Pyramidal Fractal Dimension for High Resolution Images Michael Mayrhofer-Reinhartshuber1, a) and Helmut Ahammer1, b) Institute of Biophysics, Center for Physiological Medicine, Medical University of Graz, Harrachgasse 21/IV, 8010 Graz, Austria (Dated: February 2016)
Fractal analysis (FA) should be able to yield reliable and fast results for highresolution digital images to be applicable in fields that require immediate outcomes. Triggered by an efficient implementation of FA for binary images, we present three new approaches for fractal dimension (D) estimation of images that utilize image pyramids, namely the Pyramid Triangular Prism, the Pyramid Gradient, and the Pyramid Differences Method (PTPM, PGM, PDM). We evaluated the performance of the three new and five standard techniques when applied to images with sizes up to 8192 × 8192 pixels. By using artificial fractal images created by three different generator models as ground truth, we determined the scale ranges with minimum deviations between estimation and theory. All pyramidal methods (PM) resulted in reasonable D values for images of all generator models. Especially, for images with sizes ≥ 1024 × 1024 pixels, the PMs are superior to the investigated standard approaches in terms of accuracy and computation time. A measure for the possibility to differentiate images with different intrinsic D values did not only show that the PMs are well suited for all investigated image sizes, and preferable to standard methods especially for larger images, but also that results of standard D estimation techniques are strongly influenced by the image size. Fastest results were obtained with the PDM and PGM, followed by the PTPM. In terms of absolute D values best performing standard methods were magnitudes slower than the PMs. Concluding, the new PMs yield high quality results in short computation times and are therefore eligible methods for fast FA of high-resolution images.
a)
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b)
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1
For the analysis of digital images showing similar structures at different scales, fractal analysis (FA) is regularly used. Several methods to estimate the fractal dimensions D of objects in these images have been successfully applied in a variety of scientific fields. Although reasonable outcomes were obtained, there is a lack of comparison between these different methods. Furthermore, due to improvements in digital imaging techniques, resolutions increase and D estimation algorithms have not been evaluated with the focus on their applicability to high-resolution images. Nevertheless, these methods are close to or already included in end-user applications that should offer both fast and reliable results. In this study, we present three new approaches for D estimation of grey value images that utilize image pyramids, namely the Pyramid Triangular Prism, the Pyramid Gradient, and the Pyramid Differences Method (PTPM, PGM, PDM), which were triggered by an efficient implementation of FA for binary images. We evaluated the performance of five standard (Differential Box-Counting, Fourier, Triangular Prism, 2D Higuchi, 3D Root-Mean-Square Method) and the new pyramidal techniques when applied to images having sizes from 128 × 128 pixels up to 8192 × 8192 pixels, created with three different generator models (Midpoint Displacement, Fourier Filtering, Weierstrass-Mandelbrot). All proposed pyramidal methods (PM) yielded high quality results with low deviations from optimum outcomes. For images with sizes ≥ 1024 × 1024 pixels, PMs are superior to most of the investigated standard approaches in terms of yielding more accurate absolute D values. The used measure for the possibility to differentiate structures with different intrinsic D values showed, that the PMs are well suited for all investigated image sizes and preferable to standard methods, especially for larger images. Furthermore, the results suggest that standard D estimation techniques should be used carefully because of influences due to different image sizes. The fastest methods were found to be the PDM and PGM, followed by the PTPM. In terms of absolute D values best performing standard methods were magnitudes slower than the PMs. Thus, the new PMs yield fast and high quality results and can be therefore recommended for fast fractal dimension estimation of high-resolution images.
2
I.
INTRODUCTION Fractal objects showing self similarity and scale independence are mathematically well
defined and subject of countless investigations since the pioneering work of Mandelbrot1 . Real world applications use statistical approaches in order to get quantitative values for multiscale measures, such as fractal dimensions or lacunarities. Very often digital representations of objects in form of digital images are investigated, because they can easily be gained, stored, processed and analyzed. Several methods to estimate the fractal dimension D of objects in digital images have been developed and have been successfully applied in scientific fields such as medicine, geology, geography or astronomy. The most widely used algorithm is the Box-Counting Method2 (BCM), because it straightforwardly implements the measuring of object counts in dependence on the box size, represented on the pixel grid of a digital image. Other methods using e.g. Minkowski dilation or correlation can also be applied and have partly some advantages compared to the BCM3,4 . The initially for binary (b/w) images developed methods were adapted and extended for an application to grey value images during the last decades. Besides improvements of the BCM, namely the Differential Box-Counting Method (DBCM)5,6 , other methods such as the Triangular Prism Method (TPM)7,8 , the two-dimensional Higuchi Method (HM)9,10 , the Fourier Transformation Method (FM) based on frequency analysis11,12 , or the recently developed three-dimensional Root-Mean-Square Method (3D-RMSM)13 exist. A common fact of these distinct methods is, that absolute D values gained, always differ due to the algorithms’ differences and other influences such as border effects14 . They yield reasonable results and potential advantages and disadvantages have been discussed8,15 . However, a comparison of the studies is hardly possible since the proposed methods are often applied to artificial images that have different or even not specified resolutions, that are obtained from different fractal image generation models, or insufficient parameters are given to reproduce the results. Furthermore, no influences due to different image sizes have been investigated, even though it is known that border effects and hence image sizes influence the results16 . Due to the developments in digital imaging techniques, especially in the field of life sciences such as biology and medicine, resolutions and hence the amounts of data get larger and larger. Therefore, objects of different sizes (e.g. organs - tissues - cells - sub cellular compo3
nents) can be clearly differentiated in a single image. Automated diagnosis techniques often have to include data from more than one magnitude to yield reasonable results. Hence multiscale analysis, e.g. fractal analysis (FA), is often the method of choice and D values can be used as a classification parameter in many fields of medicine. For instance, in neurosciences the applications of FA to the neurons’ topological architecture were reviewed by Di Ieva et al.17 and FA was used to categorize gross and subtle differences of microglial morphology18 . Liver images obtained by ultrasonography were used to differentiate between normal tissue, cirrhosis and hepatoma19 . FA of images obtained by contrast-enhanced computed tomography showed that with D it was possible to classify aggressive and non aggressive malignant lung tumors20 . FA was used for classifying different types of cardiac tissue21,22 as well as for quantifying tortuosity in retinal images23 . Furthermore, the methodological approaches that are necessary to measure fractal structures in medical images were extensively discussed24,25 . However, potential users of future image analysis tools that include multiscale, i.e. probably FA techniques, e.g. medical doctors, expect both fast and reliable results. Nevertheless, fractal dimension estimation algorithms have not been evaluated with the focus on their applicability to high-resolution images until the present study. Therefore, the development and evaluation of efficient and also for high-resolution images fast fractal dimension estimation algorithms was highly desirable. For binary images we found that the construction of image pyramids consisting of images with different sizes is effectively equal to the popular BCM26 . A big advantage of this new pyramidal approach is the drastically reduced computation time, because the construction of smaller versions of an image can easily be accomplished by using resampling and interpolation. So far, this new pyramidal approach has been elaborated for binary images only, but obviously more information is contained in grey value images, which are hence preferred for the application of FA techniques. In this study, we present new pyramidal methods (PMs) in order to investigate grey value images with the same advantages as for the binary images. We propose three algorithms and compare them to five popular and recent techniques (DBCM, TPM, FM, HM, 3D-RMSM) by evaluating their performance on different test image sets. Particularly, three methods for the construction of test images were used in order to increase the statistical reliability and comparability of the results. Absolute D values, their variation range, and computation times of all methods were compared. Influences due to different image sizes were investigated 4
by using sizes between 128 × 128 and 8192 × 8192 pixels. It was found that the proposed PMs yield highly accurate D values especially for image sizes ≥ 1024 × 1024 pixels, and additionally are much faster than standard approaches. The rest of the paper is divided into a Methods section, shortly describing the concept of fractal dimension estimation, the most popular and recently published methods that are used for a comparison in this study, proposing the new pyramidal algorithms, and describing the generation of the test images and the evaluation procedure. Corresponding results and the discussion are presented in the respective sections.
II.
METHODS In general, the fractal dimension D of an object is defined by the Hausdorff-Besicovitch
dimension27,28 . Although this definition can be used to calculate D for mathematically defined objects, for digital images this calculation is not possible, e.g. due to limited resolution or image size. Therefore, in practice only an estimation of D is possible. Methods for practical D estimation utilize the scaling relationship of fractals, thus they all usually have some steps in common: First, each method determines measures N (r) for varying scales (resolutions) r. Then the power law of the scaling relationship that reflects the nonlinear relationship between N and r is used. This power law reads N (r) ∝ c1 rc2 D
(1)
with constants c1 and c2 . The variable D in the exponent is the fractal dimension, c1 and c2 in Eq. (1) depend on the underlying topological dimension of the investigated object and the measure used by the method. In the next step, log(N (r)) is plotted in a doublelogarithmic plot versus log(r). If the investigated image shows fractal properties, i.e. if it exhibits (statistical) self-similarity, the plot shows an approximately linear behavior, at least in a subregion of the plot, with a slope proportional to D. Therefore, a least-squares regression line is fitted through the linear part of the plotted data. Finally, the value of the regression line’s slope together with the constants of the underlying power law can be used to estimate D of the investigated image. In practice, choosing the optimum part, i.e. the optimum scale region for the fit of the data is not trivial and different approaches for selecting the data points for the fit exist, which are treated in section III B. 5
All algorithms in this work are presented for the case of an application to square images of size 2n × 2n , n ∈ N. For these image sizes, the optimized implementations of the used algorithms prevent influences in the results due to odd sizes or aspect ratios. This approach enhances comparability of the methods within this study, as well as with results of other studies that usually use square images as well.
A.
Standard Methods for Fractal Dimension Estimation As probably the most popular ones, the DBCM5 and the FM11 were included in this
work. Furthermore, the TPM7 , which outperformed several other estimation techniques8 , and the recently developed two-dimensional HM9,10 and 3D-RMSM13 were included. All of them are outlined in the following and were evaluated in the context of large image sizes. A variety of other methods for fractal dimension estimation of grey value images exist, e.g. Minkowski Blanket3,29 , Isarithm30 , Variogram31 , Probability32 , Gray Level Difference15,33 , Variation34 , and Wavelet35,36 Methods. However, the included methods are most often used in the fractal community, hence they were chosen to act as comparison methods for this work. In the following, the investigated image is denoted as I = I(x, y), has a size of M × M pixels and consists of grey values z(x, y) at positions (x, y) (x, y, M ∈ N; x ∈ [1, M ]; y ∈ [1, M ]).
Differential Box-Counting Method (DBCM) The DBCM was proposed by Sarkar and Chaudhuri5,37 . It was tested38,39 and improved40,41 successively, resulting in the currently used version6 that was also used for this work. In this version, the image I is partitioned into tiles of size s × s. On every tile i there is a box-column with boxes of size s × s × s0 , with s0 fulfilling s0 =
s 1+3σ
and σ denoting the
standard deviation of all grey values of the investigated image6 . Then the difference between maximum grey value zi,max and minimum grey value zi,min for each tile i is calculated. The number of boxes for tile i is obtained with
ns,i =
l m i,min zi,max −z , zi,max 6= zi,min 0 s 1,
zi,max = zi,min 6
(2)
with dae the smallest integer not less than a. The sum over all tiles N (s) =
X
ns,i
(3)
i
results in the total box number N (s) and is taken as the measure6,40 . To attenuate border effects, spatially adjacent tiles overlap at their boundary, i.e. the pixels at the boundary of the boxes may contribute to the ns,i of more than one tile. Considering this overlap, r = s − 1 has to be used as the scaling variable of this method6,41 . The fractal dimension DDBC is estimated from the slope kl of the linear fit of log(N (s)) versus log(1/r) with DDBC = kl 6 .
Fourier Method (FM) Utilizing frequency analysis is a widely used approach for an estimation of D11,12,42 . For the FM, the image I is transformed into frequency space (k-space) with the (discrete) Fast Fourier Transformation (FFT) I(k) = F(I(x, y)),
(4)
k = (kx , ky ). Then I(k) is used to calculate the power spectrum P (k) with P (k) = |I(k)|2 .
(5)
The usual behavior of the power spectrum for a fractal is P (k) = c|k|−β
(6)
with a constant c. The exponent β can be obtained by analyzing the power spectrum as a q function of the distance ∆k = (kx2 + ky2 ) in k-space, usually with an upper limit which is referred to as kmax . Finally, the fractal dimension DF is estimated from the slope kl of the linear fit of log(P (k)) versus log(∆k) with12,43 DF =
8 − |kl | . 2
7
(7)
Triangular Prism Method (TPM) Originally developed for the classification of topographic surfaces7 , the TPM was improved several times44,45 , and resulted in D estimations that outperform several other techniques8 . The improved version published by Lam et al.44 , which was used during this study, is briefly described in the following. In a first step, the image I is divided into tiles of size r × r. For each tile i, the four grey values of the corner pixels of this tile are used to create a new point in the middle of this tile. The z-coordinate of this point is given by the mean grey value of the four corner pixels. By using these five points, four triangles ti,j=1,2,3,4 are created, their area is denoted as A(ti,j ). The top surface area Ai of each of the tiles P i is calculated by summing up the contributions of all four triangles Ai = j A(ti,j ). As the measure for this method, the total surface area A(r) is calculated by summing over all P prisms A(r) = i Ai . The fractal dimension DTP is estimated from the slope kl of the linear fit of log(A(r)) versus log(r) with DTP = 2 − kl 44 .
2D Higuchi Method (HM) Originally developed for the investigation of one-dimensional data streams46 , the HM was recently extended to pseudo two-dimensional42 and real 2D9,10 applications. In this study, the so-called K-fold Differences Method 10 was implemented, referred to as 2D Higuchi Method (HM). For the HM, contributions from four adjacent points, given by
r zi,j = z(n + ir, m + jr) r zi−1,j = z(n + [i − 1]r, m + jr)
(8)
r = z(n + ir, m + [j − 1]r) zi,j−1 r zi−1,j−1 = z(n + [i − 1]r, m + [j − 1]r)
are used, where r is the step width (scale) and n, m are positive integers with n, m ≤ r. Differences of these contributions are multiplied with r to create areas that are summed up 8
to An,m (r), i.e. r
r
ωn ωm X1 Ωrn,m X r r r |r − zi−1,j |r + |zi,j − zi−1,j−1 (|z r An,m (r) = 2 r i=1 j=1 4 i−1,j
(9)
r r r r + |zi,j−1 − zi−1,j−1 |r + |zi,j − zi,j−1 |r)
with the normalization factor Ωrn,m , Ωrn,m =
(N − 1)2 , b N −n cb N −m cr2 r r
r b c denoting the floor function. The upper boundaries ωnr and ωm are given by N −n N −m r r ωn = , ωm = . r r
(10)
(11)
For the HM, the measure A(r) is defined as the mean value for the different values n, m by r r 1 XX An,m (r). A(r) = 2 r n=1 m=1
(12)
The fractal dimension DH is estimated from the slope kl of the linear fit of log(A(r)) versus log(r) with DH = −kl 10 .
Three-Dimensional Root-Mean-Square Method (3D-RMSM) In the very recently developed 3D-RMSM13 , the image I is partitioned into T (r) tiles of size r × r. For each tile i, that consists of points at lateral positions (x, y)i , the root-meansquare vi (r) of the grey values z(x, y)i is given by q vi (r) = [z(x, y)i − z(x, y)i ]2
(13)
with z(x, y)i as the arithmetic mean of the grey values z(x, y) in tile i. The mean value of P the contributions of all tiles V (r) = i vi (r)/T (r) is used as the measure of this method. The fractal dimension D3D−RMS is estimated from the slope kl of the linear fit of log(V (r)) versus log(r) with D3D−RMS = 3 − kl 13 .
B.
New Pyramidal Methods for Fractal Dimension Estimation In the following, all presented pyramidal approaches for fractal dimension estimation of
grey value images are new and based on common steps: 9
Let z(x, y) denote the grey value of an image I with size M × M at position (x, y) (x, y, M ∈ N; x ∈ [1, M ]; y ∈ [1, M ]). Starting from an original image I0 with size M0 × M0 and grey values stretched to z(x, y) ∈ [0, M0 − 1], an image pyramid is created with I0 as the bottom image (base) of the pyramid. The upper layer images In of the pyramid are obtained by decreasing the size of this bottom image by using different scales sn , i.e. the images In have sizes Mn × Mn with Mn = M0 /sn . This downscaling can be carried out with different interpolation algorithms, e.g. bilinear, cubic, or nearest-neighbor interpolation. Furthermore, the downscaling can be done in two different manners. While in the recursive approach each layer In of the pyramid is created from the previous layer In−1 , in the base approach each layer In is created directly from the original image I0 (base) of the pyramid. Image pyramids represent images under investigation at multiple spatial resolutions, hence they can be used to examine their fractal dimensions. In the following, three new approaches for an estimation of D based on these image pyramids are given. The first two (PTPM, PGM) are based on area measurement approaches, while the third one is based on intensity differences calculations. By downscaling an image, the values of several neighboring pixels are merged into one single pixel value. This procedure, which is repeated several times, can be seen related to block spin transformations in solid state physics, where spin values of neighboring lattice sites are merged together into one value per block47,48 .
Pyramid Triangular Prism Method (PTPM) The good results of the TPM7,8 triggered the development of a pyramidal implementation of this method, namely the Pyramid Triangular Prism Method (PTPM). In the standard TPM approach, the grey value surface area of an investigated image is estimated by using prisms with different base areas. In contrast, for the new pyramidal approach, this surface area of each In of the image pyramid is calculated with a sum over minimum sized prisms. Therefore, for every four neighboring pixels (tiles i) of the current image In , areas Ai are calculated as described in section II A. As a measure for this method, the total grey value P surface area An = i Ai of image In is used. In contrast to the TPM, this procedure is repeated for the different images In instead of different base areas of the prisms. Since the PTPM uses the same measure as the TPM, the fractal dimension DPTP of the PTPM is also estimated from the slope kl of the linear fit of log(An ) versus log(sn ) with 10
DPTP = 2 − kl .
Pyramid Gradient Method (PGM) Several standard methods as well as the proposed PTPM utilize grey value surface areas as a measure on the way to estimate D values. However, different techniques exist to calculate surface areas, e.g. by using gradients. This approach was amongst others used by Chinga et al.49 to investigate surfaces of supercalendered paper. Here, the proposed Pyramid Gradient Method (PGM) extends this application for an estimation of D. For this method, z is interpreted as a sampled version of a continuous height function of the image with a shortest lateral distance h between the positions of two known values. The surface area Ac of this function is theoretically given by s 2 2 Z Z ∂z ∂z 1+ + dxdy. Ac = ∂x ∂y
(14)
However, for a sampled version of this function, the surface area A is calculated with v !2 !2 u Xu ∂z ∂z t1 + A = h2 + . (15) ∂x ∂y (xi ,yj ) (xi ,yj ) i,j Different approximations can be used for the partial derivatives in Eq. (15). The most simple is the centered finite difference approximation that reads z(xi+1 , yj ) − z(xi−1 , yj ) ∂z ≈ ∂x (xi ,yj ) 2h ∂z z(xi , yj+1 ) − z(xi , yj−1 ) ≈ . ∂y 2h
(16)
(xi ,yj )
The derivatives have to be calculated for every image point and successively summed up with Eq. (15) to obtain the surface area. In image processing, the estimations for the derivations are usually obtained by convolving the image I with appropriate kernels. Here, the kernels kx and ky with
0 0 0 0 −1 0 1 1 kx = −1 0 1 , ky = 0 0 0 2h 2h 0 0 0 0 1 0 can be used together with
∂z ∂x
≈ kx ∗ I and
∂z ∂y
11
≈ ky ∗ I.
(17)
Instead of this rather simple kernel also the more advanced Sobel kernel50 can be used. It is common in image analysis and includes contributions of more neighboring points. Here the Sobel kernels kxSobel and kySobel read −1 0 1 kxSobel = −2 0 8h −1 0
1 −1 −2 −1 1 Sobel = 0 0 0 . 2 , ky 8h 1 1 2 1
(18)
In our case of a pyramidal approach, h has to be replaced with sn of the current image of the pyramid. After the application of the kernels to each image of the pyramid In , the surface areas An can be estimated with Eq. (15). Since the PGM uses the same measure as the PTPM, the fractal dimension DPG of the PGM is also estimated from the slope kl of the linear fit of log(An ) versus log(sn ) with DPG = 2 − kl .
Pyramid Differences Method (PDM) In addition to the two previous methods that are based on surface calculations, also the roughness of an image, i.e. the change of grey values can be used to estimate the fractal dimension D. The approach is also referred to as a fractional Brownian motion model15 . The Pyramid Differences Method (PDM) utilizes this approach and can be seen as a modification of the HM (Pseudo 2D Direct Differences 10 ), which uses mean values of grey value differences for this roughness. In the rather simple approach of the PDM, for each image In of the pyramid, the sums of the grey value differences ∆n of horizontally and vertically neighboring pixels are calculated with ∆n =
X
|z(x + 1, y) − z(x, y)| + |z(x, y + 1) − z(x, y)|.
(19)
x,y
The fractal dimension DPD is estimated from the slope kl of the linear fit of log(∆n ) versus log(sn ) with the empirically determined equation DPD = 1 − kl .
C.
Generation of Fractal Grey Value Images For an evaluation of the new PMs, we applied the previously described D estimation
methods to artificially created grey value images. Three models, i.e. generators were used 12
to create these images, namely the Midpoint Displacement (MD)43,51 , the Fourier Filtering (FF)52,53 , and the multivariate Weierstrass-Mandelbrot function (WM)53,54 method. These algorithms are suitable since they can be used to obtain images with theoretically known fractal dimensions Dt which can be then compared to the estimated values. Furthermore, the chosen methods create stochastic shaped grey value surfaces that are statistically self-affine and are therefore comparable with natural surfaces and their scale-invariance properties. Besides the used methods for the generation of fractal grey value images, other but less frequently used methods exist, e.g. the method of Takagi surfaces29,55 , Monte Carlo generated surfaces56 , or special Weierstrass-Mandelbrot functions9 .
Midpoint Displacement Method (MD) The most recent version of the repeatedly improved MD method8,43,51 starts with an image of four pixels having random grey values. Then, in recursive steps i, new pixels are introduced between the primary pixels. Their grey values are based on the values of their neighboring pixels and random values generated from a Gaussian distribution G(µ, σi2 ) with µ = 0 and σi = 1/(22Hi ). H denotes the Hurst exponent4 with 1 ≤ H ≤ 2 and is related to the theoretical fractal dimension Dt of the image with Dt = 3 − H.
(20)
The recursive procedure is stopped once the desired image size is reached.
Fourier Filtering Method (FF) For the FF method52 , in a first step an image I(x, y) with random grey values is Fourier transformed into I(k). Thereafter, its power spectrum P (k) = |I(k)|2 is modified to show a linear decay with a specific slope β. An image with a theoretical fractal dimension Dt is obtained by applying an inverse Fourier transformation with random phases to this image. The relation between β and Dt is given by Dt =
8−β . 2
(21)
By modifying the parameter β in this procedure, an image with a specific Dt can be obtained. The images created by the FF method are also referred to as fractional Brownian motion 13
(fBm) surfaces53,57 .
Weierstrass-Mandelbrot Method (WM) The two-variable WM method uses superpositions of sinusoids with amplitudes following a power law, geometrically spaced frequencies and random phases53,54 . To enhance comparability, for this study the same form and parameters were chosen as in a recent study by Zuo et al.13 , i.e. images I with grey values z(x, y) at positions (x, y) were obtained with M0 nX Dt −2 max ln(γ) X G γ (Dt −3)n z(x, y) = L L M0 m=1 n=n L 2πγ n (x2 + y 2 )1/2 y πm −1 × cos(φm,n ) − cos cos tan − + φm,n . L x M0
(22)
The used parameters were L = 7.04, G = 2.39 · 10−7 , γ = 1.5, M0 = 10, nl = −10, and nmax = 22. φm,n denotes the phase and is given by an independent random variable uniformly distributed between 0 and 2π. Detailed descriptions of these parameters can be found in the mentioned study13 . In the continuum limit the WM approach is equivalent to fBm surfaces54 .
III.
EXPERIMENTAL SETUP
In order to test and compare the performance of the new algorithms presented in II B, an empirical study based on artificially created fractal images (8-bit depth, i.e. grey values are integers from 0 to 255) was used. All algorithms for creation and fractal dimension estimation were implemented in MATLAB (R2013a, 8.1.0.604, 64 bit) in serial computing on a standard workstation (Intel Pentium G840, 2.80 GHz, 8GB RAM, Windows 7 Enterprise 64 Bit) and optimized for image sizes 2n × 2n ; n ∈ N; n > 1.
A.
Generation of Fractal Images FF, MD and WM methods were used to create images with sizes ranging from 128 × 128
up to 8192 × 8192 pixels in steps of powers of 2. The theoretical fractal dimensions Dt of the created images were in the range from 2.1 to 2.9, with a step width of ∆Dt = 0.1. Since 14
(a) MD
(b) FF
(c) WM
Dt,MD = 2.1
Dt,FF = 2.1
Dt,WM = 2.1
Dt,MD = 2.5
Dt,FF = 2.5
Dt,WM = 2.5
Dt,MD = 2.9
Dt,FF = 2.9
Dt,WM = 2.9
FIG. 1: Exemplary fractal images created with the MD (a), the FF (b), and the WM (c) method for theoretical fractal dimensions Dt = 2.1, 2.5, 2.9. the creation methods use random numbers, images with the same theoretical Dt may have different visual appearances and hence result in different estimated D values. Therefore, 50 images were created for each method, size and Dt value in order to obtain statistics (mean: D, standard deviation of the mean: SDD ) and avoid misinterpretations due to these influences. All images were saved as TIFF files with lossless LZW compression. In Fig. 1, exemplary images created with the three different generators are shown.
B.
Estimation of Fractal Dimensions For fractal dimension estimations from the double-logarithmic plots, scales of 2i , i ∈
N were used for the DBCM and TPM (i > 0), and the pyramidal approaches (i ≥ 0), respectively. The maximum value of i started from 7 for images having a size of 128 × 128 pixels and went up to 13 for images with 8192 × 8192 pixels. For the HM, measures for step widths r = [1, 2, 3, ..., 100] were calculated, for the 3D-RMSM r = [2, 3, 4, ..., 100]. For the FM, power spectrum values ∆k of the lowest 103 , 3 · 103 , 5 · 103 , and 104 distances in k-space were used. For the investigation of the new PMs, both recursive and base approaches were evaluated. Downscaling was performed with three different interpolation algorithms, namely nearest15
neighbor (value of the nearest lattice point), bilinear (linear interpolation of four closest points; first rows, then columns), and cubic (piecewise polynomial function based on a 4 × 4 neighbourhood of points58 ) interpolation. In the PGM approach, both the simple and the Sobel kernel were used. From the double-logarithmic plots, D values were estimated based on different scale ranges, that were optimized by the procedure described in the following.
Optimization For an optimization of D estimation based on the linear fits in the double-logarithmic plots, the theoretically known Dt -values were used. The optimum scale ranges for the fits were found by minimizing the root-mean-square deviation (RMSD) between theoretical (Dt = [2.1, 2.2, ..., 2.9]) and mean estimated (D(Dt )) values. As a quantification for this agreement, a mean RMSD value RMSD was calculated based on the RMSDs of the three fractal image generator models used (MD, FF, WM): s 1 X RMSD = 3 MD,FF,WM
P
Dt
(Dt − D(Dt ))2 9
(23)
As a precondition for the fits in the double-logarithmic plots, measures starting from the smallest scale and including at least four successive scales were included. The goodness of these fits can be given by the coefficients of determination R2 , with 2
R =1−
PK
2 yresid . (K − 1)var(y)
(24)
K denotes the number of included data points y = log(N (r)), var(y) the variance of y, and yresid the residual values (yresid = |y(x) − yfit (x)|) with the linearly fitted data points yfit at positions x = log(r). In the majority of publications, this goodness of fit R2 is used instead of RMSD as the criterion for obtaining the optimum and hence included scale region, or even an arbitrary or an unspecified approach is chosen. However, to measure the quality of the obtained D values, the obtained results are always compared with Dt values of the used generator models, often by calculating RMSD. Hence, the values of the used models are treated as ground truth in these studies. Therefore, the proposed approach of minimizing RMSD over 16
all generator models is justifiable, the only difference compared to other studies is that we used values from the model images as ground truth from the beginning. Optimum scale regions were evaluated for different D estimation methods and as a function of image sizes, since the resolutions may affect the obtained results, e.g. due to border effects.
Computation Times The computation times for the actual setup, i.e. all algorithms implemented in MATLAB (R2013a, 8.1.0.604, 64 bit) for serial computing on a standard workstation (Intel Pentium G840, 2.80 GHz, 8GB RAM, Windows 7 Enterprise 64 Bit), were determined as functions of image size and the used method. This included times for calculating the measures, fitting the double-logarithmic plot, and calculating the fractal dimension D from the slope of the linear regression. Mean values and standard errors of the means were obtained by using 10 different images for each method, size and Dt -value.
Mean Variation Range Exactly estimated absolute D values, i.e. D values that coincide with theoretical Dt are not always the most important outcome in FA. Since FA is often used to classify different structures in images and hence to differentiate them based on estimated D values, it is also of high relevance that D estimation methods yield reasonable differences between images showing structures with different intrinsic D values. As such a measure, usually the fractal dimension variation range is given as the difference between the estimated D of images with highest theoretical Dt (i.e. 2.9) and the estimated D of images having the lowest theoretical Dt value (i.e. 2.1). However, for a reasonable use of D estimation techniques in image analysis, it is not only important to look at the difference at lowest and highest Dt values, but also to look at the differences at smaller intervals. In this study ∆Dt = 0.1 was used, which accurate D estimation methods should be able to reproduce, i.e. the differences ∆Di = D(Dt,i+1 ) − D(Dt,i ) should fulfill ∆Di = ∆Dt for all Dt,i = 2.0 + i · 0.1, i = [1, 2, ..., 9]. Therefore, the agreement was measured with a mean RMSD value RMSDVR for the variation range, based on the RMSDs of the three fractal image generator models 17
used (MD, FF, WM): s RMSDVR
IV.
1 = 3 MD,FF,WM X
P8
i=1
(∆Di − (∆Dt ))2 8
(25)
RESULTS
The following Figs. 2-5 show the results obtained from the standard methods and PMs with optimized scale ranges. All figures show absolute D values together with the standard deviations of the means SDD , separately for three generator models (MD, FF, WM) and seven image sizes. The dashed diagonal line represents theoretical, i.e. optimum values. The optimized scale regions for all methods and image sizes are presented in Table I. As can be seen from the table, two different groups of methods exist: For one group the optimum scale range was independent of the investigated image size (TPM, PGM), for the other larger group the optimum scale range was dependent of the image size (DBCM, FM, HM, 3D-RMSM, PTPM, PDM). In the latter one, the number of measures included for the fit of the double-logarithmic plot increased for increasing image size. An exception was the FM, for which the optimum kmax became smaller before rising to higher values for larger images.
A.
Standard Methods The fractal dimensions obtained with the DBCM (Fig. 2a) show, that for all investigated
generators and image sizes, the slopes of the obtained results are less compared to theoretical values. For MD images, the estimated D values fit well for low Dt values but show increasing deviation for higher Dt values. The results for the FF and WM images start too high for small Dt and are too low for high Dt values. For higher resolution images, DBCM performs better than for smaller images. SDD values are low for all generator models, dropping below 0.02 for the largest images. The quality of the linear fit was high (R2 > 0.999) for all image sizes and generator models. The fractal dimensions obtained with the FM (Fig. 2b) show reasonable slopes and D values for MD and FF images, while the results for WM images are worse. Estimated values from higher resolution images fit better than for smaller images. SDD values are rather high for all generator models, being worst for WM images (up to 0.1). The quality of the linear 18
TABLE I: Optimized scale ranges found by minimizing RMSD for the different fractal dimension estimation methods and different image sizes (a given image width of 128 pixels corresponds to an image size of 128 × 128 pixels). Image width (pixels)
128
256
512
Method (scale variable)
1024
2048
4096
8192
Optimized scale ranges
DBCM (s)
21 − 27
21 − 28
21 − 29
21 − 210
21 − 211
21 − 212
21 − 212
FM (kmax )
3000
3000
1000
1000
1000
5000
10000
21 − 24
21 − 24
21 − 24
21 − 24
21 − 24
21 − 24
21 − 24
HM (r)
1−6
1 − 10
1 − 21
1 − 30
1 − 29
1 − 37
1 − 42
3D-RMSM (r)
2 − 11
2 − 19
2 − 25
2 − 37
2 − 53
2 − 67
2 − 96
PTPM (sn )
20 − 23
20 − 23
20 − 24
20 − 25
20 − 25
20 − 25
20 − 26
PDM (sn )
20 − 23
20 − 23
20 − 24
20 − 25
20 − 25
20 − 25
20 − 26
PGM, simple (sn )
20 − 23
20 − 23
20 − 23
20 − 23
20 − 23
20 − 24
20 − 24
PGM, Sobel (sn )
20 − 23
20 − 23
20 − 23
20 − 23
20 − 23
20 − 23
20 − 23
TPM (r)
fit was low (R2 ≈ 0.4) for all image sizes and generator models, which is a sign for the FM being strongly dependent on the maximum number of k-space values (kmax ) used for the fit. The fractal dimensions obtained with the TPM (Fig. 2c) show reasonable slopes and D values for smaller images, while for higher resolution images results are worse, especially for lower Dt values. SDD values are low for all generator models, being worst for MD images (up to 0.035). The quality of the linear fit was high, better for smaller (R2 > 0.999) than for larger image sizes (R2 > 0.96). The fractal dimensions obtained with both, HM (Fig. 3a) and 3D-RMSM (Fig. 3b), show well fitting slopes and D values for all image sizes and generator models. D values tend to flatten out for MD images with higher Dt values. SDD values are low for both methods and all generator models, dropping below 0.015 for the largest images. With HM, the quality of the linear fit was high (R2 > 0.999) for all image sizes and generator models, with 3D-RMSM the quality of the linear fit was lower (R2 > 0.93). SDD values are increased for results obtained with some standard fractal dimension estimation methods and especially for smaller Dt values (see Figs. 2-3). Hence analysis of 19
2.9 2.8
DDBC
2.7 2.6
128x128 256x256 512x512 1024x1024 2048x2048 4096x4096 8192x8192
(a)
2.9 2.8 2.7 2.6 2.5
2.4
2.4
2.3
2.3
2.2
2.2
2.1
2.1
0.05 0.04 0.03 0.02 0.01 0
0.05 0.04 0.03 0.02 0.01 0
SDD
2.5
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, MD
2.9 2.8 2.7
DF,
2.6
128x128 256x256 512x512 1024x1024 2048x2048 4096x4096 8192x8192
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, FF
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, WM
(b)
2.9 2.8 2.7 2.6 2.5
2.4
2.4
2.3
2.3
2.2
2.2
2.1
2.1
0.10 0.08 0.06 0.04 0.02 0
0.10 0.08 0.06 0.04 0.02 0
SDD
2.5
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, MD
2.9 2.8 2.7
DTP
2.6
128x128 256x256 512x512 1024x1024 2048x2048 4096x4096 8192x8192
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, FF
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, WM
(c)
2.9 2.8 2.7 2.6 2.5
2.4
2.4
2.3
2.3
2.2
2.2
2.1
2.1
0.05 0.04 0.03 0.02 0.01 0
0.05 0.04 0.03 0.02 0.01 0
SDD
2.5
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, MD
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, FF
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, WM
FIG. 2: Absolute D values and standard deviations SDD as functions of theoretical Dt values, obtained by the DBCM (a), the FM (b), and the TPM (c) for three different generator models (MD, FF, WM) and image sizes from 128 × 128 to 8192 × 8192 pixels.
20
2.9 2.8 2.7
DH
2.6
128x128 256x256 512x512 1024x1024 2048x2048 4096x4096 8192x8192
(a)
2.9 2.8 2.7 2.6 2.5
2.4
2.4
2.3
2.3
2.2
2.2
2.1
2.1
0.05 0.04 0.03 0.02 0.01 0
0.05 0.04 0.03 0.02 0.01 0
SDD
2.5
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, MD
2.9 2.8
D3D-RMS
2.7 2.6
128x128 256x256 512x512 1024x1024 2048x2048 4096x4096 8192x8192
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, FF
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, WM
(b)
2.9 2.8 2.7 2.6 2.5
2.4
2.4
2.3
2.3
2.2
2.2
2.1
2.1
0.05 0.04 0.03 0.02 0.01 0
0.05 0.04 0.03 0.02 0.01 0
SDD
2.5
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, MD
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, FF
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, WM
FIG. 3: Absolute D values and standard deviations SDD as functions of theoretical Dt values, obtained by the HM (a) and the 3D-RMSM (b) for three different generator models (MD, FF, WM) and image sizes from 128 × 128 up to 8192 × 8192 pixels. variance tests and post-hoc pairwise tests were applied to results obtained in these Dt regions. For images with sizes ≥ 1024 × 1024 pixels, the tests yielded significantly different (p < 0.05) values for all tested pairs except for the FM applied to images created with the MD and WM generator models. The FM also showed highest SDD values of all investigated standard methods.
21
B.
Pyramidal Methods From all combinations of base and recursive approaches, and the different interpolation
algorithms for downscaling of the pyramid images (see section II B), the base approach together with nearest-neighbor interpolation yielded best results. Hence these settings were used for all PMs. Every new pyramidal approach, PTPM (Fig. 4a), PDM (Fig. 4b), PGM simple (Fig. 5a), and PGM Sobel (Fig. 5b), yielded D values that correlate well with theoretical Dt values for all generator models. Estimated D values reproduce Dt values in general better for higher resolution images. The only and relatively mild exceptions are results obtained with the PTPM applied to WM images and with the PGM (simple, Sobel) applied to MD images. Furthermore, D values seem to converge towards optimum values with increasing image size, especially with the PDM. For smaller images < 512 × 512 pixels, D estimations exhibit larger deviations from Dt values, especially for the PGM and images created by FF. SDD values are low for all PMs and generator models, dropping below 0.015 for the largest images. For all proposed methods, the quality of the linear fit was high (PDM, PGM simple, PGM Sobel: R2 > 0.999; PTPM: R2 > 0.98) for all image sizes and generator models. Analysis of variance tests and post-hoc pairwise tests were applied to the results showing increased SDD values, i.e. to results from images with smaller Dt values. For images with sizes ≥ 1024×1024 pixels, the tests yielded significantly different (p < 0.05) values for all tested pairs. In Fig. 6, exemplary double-logarithmic plots of measures obtained by the new pyramidal approaches are shown for images of size 8192 × 8192 pixels and Dt = 2.1, 2.5, 2.9. Linear behavior due to the fractal character of the images and appropriate measures of the PMs is clearly observable. The linear fits used to obtain D values are depicted as straight lines, data points used for the fit are indicated by circles. The obviously linear behaviour of the measures obtained by the new PMs is an indication for the quality of the proposed approaches.
22
2.9 2.8
DPTP
2.7 2.6
128x128 256x256 512x512 1024x1024 2048x2048 4096x4096 8192x8192
(a)
2.9 2.8 2.7 2.6 2.5
2.4
2.4
2.3
2.3
2.2
2.2
2.1
2.1
0.05 0.04 0.03 0.02 0.01 0
0.05 0.04 0.03 0.02 0.01 0
SDD
2.5
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, MD
2.9 2.8 2.7
DPD
2.6
128x128 256x256 512x512 1024x1024 2048x2048 4096x4096 8192x8192
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, FF
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, WM
(b)
2.9 2.8 2.7 2.6 2.5
2.4
2.4
2.3
2.3
2.2
2.2
2.1
2.1
0.05 0.04 0.03 0.02 0.01 0
0.05 0.04 0.03 0.02 0.01 0
SDD
2.5
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, MD
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, FF
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, WM
FIG. 4: Absolute D values and standard deviations SDD as functions of theoretical Dt values, obtained by the PTPM (a) and the PDM (b) for three different generator models (MD, FF, WM) and image sizes from 128 × 128 up to 8192 × 8192 pixels.
23
2.9 2.8
DPG, simple
2.7 2.6
128x128 256x256 512x512 1024x1024 2048x2048 4096x4096 8192x8192
(a)
2.9 2.8 2.7 2.6 2.5
2.4
2.4
2.3
2.3
2.2
2.2
2.1
2.1
0.05 0.04 0.03 0.02 0.01 0
0.05 0.04 0.03 0.02 0.01 0
SDD
2.5
2.9 2.8
DPG, Sobel
2.7 2.6
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 128x128 256x256 512x512 1024x1024 2048x2048 4096x4096 8192x8192
Dt, MD (b)
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, FF
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, WM
2.9 2.8 2.7 2.6 2.5
2.4
2.4
2.3
2.3
2.2
2.2
2.1
2.1
0.05 0.04 0.03 0.02 0.01 0
0.05 0.04 0.03 0.02 0.01 0
SDD
2.5
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, MD
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, FF
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
Dt, WM
FIG. 5: Absolute D values and standard deviations SDD as functions of theoretical Dt values, obtained by the PGM (simple) (a) and the PGM (Sobel) (b) for three different generator models (MD, FF, WM) and image sizes from 128 × 128 up to 8192 × 8192 pixels.
24
log(An) log(An) log(An) log(Δn)
25 24 23 22 21 20 19 18 17 25 24 23 22 21 20 19 18 17 25 24 23 22 21 20 19 18 17 24 22 20 18 16 14 12 10 8 6 4 2
2.1 2.5 2.9 Dt measures + for lin. regression + linear regression
(I)(a) - MD, PTPM
(II)(a) - FF, PTPM
(I)(b) - MD, PGM (simple)
(II)(b) - FF, PGM (simple)
(I)(c) - MD, PGM (Sobel)
(II)(c) - FF, PGM (Sobel)
(I)(d) - MD, PDM 0
1
2
3
4
(II)(d) - FF, PDM 5
log(sn)
6
7
8
0
1
2
3
4
5
log(sn)
6
7
8
(III)(a) - WM, PTPM
25 24 23 22 21 20 19 18 17
(III)(b) - WM, PGM (simple)
25 24 23 22 21 20 19 18 17
(III)(c) - WM, PGM (Sobel)
25 24 23 22 21 20 19 18 17
(III)(d) - WM, PDM
24 22 20 18 16 14 12 10 8 6 4 2
0
1
2
3
4
5
log(sn)
6
7
8
FIG. 6: Exemplary double-logarithmic plots of the measures (An , ∆n ) of the new pyramidal approaches PTPM (a), PGM simple (b), PGM Sobel (c), and PDM (d) as a function of the scales sn (base approach, nearest-neighbor interpolation). The measures shown in the plots resulted from nine single images (8192 × 8192 pixels), created with the MD (I), FF (II), and WM (III) generators, and having theoretical fractal dimensions of Dt = 2.1 (plus, blue), Dt = 2.5 (cross, green), and Dt = 2.9 (star, red). Measures of the optimized scale regions for the linear fits (solid lines) are indicated by circles. The coefficients of determination of the linear fits were R2 > 0.98 for the PTPM (a) and R2 > 0.999 for PGM (b,c) and PDM (d).
25
22 (b) 20 18 16 14 12 10 8 6 4 2 0
non-pyramidal
new pyramidal
M
el
ob
PD
si
m
PG ple S M
M
Methods
3D HM -R M SM PT PM
TP
BC
M
108
D
105 107 106 Image size / pixels
FM
RMSD / (10-2)
PTPM DBCM PGM, simple FM TPM + PGM, Sobel PDM HM 3D-RMSM
RMSD / (10-2)
22 (a) 20 18 16 14 12 10 8 6 4 2 0 104
FIG. 7: Mean RMSD of absolute D values (RMSD) for different D estimation methods as a function of image size (a) and as a detailed plot for images with size 8192 × 8192 pixels (b).
C.
Performance Evaluation In Fig. 7, the mean RMSD of absolute D values, RMSD, for the different D estimation
methods is shown as a function of image size (a) and as a detailed plot for images of size 8192 × 8192 pixels (b). Exact values obtained for RMSD together with corresponding standard deviations can be found in Table II in the Appendix. In general, the new PMs show low RMSD values. While the PTPM performed well for all image sizes, results for PGM (simple, Sobel) and PDM are best for higher resolution images. For small image sizes, several standard methods (TPM, HM, 3D-RMSM) show similar or even lower values than the PMs. However, for larger image sizes (≥ 2048 × 2048 pixel), results for the TPM get worse and only HM and 3D-RMSM yielded results of comparable quality. DBCM and FM yielded worst absolute D and hence high RMSD values. In Fig. 8, the mean RMSD of the variation range (RMSDVR ) for the different D estimation methods is shown as a function of image size (a) and as a detailed plot for images of size 8192 × 8192 pixels (b). Exact values obtained for RMSDVR together with corresponding standard deviations can be found in Table III in the Appendix. In general, the new PMs show low RMSDVR values. For smaller image sizes, all standard methods except the DBCM are in the same region as the PMs. For higher resolution images (≥ 1024 × 1024 pixel) 26
new pyramidal
6
5 4 3 2 1
5 4 3 2
M
el
PD
ob
m
PG ple S M
PM
si
PT
SM
H M
M
-R
Methods
3D
M
0
108
TP
105 107 106 Image size / pixels
FM
1 D BC
0 104
non-pyramidal
7
M
RMSDVR / (10-2)
6
(b)
PTPM DBCM PGM, simple FM TPM + PGM, Sobel PDM HM 3D-RMSM
RMSDVR / (10-2)
(a) 7
FIG. 8: Mean RMSD of the variation range (RMSDVR ) for different D estimation methods as a function of image size (a) and as a detailed plot for images with 8192 × 8192 pixels (b).
the PMs, especially the PDM and the PGM (Sobel) performed best while the TPM again shows the worst trend for increasing image size. DBCM yielded the worst agreement with theoretical slopes of D values and hence high RMSDVR values. Fig. 9 shows computation times for the different D estimation methods with optimized scale ranges (see Table I) as a function of image size (a) and as a detailed plot for images of size 8192 × 8192 pixels (b). Fastest results were obtained with the PDM and PGM (simple, Sobel), with 2.7 s ± 0.1 s (PDM) and 3.1 s ± 0.1 s (PGM) for an image having 8192 × 8192 pixels and the described setup. Also the PTPM with 12.4 s±0.2 s, the FM with 7.8 s±0.2 s, and the TPM with 12.6 s ± 0.1 s resulted in rather low computation times while the DBCM with 320 s ± 2 s, the HM with 144 s ± 5 s, and the 3D-RMSM with 1374 s ± 3 s were slowest for an image of the largest investigated size. The advantage of the PMs in terms of computation time becomes clear when taking a look at the computational effort necessary P for each method. All PMs are based on calculations of the order N i=1 O(ni ) with N scale values i, n0 the number of pixels of the original image, and ni = n0 /2i the number of pixels of an image at scale i. Effort for the TPM is of the same order, hence the resulting times were similar. Also the FM yielded results in quite fast computation times since it needs only one run with a slightly larger effort of O(n0 log(n0 )). DBCM, HM and 3D-RMSM are based on algorithms requiring similar efforts for each scale (N O(n0 )), hence they are slower. 27
DBCM FM TPM HM
3D-RMSM PTPM PGM PDM
(a)
1000
new pyramidal
M
ob el
PD
m
PG ple S M
PM
M
SM
si
PT
M
-R
Methods
3D
0.01
108
D
107 106 Image size / pixels
0.1
M
0.1
1
H
1
10
FM
10
TP
Computation time / s
Computation time / s
non-pyramidal
100
100
0.01 5 10
(b)
BC M
1000
FIG. 9: Computation times for different D estimation methods as a function of image size (a) and as a detailed plot for images with 8192 × 8192 pixels (b). All methods implemented in MATLAB (R2013a, 8.1.0.604, 64 bit) in serial computing on a standard workstation (Intel Pentium G840, 2.80 GHz, 8GB RAM, Windows 7 Enterprise 64 Bit). Differences in computation times of methods with similar computational efforts arise from different numbers of functions necessary for the different algorithms.
28
(a) control
(b) fibrosis
FIG. 10: Exemplary images for the control group (a) and the fibrosis group (b) that were used for the demonstration of the applicability of the new pyramidal methods.
V.
APPLICATION TO HISTOLOGICAL IMAGES In the following, we give a real-world example to illustrate the power of the new pyramidal
methods, especially in terms of computation time. A possible application of fractal analysis is the investigation of different pathologies observable in medical images. Histological samples can be digitalized by using whole-slide scanners and typically result in high-resolution images. We applied the PDM and the PGM (Sobel) to histological images showing parts of the human interventricular septum of patients that suffered from ischemic heart disease. These images were used in a previous study22 , where further details can be found. For comparison, we also applied the DBCM and the HM to these images. The first group included five images showing healthy myocardial tissue (control group, see Fig. 10(a)), the second group included five images showing enhanced myocardial fibrosis (fibrosis group, see Fig. 10(b)). All images had a resolution of 4096 × 4096 pixels. Since the tissue was stained with Masson’s trichrome, i.e. fibrotic tissue was stained blue, all D estimation methods were applied to the blue channel of the images only. For all methods, optimum scale ranges were used (see Table I). The pyramidal methods were applied in the base approach with nearest-neighbor interpolation. Mean values for the estimated fractal dimension of the control (Dcon ) and the fibrosis group (Dfib ) and corresponding standard deviations of the means were calculated. The PDM yielded Dcon = 2.24±0.05 and Dfib = 2.38±0.07, with a total computation time of 17 s for all ten images. Dcon = 2.27 ± 0.06 and Dfib = 2.46 ± 0.11 were obtained with the PGM (Sobel), with a computation time of 21 s. The DBCM resulted in Dcon = 2.83 ± 0.02 and Dfib = 2.84 ± 0.03 in 815 s. HM yielded Dcon = 2.30 ± 0.05 and Dfib = 2.42 ± 0.07 in 29
410 s. The results of all applied methods show, although their different absolute values, the same tendency of images having smaller fractal dimensions for the control group compared to the fibrosis group. Student’s t-tests showed statistically significant differences of Dcon and Dfib obtained with the PDM, the PGM, and the HM (p < 0.05). Differences of results obtained with the DBCM results were non-significant (p > 0.28). However, due to the small number of images in each group, these results should be treated with care. While a more extensive study with a larger image set is necessary for a valueable conclusion in terms of fibrosis classification, this example clearly shows a possible real-world application and the advantage of the new pyramidal methods in terms of computation time.
30
VI.
DISCUSSION AND CONCLUSION
In this study, we evaluated the performance of three new pyramidal (PTPM, PDM, PGM) and five standard (DBCM, FM, TPM, HM, 3D-RMSM) fractal dimension estimation techniques when applied to grey value images with sizes between 128 × 128 and 8192 × 8192 pixels and theoretical fractal dimensions Dt = [2.1, 2.2, ..., 2.9]. We used artificially created fractal images obtained from three different models (MD, FF, WM) as ground truth (50 images per size, Dt , and model) to find optimum scale ranges for all investigated methods, i.e. the scale ranges that yielded a minimum RMSD between estimated D and theoretical Dt values, averaged over all generator models. The measures of the proposed methods show clearly linear behavior in the doublelogarithmic plots with coefficients of determination R2 > 0.999 (PDM,PGM) and R2 > 0.988 (PTPM), which is an indication for the quality of the proposed approaches. Based on the optimum scale ranges found for each method (Table I), two groups of methods were identified, one with optimum scale ranges being independent of the investigated image size (TPM, PGM) and a larger group being dependent on the image size (DBCM, FM, HM, 3D-RMSM, PTPM, PDM). For the methods included in the latter one, the number of measures to be included in the fit of the double-logarithmic plot, i.e. the number of measures from different scales that had to be included, had to be higher for larger images to obtain best results. The found values may act as an indication for the number of scale ranges that have to be included in future fractal analysis applications, especially when different methods should be compared. Nevertheless, in a particular task the appearance of the double-logarithmic plot should always be taken into account. All proposed PMs (PTPM, PGM, PDM) in the base approach, i.e. where all layers of the image pyramid were created directly from the original image, together with nearest-neighbor interpolation, yielded reasonable D values for images of all generator models. Especially for larger images, mean RMSD values (see Fig.7, Table II) show that the PMs are superior to most of the investigated standard approaches (DBCM, FM, TPM) in terms of yielding more accurate absolute D values, only the HM and the 3D-RMSM yielded comparable results. For images of sizes ≥ 1024 × 1024 pixels, best results were obtained with the PDM and the PGM (Sobel), results of the PDM seem to converge towards optimum values with increasing image size. Also SDD values were low for these methods, showing similar values as obtained 31
with the best standard methods. Interestingly, the popular DBCM and FM yielded worst results, especially the FM also suffered from high SDD values and is strongly dependent on the used parameters. They are regularly used in image analysis studies as well as for comparison for new fractal methods10,15,19–21 , but should be treated and used carefully in future applications. The good results for the TPM obtained in previous studies8,13 , could only be reproduced for images with sizes up to 1024 × 1024 pixels. TPM’s drawback of yielding far too low D values for low Dt regions, which was already shown in previous studies8,44 , became worse for larger image sizes. Hence, the TPM is not an adequate choice for larger images, especially if the aim is to differentiate between structures in images having low intrinsic D values. HM and 3D-RMSM resulted in accurate D values that slightly flattened out in high Dt regions. In addition to the calculation of RMSD values, which quantify the agreement between theoretical Dt and estimated D values, analysis of variance and post-hoc pairwise tests were applied to results showing regions with increased SDD values. The results of these tests indicate that also images with different theoretical fractal dimensions in these regions could be statistically reliably separated from each other with all new PMs and all investigated standard methods except with the FM. By evaluating the mean RMSD of the variation range (RMSDVR , see Fig. 8, Table III), it was shown that the PMs yield rather accurate variation ranges throughout the whole region of Dt values. Again the TPM showed a large dependency on the image size and was better for smaller images ≤ 1024 × 1024 pixels. This found dependency exemplifies the importance of investigating the influences due to different image sizes, and hence the importance of the present study. In terms of computation times (see Fig. 9), fastest results were obtained with the PDM and PGM, which were about the factor of 4× faster than the PTPM. From the standard methods, FM was the fastest, still about the factor of 2 − 3× slower than PDM and PGM. All other methods were slower than the PMs (TPM ≈ 4×, HM ≈ 50×, DBCM ≈ 100×, 3DRMSM ≈ 450× slower than the PDM). In absolute numbers, for an image size of 8192×8192 pixels and the used hardware/software setup, the fastest PDM yields a D estimation value in 2.7 s ± 0.1 s, the second placed PGM in 3.1 s ± 0.1 s. By using smaller scale ranges or non-consecutive scale values, computation times of standard as well as pyramidal methods could be reduced further, however, this would result in worse absolute D values or would negatively affect the robustness of the methods. An advantage of computation time of 32
pyramidal approaches was also observed with the binary pyramidal method when compared to the binary BCM26 . Since FA is often used for multiscale analysis of high-resolution images, e.g. in medicine, these results propose the application of the new PMs whenever fast results are required. We demonstrated the applicability and power of the new PMs in such a real-world example, in which the methods were applied to images showing myocardial fibrosis or healthy tissue and showed their advantage in computation time. GPU and parallel computing may be able to decrease computation times of the PMs even further since the concepts of their algorithms, i.e. the downscaling of the images in the base approach and the calculation of the corresponding measures, seem well suited for these future enhancements. However, also the presented standard approaches may get faster since the calculation of their measures can be parallelized too. A possible limitation of the proposed PMs is their intrinsic isotropy, i.e. they yield a single D value for the whole two-dimensional grey value image. In practice there exist also FA applications, in which one is e.g. interested in D values of anisotropic surfaces, i.e. different D values for different directions. However, existing algorithms like the one-dimensional HM59 cover this area and the present study did not aim to improve these approaches. A possible limitation of the study is the fact, that only square images were used to test the algorithms. Since in most cases of FA of grey value images, it is possible to choose square (sub)regions of the image for analysis, possible influences due to an application to rectangular images seem to be of minor importance and were therefore beyond the scope of this work. All presented fractal methods (standard and new pyramidal) can rather easily be adapted to handle images with different aspect ratios. However, the PMs have the advantage of not suffering from restrictions like e.g. the DBCM or the TPM with their fixed box sizes or base areas, with which an image has to be covered as good as possible. A future study on grey value images may elaborate possible influences due to these different aspect ratios. For binary images it was shown that these influences are rather small26 . A further limitation might be seen in the artificially images generated and interpreted as ground truth, i.e. representations of fractals with well known dimension. Due to the finite size and discrete grid of digital images, it is not possible to represent self-similar structures until infinity, hence the images itself have to be seen as approximations and exact theoretical values cannot be expected. However, we included images from three different generator models and optimized and evaluated the included methods for all of them simultaneously. 33
This is an advantage to other studies, where often images of just a single generator model are included or FA methods are evaluated separately for images of different generators6,9,12,13,15 . Concluding, it can be stated that the new PMs yield high quality results in short computation times and are therefore well suited for fast fractal dimension estimation of images, especially with sizes ≥ 1024 × 1024 pixels. Due to the presented advantages of the new PMs, they were also included into IQM, a completely free and open source image and signal analysis software that was published by our group recently60 .
ACKNOWLEDGMENTS The authors wish to thank Dr. D. S´anchez-Quintana, from the Department of Anatomy and Cell Biology, University of Extremadura, Spain for having kindly provided the histological images. We further acknowledge K. Zorn-Pauly, Institute of Biophysics, Medical University of Graz, Austria for fruitful discussions and helpful guidance regarding statistics.
34
APPENDIX: EXACT RMSD AND RMSDVR VALUES
TABLE II: RMSD of D (mean absolute D values) for the different fractal dimension estimation methods (optimized scale ranges as given in Table I) and different image sizes (a given image width of 128 pixels corresponds to an image size of 128 × 128 pixels). Best, i.e. minimum values for each image size are indicated by boldfaced numbers. Image width (pixels)
128
256
512
1024
2048
4096
8192
RMSD (10−2 )
Method DBCM
20.1 ± 3.7 17.6 ± 3.6 16.1 ± 2.8 14.8 ± 2.4 14.1 ± 1.6 13.2 ± 1.2 13.9 ± 0.7
FM
18.7 ± 9.0 17.0 ± 9.9 15.6 ± 2.6 23.2 ± 9.8 14.5 ± 1.4 9.3 ± 1.0 9.8 ± 2.9
TPM
9.4 ± 3.8 5.2 ± 3.2 5.3 ± 2.3 7.1 ± 2.2 9.3 ± 2.8 11.8 ± 3.7 14.4 ± 4.8
HM
8.8 ± 3.2 7.7 ± 2.5 6.3 ± 1.5 5.9 ± 0.9 5.8 ± 0.9 5.3 ± 0.7 5.0 ± 0.5
3D-RMSM
8.9 ± 3.1 7.8 ± 2.4 7.1 ± 2.0 6.7 ± 1.6 6.8 ± 1.1 7.0 ± 1.1 7.1 ± 0.6
PTPM
8.0 ± 3.9 6.3 ± 2.7 5.4 ± 1.6 5.5 ± 1.3 5.8 ± 0.9 6.2 ± 0.9 6.2 ± 0.7
PDM
11.8 ± 2.6 8.5 ± 2.0 7.1 ± 0.9 5.9 ± 1.0 5.3 ± 1.1 5.0 ± 1.3 4.5 ± 1.1
PGM, simple
12.5 ± 5.6 8.3 ± 3.8 6.3 ± 2.4 5.6 ± 1.5 5.7 ± 0.9 5.9 ± 0.6 6.0 ± 0.4
PGM, Sobel
13.0 ± 6.6 8.8 ± 4.8 6.5 ± 3.2 5.2 ± 1.9 4.9 ± 1.0 4.9 ± 0.3 5.1 ± 0.2
35
TABLE III: RMSDVR of obtained variation range for the different fractal dimension estimation methods (optimized scale ranges as given in Table I) and different image sizes (a given image width of 128 pixels corresponds to an image size of 128 × 128 pixels). Best, i.e. minimum values for each image size are indicated by boldfaced numbers. Image width (pixels)
128
256
512
1024
2048
4096
8192
RMSDVR (10−2 )
Method DBCM
5.4 ± 1.0 5.2 ± 0.8 5.0 ± 0.6 4.9 ± 0.6 4.8 ± 0.6 4.7 ± 0.7 4.5 ± 0.7
FM
3.7 ± 2.8 1.9 ± 0.9 1.7 ± 0.1 2.7 ± 0.7 3.0 ± 1.1 2.5 ± 1.0 2.7 ± 1.8
TPM
2.6 ± 1.3 2.4 ± 1.0 2.4 ± 0.7 2.6 ± 0.5 2.9 ± 0.6 3.2 ± 0.8 3.8 ± 1.1
HM
3.4 ± 1.2 3.1 ± 0.9 2.7 ± 0.6 2.5 ± 0.4 2.5 ± 0.4 2.3 ± 0.3 2.2 ± 0.2
3D-RMSM
3.6 ± 1.1 3.3 ± 0.9 3.1 ± 0.7 2.9 ± 0.6 2.8 ± 0.4 2.8 ± 0.4 2.8 ± 0.3
PTPM
2.9 ± 1.1 2.5 ± 0.9 2.3 ± 0.6 2.3 ± 0.5 2.3 ± 0.5 2.3 ± 0.5 2.2 ± 0.5
PDM
2.9 ± 1.1 2.7 ± 0.9 2.2 ± 0.7 2.1 ± 0.6 2.0 ± 0.5 1.9 ± 0.6 1.8 ± 0.5
PGM, simple
3.3 ± 1.1 2.6 ± 0.9 2.3 ± 0.7 2.2 ± 0.6 2.3 ± 0.4 2.2 ± 0.4 2.1 ± 0.4
PGM, Sobel
3.7 ± 1.0 2.7 ± 0.8 2.2 ± 0.6 2.0 ± 0.4 2.0 ± 0.3 1.9 ± 0.2 1.9 ± 0.2
36
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