstrap Joint Testing Simon Vandekar N Procedures for Neuroimaging penn115 0.1.1 Template MRI Scan from 115 University John Muschelli N of Pennsylvania Patients PTAk 1.2-12 Principal Tensor Analysis on k Modes Didier G. Leibovici C qMRI 1.0.1 Methods for Quantitative Magnetic Karsten Tabelow C Resonance Imaging (qMRI) RAVEL 1.1.1 Removal of Artificial Voxel Effect by Jean-Philippe N Linear Regression Fortin rcamino 0.6.3 Port of the Camino Software John Muschelli N RNifti 0.10.0 Fast R and C++ Access to NIfTI Images Jon Clayden NC RNiftyReg 2.6.4 Image Registration Using the “NiftyReg” Jon Clayden NC Library (continued) Appendix B: Resources for Neuroimaging in R 195

Table B.1 (continued) Package Version Title Maintainer robex 1.2.4 Robust Brain Extraction (“ROBEX”) John Muschelli N ROpenCVLite 0.1.34.3 Install OpenCV Within R Simon Garnier N Rxnat 0.0.0.9 Query/retrieve Images from XNAT Adi Gherman N Public/Private Datasets Rvcg 0.18 Manipulations of Triangular Meshes Stefan Schlager C Based on the “VCGLIB” API smri.process 0.7.11 Processing of Structural Magnetic John Muschelli N Resonance Imaging spant 0.13.9 MR Spectroscopy Analysis Tools Martin Wilson N spm12r 2.8.0 Wrapper Functions for “SPM” John Muschelli N (Statistical Parametric Mapping) Version 12 from the “Wellcome” Trust Centre for “Neuroimaging” sri24 0.1.1 SRI24 MRI Atlas for Normal Adult John Muschelli N Brain Anatomy stapler 0.6.5 Simultaneous Truth and Performance John Muschelli N Level Estimation sublime 1.3 Automatic Lesion Incidence Estimation Elizabeth M. N and Detection using Multi-Modality Sweeney Longitudinal Magnetic Resonance Images tractor.base 3.3.0 Read, Manipulate and Visualise Jon Clayden C Magnetic Resonance Images voxel 1.3.5 Mass-Univariate Voxelwise Analysis of Angel Garcia de la N Medical Imaging Data Garza waveslim 1.8.0 Basic Wavelet Routines for One-, Two-, Brandon Whitcher NC and Three-Dimensional Signal Processing WhiteStripe 2.3.1 White Matter Normalization for John Muschelli N Magnetic Resonance Images using WhiteStripe

• adimpro (Tabelow and Polzehl 2019a) mainly provides functions for the manip- ulation and visualization of 2D images. The package is used several times in this book. • AnalyzeFMRI (Marchini and Lafaye de Micheaux 2019) is a package originally written for the processing and analysis of large structural and functional MRI datasets under the ANALYZE format. It includes functionality for spatial/temporal ICA (Independent Components Analysis) via a graphical user interface (GUI), cross-platform visualization based on Tcl/Tk components and complete NIfTI input/output. • brainwaver (Achard 2012) provides basic wavelet analysis of multivariate time series, via waveslim (Whitcher 2018), with a visualization and parametrization 196 Appendix B: Resources for Neuroimaging in R

using graph theory. The package focuses on functional connectivity, the construc- tion of adjacency matrices, and their analysis using graph theory. • cifti (Muschelli 2019a) contains functions for the input/output and visualization of medical imaging data in the form of “CIFTI” files (NITRC 2019a). • dcemriS4 (Whitcher and Schmid 2011) provides a collection of functions to per- form quantitative analysis from a DCE-MRI (or diffusion-weighted MRI) acqui- sition on a voxel-by-voxel basis. Data management capabilities include read/write for NIfTI extensions, full audit trail, and improved visualization. The pipeline to quantify DCE-MRI consists of motion correction and/or co-registration, T1 esti- mation, conversion of signal intensity to gadolinium contrast-agent concentration, and kinetic parameter estimation. • dcm2niir (Muschelli 2019b) is a simple wrapper for the “dcm2nii” and “dcm2niix” functions by Chris Rorden at “USC”, which convert DICOM data to NIfTI formats. • dcmsort (Muschelli 2019c) provides wrapper functions to sort DICOM images. • dcmtk (Muschelli 2019d) contains a set of functions porting the digital imaging and communications in medicine (“DICOM”) toolkit. • dti (Tabelow and Polzehl 2019b) provides an environment to process and ana- lyze diffusion-weighted data. The functionality includes diffusion tensor imaging (DTI), diffusion kurtosis imaging (DKI), modeling for high angular resolution diffusion-weighted imaging (HARDI) using Q-ball-reconstruction and tensor mix- ture models, several methods for structural adaptive smoothing including POAS and msPOAS, and a streamline fiber tracking for tensor and tensor mixture models. The package provides functionality to manipulate and visualize results in 2D and 3D, via adimpro and rgl (Adler et al. 2019). The package is used in Chap. 5. • extrantsr (Muschelli 2018a) extends the “ANTsR” package with complex pro- cessing streams, e.g., brain extraction, segmentation, registration, and bias field correction. • fmri (Tabelow and Polzehl 2019c) contains functions to perform single sub- ject fMRI analyses. Functionality includes modeling of task-based experiments using linear models, adaptive spatial smoothing, hypothesis testing using differ- ent approaches, and the analysis of resting state experiments using independent component analysis, see also Chap. 4. • gdimap (da Silva 2015)—The package implements algorithms to estimate and visualize the orientation of neuronal pathways in model-free methods (q-space imaging methods). Fiber orientation estimation is based on mixtures of von Mises– Fisher (vMF) distributions and on local peak detection based on a ODF (Aganj et al. 2010; Yeh et al. 2010; Garyfallidis 2012) reconstruction. • ggBrain (Fisher 2018) enables plotting of brain images with ggplot2 (Wickham 2016). • ggneuro (Muschelli 2019f) contains plotting functions, especially orthographic views of NIfTI images. • gifti (Muschelli 2019g) provides functions to read files stored in the geometry for- mat called “GIfTI” (NITRC 2019b). These files contain surfaces of brain imaging data. Appendix B: Resources for Neuroimaging in R 197

• itksnapr (Muschelli 2019i)—This package wraps the ITK-SNAP software for viewing of NIfTI objects. • mni (Muschelli 2019h) provides functions for downloading magnetic resonace imaging (“MRI”) templates of humans. • MNITemplate (Fonov et al. 2009, 2011; Collins et al. 1999) enables access to MNI152 Templates as NIfTI-objects. • mritc (Feng and Tierney 2011) contains tools for MRI tissue classification using normal mixture models and hidden Markov normal mixture models fitted by vari- ous methods. Functions to obtain initial values and spatial parameters are available. Facilities for visualization and evaluation of classification results are provided. The package is used in Sect. 4.1.7. • neurohcp (Muschelli 2018b) enables downloads and reads data from the Human “Connectome” Project database (WU-Minn HCP Consortium 2019a) using Ama- zon Web Services (“AWS”) “S3” buckets. • neuRosim (Welvaert et al. 2011)—the package allows users to generate fMRI time series or 4D data. Some high-level functions with only a few arguments enable easy definition of activation areas, noise specification, and fast data generation, see also Sect. 4.2.3. • nitrcbot (Gherman and Muschelli 2018) parses and downloads images from various “NeuroImaging Tools and Resources Collaboratory” (NITRC 2019c) resources. • oro.dicom (Whitcher et al. 2011) provides input/output functions for data that conform to the digital imaging and communications in medicine (DICOM) stan- dard. The package is part of the Rigorous Analytics bundle. The package is used in Chap. 3. • oro.nifti (Whitcher et al. 2011) contains functions for the input/output and visual- ization of medical imaging data that follow either the ANALYZE, NIfTI, or AFNI formats. This package is part of the Rigorous Analytics bundle. It is used in several chapters of this book. • qMRI (Tabelow and Polzehl 2019d) implements methods for estimation of quanti- tative maps from multiparameter mapping (MPM) acquisitions including adaptive smoothing methods in the framework of the ESTATICS model, see Chap. 6. • RNifti (Clayden et al. 2019a) provides very fast read and write access to images stored in the NIfTI-1 and ANALYZE-7.5 formats, with seamless synchronization between compiled C and interpreted R code. Also provides a C/C++ API that can be used by other packages. • RNiftyReg (Clayden et al. 2019b) provides an R interface to the “NiftyReg” image registration tools (Modat et al. 2014). Linear and nonlinear registrations are supported in two and three dimensions. • Rxnat (Gherman 2019) enables to query and extract subsets of images from a private/public XNAT dataset. • tractor.base (Clayden et al. 2011) contains functions for reading and writing of popular file formats (DICOM, ANALYZE, NIfTI-1, NIfTI-2, and MGH), interac- tive and noninteractive visualization, flexible image manipulation, metadata, and sparse image handling. 198 Appendix B: Resources for Neuroimaging in R

• waveslim (Whitcher 2012) provides basic wavelet routines for time series (1D), image (2D), and array (3D) analysis. • WhiteStripe (Shinohara et al. 2014) implements an intensity-based normalization of T1 and T2 images, where normal appearing white matter performs well but requires segmentation. This method performs white matter mean and standard deviation estimates on data that has been rigidly registered to the “MNI” template and uses histogram-based methods.

B.2 Open Neuroimaging Data Archives

Following recent discussions on the reproducibility crisis in the neuroscience (Ioannidis 2005) or in psychology (Open Science Collaboration 2015), the neu- roimaging community made major attempts to openly share the data, together with metainformation on experimental conditions and processing pipelines, used to achieve the scientific results, see Nichols et al. (2017), and McNutt (2014). This is an important step to enable reproducibility of the results and to reuse data for new sci- entific questions. Here, we give a short overview on some open data archives. Access to the data is either public or provided for scientific purposes after registration. For a more comprehensive overview see, e.g., the editorial Eickhoff et al. (2016) and the special issues Turner et al. (2016, 2017). • OpenNeuro (Stanford Center for Reproducible Neuroscience 2019) is a free and open platform for analyzing and sharing neuroimaging data. It was established by the openfMRI project (Poldrack et al. 2013; Poldrack and Gorgolewski 2017)at the Stanford Center for Reproducible Neuroscience. The Archive contains exten- sive data from more than 180 neuroimaging studies in standardized form (BIDS format). In addition to data the project also offers standardized pipelines for data processing. Data from the OpenNeuro archive can be downloaded from Amazon Web Services S3 storage using AWS CLI tool (Amazon Web Services 2019), see Appendix C.1.2 for the code used for our examples, or directly from the archive website. • Alzheimer’s disease neuroimaging initiative (ADNI) (Alzheimer’s Disease Neu- roimaging Initiative 2019) refers to a collaborative effort to investigate the pro- gression of Alzheimer’s disease using MRI, positron emission tomography (PET), genetics, cognitive tests, cerebrospinal fluid (CSF), and blood biomarkers as pre- dictors of the disease. The repository contains data from a large number of studies. • The WU-Minn Human Connectome Project (WU-Minn HCP Consortium 2019a, b) “has tackled one of the great scientific challenges of the 21st century: mapping the human brain, aiming to connect its structure to function and behavior”2 in large multimodal imaging and genomics studies (Van Essen et al. 2012, 2013; Hodge et al. 2016). The most prominent studies are

2Cited from WU-Minn HCP Consortium (2019b). Appendix B: Resources for Neuroimaging in R 199

– WU-Minn HCP data—The final data release consists of extensive multimodal 3T MR imaging (structural, resting state, task-based fMRI, and dMRI) and behavioral data of 1200 healthy young adults. This is accompanied with 7T MR imaging data for some of the subjects. Genomewide association study analyzed data for all subjects is supposed to be soon available. – WU-Minn HCP Lifespan Pilot Data—The project acquires multimodal imaging data in four age groups (age 20–44 weeks, age 0–5, age 5–21, and age 36–100+). The project also provides open-source tools and analysis pipelines used in the original analyses. • XNAT CENTRAL (Herrick et al. 2016) is a publicly accessible data-sharing por- tal at Washington University Medical School. XNAT provides neuroimaging data through a web interface and a customizable open-source platform. XNAT facili- tates data uploads and downloads for data sharing, processing, and organization. Data can be downloaded using the package Rxnat (Gherman 2019). • The International Neuroimaging Data-sharing initiative (Child Mind Institute 2019) provides access to, e.g., data of the 1000 Functional Connectomes Project consisting of more than 1200 resting state fMRI datasets independently collected at 33 sites (1000 Functional Connectomes 2019). • The Neuroimaging Tools & Resources Collaboratory (NITRC) (NITRC 2019c) provides imaging, atlas, and test data, together with analysis tools. The atlas data used in Chap. 4, see also Sect.C.1.5, can be downloaded from this archive. Appendix C Data, Software, and Hardware Resources

C.1 How to Acquire and Organize the Example Data

We shortly describe how the example data can be obtained and how they should be organized in order to reproduce the examples in this book. As a first step, specify a directory to host the datasets, e.g., a directory data that resides directly within the working directory. dataDir <- "data" if(!dir.exists(dataDir)) dir.create(dataDir)

C.1.1 Data from the “Kirby21” Reproducibility Study

These data are provided via the Neuroconductor project. Access is realized using the kirby21.base, kirby21.t1, kirby21.t2, and kirby21.flair packages (Landman et al. 2011). To download the T1 weighted, T2 weighted, and structural fluid attenuation inversion recovery (“FLAIR”) magnetic resonance imaging (“MRI”) data used in Chap. 2 execute dataDir <- "data" library(kirby21.base) download_kirby21_data(modality = "T1", outdir = file.path(dataDir, "Kirby21", "T1")) download_kirby21_data(modality = "T2", outdir = file.path(dataDir, "Kirby21", "T2")) download_kirby21_data(modality = "FLAIR", outdir = file.path(dataDir, "Kirby21", "FLAIR"))

© Springer Nature Switzerland AG 2019 201 J. Polzehl and K. Tabelow, Magnetic Resonance Brain Imaging, Use R!, https://doi.org/10.1007/978-3-030-29184-6 202 Appendix C: Data, Software, and Hardware Resources

C.1.2 Data from openNeuro

The example data used in Chaps. 4 and 5 are available from the openNeuro archive (Stanford Center for Reproducible Neuroscience 2019; Poldrack et al. 2013; Poldrack and Gorgolewski 2017), see also Appendix B.2. They are best downloaded from Amazon Web Services S3 storage using the AWS CLI tool.3 Conducting dataDir <- "data" dn01a <- file.path("ds000105", "sub-1", "func") dn1a <- file.path(dataDir, dn01a) cmd <- paste0("aws s3 sync --no-sign-request ", "s3://openneuro.org/", dn01a, "", dn1a) system(cmd) dn01b <- file.path("ds000105", "sub-1", "anat") dn1b <- file.path(dataDir, dn01b) cmd <- paste0("aws s3 sync --no-sign-request ", "s3://openneuro.org/", dn01b, "", dn1b) system(cmd) provides a subset of the dataset ds000105 described in Sect. 4.1.1 and stores it in a correspondingly named subfolder. The dataset ds000117 also introduced in Sect. 4.1.1 can be obtained using dn02 <- file.path("ds000117", "sub-01", "ses-mri", "func") dn2 <- file.path(dataDir, dn02) cmd <- paste0("aws s3 sync --no-sign-request ", "s3://openneuro.org/", dn02, "", dn2) system(cmd) while the data, including the JSON files containing DICOM header information, of sessions 105 and 106 of the MyConnectome study (Poldrack et al. 2015), see Sects. 4.1.1 and 5.1.2, are downloaded by dataDir <- "data" dn03 <- file.path("sourcedata", "dicom_headers", "sub-01", "ses-106") dn3 <- file.path(dataDir,"MyConnectome",dn03) cmd <- paste0("aws s3 sync --no-sign-request ", "s3://openneuro.org/ds000031/", dn03, "", dn3) system(cmd) dn04 <- file.path("sub-01", "ses-105") dn4 <- file.path(dataDir,"MyConnectome", dn04)

3To install the Amazon Web Services CLI tools please follow the instructions at Amazon Web Ser- vices (2019). Appendix C: Data, Software, and Hardware Resources 203 cmd <- paste0("aws s3 sync --no-sign-request ", "s3://openneuro.org/ds000031/", dn04, "", dn4) system(cmd) dn04 <- file.path("sub-01", "ses-106") dn4 <- file.path(dataDir,"MyConnectome", dn04) cmd <- paste0("aws s3 sync --no-sign-request ", "s3://openneuro.org/ds000031/", dn04, "", dn4) system(cmd)

Alternatively, the data can be directly obtained from the openNeuro archive (Stanford Center for Reproducible Neuroscience 2019).

C.1.3 DICOM Example Data

The DICOM data used in Chap. 3 can be cloned from a git repository (Hanke 2018) using dn5 <-file.path(dataDir,"example-dicom") cmd <- paste("git clone", "https://github.com/datalad/example-dicom-structural", dn5) system(cmd)

C.1.4 MPM Data Example

The example MPM data analyzed in Chap. 6 can be accessed by dataDir <- "data" srcurl <- paste0("https://owncloud.gwdg.de/index.php/", "s/iv2TOQwGy4FGDDZ/download?path=", "%2F&files=hmri_sample_dataset.zip") download.file(srcurl, file.path(dataDir, "MPM.zip")) unzip(file.path(dataDir, "MPM.zip"))

C.1.5 Atlas Data

Access information for the Montreal Neurological Institute brain atlases can be obtained using package mni (Muschelli 2019h). The call 204 Appendix C: Data, Software, and Hardware Resources library(mni) mni_datasets() provides on overview on 30 atlases. The MNI ICBM 152 linear atlas data used in Chap. 4 can be downloaded executing dataDir <- "data" srcurl <- mni_datasets("nifti")[[5]][1] download.file(srcurl, file.path(dataDir, "mni_icbm152_lin_nifti.zip")) unzip(file.path(dataDir, "mni_icbm152_lin_nifti.zip"))

The functional brain atlas from Finn et al. (2015) can be accessed from the Neu- roimaging Tools & Resources Collaboratory BioImage Suite (NITRC 2019c)using atlasd <- file.path(dataDir,"atlas") if(!dir.exists(atlasd)) dir.create(atlasd) fnshen <- file.path(atlasd, "shen_1mm_268_parcellation.nii.gz") download.file(paste0("https://www.nitrc.org/frs/", "download.php/7976/shen_1mm_268_parcellation", ".nii.gz"), fnshen)

The total size of all downloaded datasets is about 4 GB.

C.2 Packages and Software to Install

As a prerequisite to execute the example code in this book, some software needs to be installed. This includes a recent version of R, we used version 3.5.2, the FMRIB Software Library (FSL) (version 5.0.11), available from FMRIB Analysis Group, Oxford, UK (2018) and the FreeSurfer (Fischl 2012) software package (version 5.3.0), available from Laboratory for Computational Neuroimaging at the Athinoula A. Martinos Center for Biomedical Imaging (2019). TableC.1 lists the R packages that need to be installed, together with their dependencies, from either CRAN or Neuroconductor.

C.3 System Requirements

The example code has been tested using R version 3.5.2 on macOS High Sierra Version 10.13.6 (32 GB, Intel Core I7, AMD Radeon R9 M395X) and Ubuntu 16.04 Appendix C: Data, Software, and Hardware Resources 205

Table C.1 Packages loaded by the R code in the book Package name Package version Available from adimpro 0.9.0 CRAN ANTsR 0.4.5 Neuroconductor ANTsRCore 0.5.7.2 Neuroconductor aws 2.2-2 CRAN dti 1.4.1 CRAN/Neuroconductor fmri 1.9.1 CRAN/Neuroconductor freesurfer 1.6.2.9001 (CRAN)/Neuroconductor fslr 2.22.0 CRAN/Neuroconductor igraph 1.2.4 CRAN ITKR 0.4.16.1 Neuroconductor jsonlite 1.6 CRAN KernSmooth 2.23-15 CRAN kirby21.base 1.7.0 CRAN/Neuroconductor kirby21.flair 1.7.0 Neuroconductor kirby21.t1 1.7.0 CRAN/Neuroconductor kirby21.t2 1.7.0 Neuroconductor knitr 1.21 CRAN mritc 0.5-1 CRAN mni 0.2.0 Neuroconductor neuRosim 0.2.12 CRAN oro.dicom 0.5.0 CRAN/Neuroconductor oro.nifti 0.9.1 CRAN/Neuroconductor qMRI 1.0.1 CRAN rgl 0.99.16 CRAN XML 3.98-1.17 CRAN

(32 GB, Intel Core I7, Intel HD Graphics 620). Parts of the code, especially in Chaps. 5 and 6, are computationally demanding in both memory usage, mainly due to the size of data objects involved, and computing time, due to the large number of voxel involved in the analysis. To level the latter, MPI and openMP parallelization are used where appropriate. In case of limited main memory, we recommend to restrict evaluations to sub-cubes of the data, using the arguments xind, yind and zind in the functions that read the imaging data, or to use more restrictive brain masks. The calls to fslr functions xfibres and probtrackx in Chap. 5 were exe- cuted on compute servers equipped with extensive main memory. These calculations, running exclusively in FSL, took up to 230 GB of memory and 2 weeks of comput- ing time on a single core. The resources needed may be significantly reduced using restrictive seed masks. References

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A Connectivity Acquisition parameter, 13, 88, 89, 102 functional, 2, 11, 75, 78, 80, 196 Adaptation bandwidth, 62, 63, 66, 139, 165, network, 78–80, 143–145 183 structural, 2, 143, 145 Adaptive weights, 165, 177, 178, 186 Contrast of parameters, 48 Angular momentum, 6 Apparent diffusion coefficient, 107, 112, 118, 123 D Artifacts, 25, 44, 72, 81, 84, 85, 99, 101 Diffusion equation, 81–83 Atomic nucleus, 5 Diffusion gradient, 2, 12, 13, 83, 84, 87, 91– 93, 97, 107, 109, 112, 123, 127, 129, 137 B Diffusion kurtosis imaging, 117, 196 Barycentric coordinates, 114, 115 Diffusion magnetic resonance imaging, 2, 3, Bias 12, 13, 26, 81, 83, 92, 94, 96, 97, instrumental, 147, 157 102, 108, 117, 123, 131, 132, 135– Rician, 101, 110, 159 137, 140, 147, 159, 192 BIDS standard, x, 23, 26, 198 Diffusion propagator, 81, 83, 101, 108, 120, Bloch equations, 8 122, 128 Blood oxygenation level-dependent effect, Diffusion spectrum imaging, 123 11, 31, 40–45, 47–50, 54, 75, 190 Diffusion tensor, 82, 83, 107–114, 117–119, Bonferroni correction, 53, 54, 56 121, 128, 131, 137, 140, 142, 159, Brain atlas, 33, 36–38, 191–193, 195 160 Harvard-Oxford, 37 Diffusion tensor imaging, 108, 196 MNI ICBM 152, 204 Diffusion tensor model, 196 Shen268, 37, 38, 204 Distribution Talairach, 32, 75, 76 angular central Gaussian, 132 Brain extraction, 34, 51, 195, 196 Gaussian, 63, 82, 100, 110, 138, 155, Brain mask, 34, 35, 48, 52, 53, 56, 58, 77, 159, 172, 186 90, 94, 98, 102, 110, 142, 150 noncentral Chi, 11, 101 Brodmann areas, 36, 76, 77, 80 Rician, 10, 100 B-value, 13, 84–87, 91, 92, 95, 107, 127, 129, 187 E Echo time, 9, 12, 14, 24, 25, 83, 88, 89, 148, C 150, 152, 155, 159 Cluster thresholds, 57 Eddy current, 84, 85, 91, 101 © Springer Nature Switzerland AG 2019 227 J. Polzehl and K. Tabelow, Magnetic Resonance Brain Imaging, Use R!, https://doi.org/10.1007/978-3-030-29184-6 228 Index

Effective number of coils, 138, 139 non-diffusion weighted, 13, 84, 87–89, Effective order, 133, 134 93, 96, 98, 99, 102–104, 109, 110, 112, Ensemble average propagator, 127 137 Ernst equation, 148 Image formats ESTATICS model, 187 AFNI, 23, 194, 197 Estimate ANALYZE, 15, 19, 20, 92, 191, 194, 195, kernel, 173, 174, 181 197 least squares, 47, 108, 118, 174, 178 CIfTI, 191, 196 local parametric, 173 DICOM, 15–20, 23, 92, 94, 191, 192, maximum-likelihood, 47, 176, 178 194, 196, 197, 202, 203 Nadaraya–Watson, 173 GIfTI, 192, 196 quasi-likelihood, 111, 136, 139 NIfTI, x, 15, 16, 19–21, 23, 27, 29, 34, Euler characteristic, 54 36, 50, 85, 88, 90, 92, 104, 129, 142– Expectation value, 101, 110, 111 144, 150, 191, 194–197 Independent component analysis, 43, 70, 195, 196 F Information criterion False discovery rate, 56, 57 Akaike, 132 Fick’s law of diffusion, 82 Bayes, 132 Field of view, 25 In vivo, 1, 5, 12, 14, 81, 147 Filter Gaussian, 39, 40, 47, 54, 56, 60, 62, 63, 135, 136 J nonlocal means, 135, 175 JSON file, 23, 24, 88, 202 Flip angle, 14, 24–26, 148–150, 157 Fourier transform, 39, 83, 100, 122 Fractional anisotropy, 112, 113, 115, 131, K 134–136, 140, 159 Kernel function, 60, 62, 63, 138, 165, 174, partial volume corrected, 133, 134 177, 179 k Functional magnetic resonance imaging, 2, -space, 10, 11, 99, 100 3, 11, 12, 15, 16, 23, 25, 27–30, 32– Kullback–Leibler divergence, 63, 138, 178, 34, 38–40, 42–44, 47, 48, 50, 53–58, 179, 181 60, 62, 64–67, 70, 188–192, 194, 196, Kurtosis 197, 199 apparent, 118, 119 fractional, 119 mean, 119 tensor, 118, 119 G Geodesic anisotropy, 113, 115 GRAPPA, 11, 101 L Gyromagnetic ratio, 6 Larmor frequency, 7, 9 Local hypothesis, 50

H Hemodynamic response function, 40–42, 48 M Homogeneity structure, 175, 177, 178 Magnetic field, 1, 5–7, 9, 11, 12, 35, 36, 72, Hypothesis test, 51–54, 56, 58, 60, 65, 66 83, 84 Magnetic resonance imaging, 1, 2, 5, 194, 195, 201 I Magnetization transfer, 14 IC fingerprints, 72, 73 Magnetization transfer saturation, 14, 147 Image Mass-univariate approach, 50 diffusion weighted, 12, 13, 84, 91, 92, 96, Mean diffusivity, 112, 118–120, 130 98, 110, 132, 136, 137 Medical imaging, 15, 40, 195, 197 Index 229

Model O ball-and-stick, 128, 129, 140, 142 Orientation distribution function, 120–127, biophysical, 14, 147 133, 196 diffusion kurtosis, 117–119 diffusion tensor, 108, 117, 132, 135, 136, 139–142 P ESTATICS, 153–155, 160, 164, 165, 197 Packages higher order tensor, 117 adaptsmoFMRI, 190 local constant, 167, 175 adimpro, 17, 94, 96, 115, 125, 156, 174, orientation distribution function, 120– 195, 196, 205 124, 128, 132 afnir, 189 tensor mixture, 128, 129, 132–134, 141, AnalyzeFMRI, 70, 195 142, 144, 145 ANTsR, 28–31, 35, 38, 44, 70, 71, 75, Motion, 28, 29, 31, 33, 38, 53, 63, 71, 72, 85, 144, 152, 157, 189, 205 84, 85, 91, 99, 101, 196 ANTsRCore, 31, 35, 189, 205 Motion correction, 29 arf3DS4, 40 MR contrast aws, viii, 39, 136, 174, 175, 179, 186, FLAIR, 9, 30, 193, 201 187, 205 MT w, 147–149, 151, 155 brainwaver, 195 PDw, 9, 13, 30, 147, 149, 151, 152 cifti, 196 T1w, 9, 13, 14, 30, 31, 36, 147, 149–151, dcemriS4, 196 201 dcm2niir, 196 T2w, 9, 13, 30, 201 dcmsort, 196 MR sequence, 9, 11, 13 dcmtk, 196 FLASH, 14, 148, 149 dti, viii, 92, 94–98, 102, 103, 109–112, gradient echo, 11, 14, 147 114–116, 118, 119, 121, 124, 127, 132, multi-echo, 14, 147, 149 133, 135, 137, 138, 140, 144, 185–188, pulsed gradient spin echo, 12, 13, 83 196, 205 spin echo, 12, 83 extrantsr, 196 MR software fastICA, 72 AFNI, 23, 40, 58, 191 fmri, vii, viii, 19, 23, 27, 28, 40, 42–45, ANTs, 70, 189 47, 54, 57, 58, 64, 66, 68, 70, 185, 196, Brainvoyager, 28, 33 205 freesurfer, 34, 189, 192 freesurfer, 34, 51, 189, 205 FSL, 28, 33, 35–37, 40, 59, 85, 87, 90– fslr, 28, 35, 36, 76, 85, 87, 91, 98, 104, 92, 98, 99, 129, 142, 143, 146, 150, 190, 129, 142, 143, 150, 190, 205 192 gdimap, 196 ITK, 28, 31, 70, 85, 189, 192, 197 ggBrain, 196 SPM, 19, 28, 33, 40, 59, 157, 190, 195 ggneuro, 196 Multiparameter mapping, 2, 11, 13–15, 147– ggplot2, 196 150, 154, 159, 160, 197, 203 gifti, 196 Multiple comparison problem, 51, 58, 65 glasso, 78 gsl, 100, 119 Multivariate pattern analysis, 68 igraph, 78, 79, 146, 205 ITKR, 189, 205 itksnapr, 197 N jsonlite, 24, 205 Noise, 2, 10, 39, 47–49, 60, 64, 84, 99, 100, KernSmooth, 205 102–104, 110, 118, 135, 139, 159, kirby21.base, 201, 205 160, 164, 166, 174, 194, 197 kirby21.flair, 201, 205 Normalization, 31–34 kirby21.t1, 201, 205 Nuclear magnetic resonance, 5, 6, 9 kirby21.t2, 201, 205 Nuclear spin, 1, 6, 7 knitr, x, 205 230 Index

mni, 197, 203, 205 nonparametric, 171 MNITemplate, 197 quasi-likelihood, 110 mritc, 36, 197, 205 Relaxation time, 2, 8, 9, 11, 13, 14, 153 neurohcp, 197 Repetition time, 8, 11, 14, 15, 24, 25, 27, 42, neuRosim, 48, 50, 197, 205 48, 148, 150, 154, 155 nitrcbot, 197 Resting state functional magnetic resonance oro.dicom, 16–20, 197, 205 imaging, 26, 30, 37, 40, 70, 75–77, oro.nifti, 20, 22, 23, 27, 28, 34, 39, 49, 194, 196, 199 76, 86, 92, 197, 205 qMRI, viii, 150, 154, 157, 159, 185–187, 197, 205 S quadprog, 118 SENSE, 11, 101, 160 R-base, 78 rgl, 96, 97, 118, 121, 196, 205 Sigma filter, 177 RNifti, 197 Signal attenuation, 13, 96, 107, 122, 123 RNiftyReg, 85, 197 Signal-to-noise ratio, 60, 101, 135, 159 Rxnat, 197, 199 Simulated data, 48, 49 spm12r, 190 Slice time correction, 27, 28 tractor.base, 19, 197 Smoothing waveslim, 195, 198 adaptive weights, 167, 178, 179, 182, WhiteStripe, 198 186, 187, 191 XML, 146, 205 Gaussian, 39, 136, 164 Parallel imaging, 10, 138 (multi-shell) position-orientation adap- Permutation test, 60 tive, 136–139, 187, 196 Precession, 6, 7 nonadaptive smoothing, 62–64 Preprocessing, 2, 25, 27, 28, 30, 32, 39, 40, patchwise adaptive weights, 167, 182– 53, 63, 71, 72, 81, 84, 85, 91, 92, 98, 184, 186, 187 101, 135, 137 structural adaptive, 2, 62, 64–66, 81, 164 Pre-whitening, 46–48, 51 Space Propagation condition, 63, 64, 66, 139, 180, MNI, 33, 34, 37, 38, 75, 76, 146 181 subject, 34, 37, 38, 76, 144 Propagation-separation approach, 2, 62, 104, Talairach, 33 135–137, 164 Spatial correlation, 40, 53, 63, 101, 188 Proton density, 9, 13, 14, 147, 148, 156 Standard deviation, 10, 39, 47, 60, 68, 100, P-value, 51, 54–59, 64, 66 130, 139 Statistical penalty, 62, 64, 104, 137, 138, 165, 167, 182, 183, 187, 188 Q Stimulus, 27, 28, 40–42, 48, 50, 52, 67, 70 Q-ball imaging, 122, 196 Structural adaptive segmentation, 65, 66 Quantitative magnetic resonance imaging, 2, Susceptibility, 84, 85, 91, 92, 99, 101 13, 14, 147, 174, 188 Quantum mechanics, 5–7 T R Temporal correlation, 47 Radio-frequency pulse, 5, 7 Tensor mixture model, 196 Random field theory, 54, 64 Tissue segmentation, 30, 35, 102, 191–193, Region of interest, 93, 150 196 Registration, 28, 30, 31, 33, 85, 99, 110, 157, Tractography, 2, 81, 112, 140, 143 189, 194, 196, 197 probabilistic fiber tracking, 142 Regression, 171 streamline fiber tracking, 141–145 least squares, 172, 176 Transformation local parametric, 173 affine, 28, 31, 76, 85, 144 nonlinear, 109, 110, 132, 154 rigid body, 28, 29, 33, 85, 151 Index 231

V X X-ray computed tomography, 5, 192 Variance reduction, 63, 64, 66, 176, 179, 180 Z Voxelwise signal detection, 51–53, 57 Zeeman effect, 5, 6