CN107909099A - A kind of threedimensional model identification and search method based on thermonuclear - Google Patents

Abstract

The invention discloses a kind of threedimensional model identification based on thermonuclear and search method, including step, the thermonuclear feature unrelated with size of threedimensional model is extracted；Thermonuclear feature is improved, makes thermonuclear feature that there is change of scale consistency, obtains improved thermonuclear feature NSI HKS (x)；The amplitude for choosing second to the 6th low frequency component of NSI HKS (x) is sampled as local feature；K mean cluster processing is carried out to the thermonuclear feature of sampling, obtains the cluster belonging to each data object；Threedimensional model similarity is measured, threedimensional model is identified and is retrieved.The beneficial effects of the invention are as follows：Utilize thermonuclear feature, carry out change of scale consistency improvement, then the selection to data sample is improved, then k average value processings are carried out, feature between different models is compared, finally use based on bright Cowes cardinal distance comparative approach with a distance from, improve identification and effectiveness of retrieval and accuracy rate.

Description

Three-dimensional model identification and retrieval method based on thermonuclear

Technical Field

The invention relates to the technical field of computers, in particular to a three-dimensional model identification and retrieval method based on thermonuclear.

Background

With the development of computer technology and the improvement of computer hardware technology, the technology for acquiring three-dimensional models is rapidly developed. Three-dimensional models have not only grown dramatically in number, but they are also becoming more and more widely used. The main application fields include various aspects of industrial product design, virtual reality, three-dimensional games, building design, movie animation, medical diagnosis, molecular biological research and the like. Due to the rapid increase of the application demand of three-dimensional models, more and more three-dimensional model libraries, such as industrial solid model libraries, three-dimensional game model libraries, building model libraries, vehicle model libraries, protein molecule model libraries and the like, are generated. A large number of industries use three-dimensional models, and a large amount of time and energy are consumed for creating a high-fidelity three-dimensional model. Sometimes, the existing three-dimensional model can be used only by locally modifying, and statistics shows that more than 85% of new products are updated and modified on the basis of original products. It is very important to effectively manage these massive model information, so as to facilitate retrieval, query and reuse. Therefore, the rapid identification and retrieval of the three-dimensional model become an urgent problem to be solved.

Disclosure of Invention

In order to solve the above problems, the present invention provides a thermonuclear-based three-dimensional model identification and retrieval method, comprising the steps of:

s100) extracting the size-independent thermonuclear characteristics of the three-dimensional model, and expressing the thermonuclear characteristics of the three-dimensional model as a function HKS (x) in a time domain:

wherein λ_iAnd phi_i(x) The ith eigenvalue and the characteristic function of the Laplace-Beltrami operator of the shape;

s200) improving the thermonuclear characteristics to enable the thermonuclear characteristics to have scale transformation invariance, so as to obtain improved thermonuclear characteristics NSI-HKS (x);

s300) selecting amplitudes of second to sixth low-frequency components of NSI-HKS (x) as local features for sampling;

s400) carrying out K-means clustering processing on the sampled thermonuclear characteristics to obtain a cluster to which each data object belongs;

s500) measuring the similarity of the three-dimensional model, and identifying and retrieving the three-dimensional model.

Preferably, the step of extracting size-independent thermonuclear features of the three-dimensional model comprises:

s110) calculating Laplace-beltrami operator delta according to a formula_X＝A^-1W, wherein A and W are an area normalization matrix and a cosine weight matrix respectively;

s120) carrying out characteristic decomposition on the Laplace-beltrami operator to obtain lambda_iIs phi_iRespectively, the ith eigenvalue and eigenvector.

Preferably, the step of improving the thermonuclear characteristics comprises:

s210) for each point x on the model, α with a time t^τThe thermal characteristics are sampled, and the discrete function is shown as formula (1):

h_τ＝h(x,α^τ) (1)

the scaling β of the S220 model is converted into a time shift S of 2log_αβ and amplitude scaling β²As shown in formula (2):

h′_τ＝β²h_τ+s(2)；

s230) h takes a logarithmic form, and then takes a discrete derivative to eliminate the constant β²As shown in formula (3):

wherein,

namely, it is

S240) pairsPerforming discrete-time Fourier transform, as shown in equation (4):

H′(ω)＝H(ω)e^2πωs(4)

wherein H and H' are each independentlyAndis the Fourier transform of (a), omega ∈ [0,2 π ∈]

S250) then eliminate e by taking the modulus^2πωsThat is, | H' (ω) | H (ω) |, and | H (ω) | is denoted as NSI-hks (x).

Preferably, τ is selected as τ e [ τ ] in the characteristic function NSI-HKS (x)_min,τ_max]The amplitudes of the second to sixth low frequency components of NSI-HKS (x) are sampled as local features.

Preferably, the K-means clustering process includes the steps of:

s410) randomly selecting K objects as initial clustering centers;

s420) calculating the distance between each object and each seed cluster center, and allocating each object to the cluster center closest to the object, wherein the cluster center and the objects allocated to the cluster center represent a cluster;

s430) after all the objects are distributed, the distribution center of each cluster is recalculated according to the existing objects in the cluster;

s440) continuously repeating the steps S420 and S430 until the data members of each cluster are not changed;

s450) obtaining the cluster to which each data object belongs.

Preferably, the value of K in the K-means clustering process is 60.

Preferably, the similarity measure method comprises a Minicosky distance comparison method.

Preferably, the formula of the Minkowski distance is as follows:

preferably, the step of measuring the similarity includes:

s510) comparing the distance between the first point of the model A and all the points of the model B, and selecting one group with the minimum distance for matching;

s520, comparing the Minicossian distances between the second point of the model A and all points of the model B (except matched points), and selecting one group with the minimum distance for matching until all 60 points are matched;

s530) calculating the average value of the distances among all matched points, wherein the smaller the average value is, the more similar the samples are, and the smaller the difference is; the larger the average, the more dissimilar the samples, and the greater the degree of difference.

The invention has the beneficial effects that: the invention provides a three-dimensional model identification and retrieval method based on thermonuclear, which is characterized in that the thermonuclear characteristics are utilized to improve the invariance of scale transformation, then the acquisition of data samples is improved, then k-means processing is carried out, so that the characteristics among different models can be compared, and finally, a distance comparison method based on the Minicos distance is adopted, so that the efficiency and the accuracy of identification and retrieval are improved.

Drawings

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without inventive exercise.

FIG. 1 is a flow chart of a method for thermonuclear-based three-dimensional model identification and retrieval;

FIG. 2 is a flow chart illustrating the extraction of size-independent thermonuclear features of a three-dimensional model;

FIG. 3 is a flow diagram of an improved scale-independent method of processing thermonuclear features;

FIG. 4 is a flow chart of the K-means clustering process;

fig. 5 is a flowchart illustrating the steps of the similarity measure.

Detailed Description

The conception, the specific structure, and the technical effects produced by the present invention will be clearly and completely described below in conjunction with the embodiments and the accompanying drawings to fully understand the objects, the features, and the effects of the present invention. It is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all embodiments, and those skilled in the art can obtain other embodiments without inventive effort based on the embodiments of the present invention, and all embodiments are within the protection scope of the present invention. All technical characteristics in the invention can be interactively combined on the premise of not conflicting with each other.

Fig. 1 is a flowchart illustrating a method for identifying and retrieving a three-dimensional model based on thermonuclear according to an embodiment of the present invention, where the method comprises the following steps:

s100) extracting the size-independent thermonuclear characteristics of the three-dimensional model, and expressing the thermonuclear characteristics of the three-dimensional model as a function HKS (x) in a time domain:

wherein λ_iAnd phi_i(x) The ith eigenvalue and the characteristic function of the Laplace-Beltrami operator of the shape;

s200) improving the thermonuclear characteristics to enable the thermonuclear characteristics to have scale transformation invariance, so as to obtain improved thermonuclear characteristics NSI-HKS (x);

s300) selecting amplitudes of second to sixth low-frequency components of NSI-HKS (x) as local features for sampling;

s400) carrying out K-means clustering processing on the sampled thermonuclear characteristics to obtain a cluster to which each data object belongs;

s500) measuring the similarity of the three-dimensional model, and identifying and retrieving the three-dimensional model.

FIG. 2 is a flow diagram illustrating the extraction of size-independent thermonuclear features of a three-dimensional model, according to one embodiment of the invention, comprising the steps of:

s110) calculating Laplace-beltrami operator delta according to a formula_X＝A^-1W, wherein A and W are an area normalization matrix and a cosine weight matrix respectively;

s120) carrying out characteristic decomposition on the Laplace-beltrami operator to obtain lambda_iIs phi_iRespectively, the ith eigenvalue and eigenvector.

FIG. 3 is a flow diagram of an improved scale-independent thermonuclear feature processing method, according to a method embodiment of the invention, comprising the steps of:

s210) for each point x on the model, α with a time t^τThe thermal characteristics are sampled, and the discrete function is shown as formula (1):

h_τ＝h(x,α^τ) (1)

s220) the scaling β of the model is converted into a time shift S of 2log_αβ and amplitude scaling β²As shown in formula (2):

h_τ′＝β²h_τ+s(2)；

s230) h takes a logarithmic form, and then takes a discrete derivative to eliminate the constant β²As shown in formula (3):

wherein,

namely, it is

S240) pairsPerforming discrete-time Fourier transform, as shown in equation (4):

H′(ω)＝H(ω)e^2πωs(4)

wherein H and H' are each independentlyAndis the Fourier transform of (a), omega ∈ [0,2 π ∈]

S250) then eliminate e by taking the modulus^2πωsThat is, | H' (ω) | H (ω) |, and | H (ω) | is denoted as NSI-hks (x).

Fig. 4 is a flowchart illustrating the K-means clustering process, according to an embodiment of the present invention, including the steps of:

s410) randomly selecting K objects as initial clustering centers;

s420) calculating the distance between each object and each seed cluster center, and allocating each object to the cluster center closest to the object, wherein the cluster center and the objects allocated to the cluster center represent a cluster;

s430) after all the objects are distributed, the distribution center of each cluster is recalculated according to the existing objects in the cluster;

s440) continuously repeating the steps S420 and S430 until the data members of each cluster are not changed;

s450) obtaining the cluster to which each data object belongs.

Fig. 5 is a flowchart illustrating the steps of the similarity measure, according to an embodiment of the present invention, the steps include:

s510) comparing the distance between the first point of the model A and all the points of the model B, and selecting one group with the minimum distance for matching;

s520) comparing the Minicosky distance between the second point of the model A and all the points of the model B (except the matched points), and selecting one group with the minimum distance for matching till 60 points are all matched;

s530) calculating the average value of the distances among all matched points, wherein the smaller the average value is, the more similar the samples are, and the smaller the difference is; the larger the average, the more dissimilar the samples, and the greater the degree of difference.

According to one embodiment of the present invention, the thermonuclear feature is further described below, and for a compact riemann fluid (possibly with boundaries) M, the heat diffusion process at M depends on thermal equation (5):

wherein Δ_MIs on MLaplace-beltrami operator. If M is a bounded riemann fluid, u is also required to satisfy the dirichlet boundary condition, i.e., u (x, t) ═ 0 is required to satisfy all points x on the fluid M for all times t. Given an initial heat distribution f: M → R, let H_t(f) Represents the heat distribution of the fluid M at time t, and therefore H_t(f) Thermal equation (6) is satisfied for all times t, and:

wherein H_tIs a thermal operator. H_tAnd Δ_MAre both real function operators defined on the fluid M, so it is easy to prove that the two operators satisfy the relationSince the two operators share the same eigenfunction, if λ is Δ_MOne of the characteristic values, then e^-λtIs H_tOne of the characteristic values. For an arbitrary Riemann fluid M, there is a function k_t(x,y):R⁺XM M → R is such that:

where dy is the convolution form of y ∈ M. Minimum value k satisfying formula (7)_t(x, t) is the thermal kernel and can be viewed as the amount of heat conducted from point x to point y in a given time t. That is to say k_t(x,·)＝H_t(δ_x) Wherein δ_xIs a function of δ (dirac), i.e. δ is given for any z ≠ x_x(z) 0 andfor a compact fluid M, the thermal nuclei decompose with the following characteristics:

thermal nucleation can also be understood as a transfer density function of brownian motion on the riemann fluid M, which means for any polaier subset of the riemann fluid Mthe transition probability of the brownian motion starting from point x at time t isBrownian motionIs the most basic continuous-time Markov process of Riemann fluid M, which well explains the abundant characteristic information of the model contained in the thermonuclear. Since the transition probability of brownian motion is related not only to the shortest path between two points, but also to the weighted average of all possible paths at time t, the thermokernel contains more information than the shortest distance between two points. The thermonuclear contains a lot of redundant information because the thermal diffusion process depends on the thermal equation (1), which means that the characteristic equation of the spatial domain varies with time. To overcome the above difficulties, Sun et al propose a thermonuclear signature (HKS) that discards all information of the spatial domain of the thermonuclear. For a point x on the riemann fluid M, the thermonuclear signature is defined as a function hks (x) over a time domain:

wherein λ_iAnd phi_i(x) The ith eigenvalue and eigenfunction of the Laplace-Beltrami operator for that shape.

According to another embodiment of the present invention, most of the signal information is contained in the low frequency part after fourier transform, so a compact local descriptor is created by sampling | H (ω) | low frequency, and therefore the amplitudes of the first six low frequency components are taken as local features.

According to another embodiment of the present invention, different choices of the time parameter affect the characteristics of the models of different scales, and in order to better adapt the time parameter to the models of different scales, the time parameter, i.e., τ, is determined from the median value of the semi-sphere radii_min＝floor(lbt_min)，τ_max＝ceil(lbt_max) When τ > lbt_maxThen HKS will not change, and for models of different dimensions, t_maxIn contrast, the larger the scale of the model, t_maxThe larger.

According to another embodiment of the present invention, further describing the K-means clustering algorithm, the K-means clustering algorithm is to randomly select K objects as initial clustering centers, then calculate the distance between each object and each seed clustering center, and assign each object to the nearest clustering center. The cluster centers and the objects assigned to them represent a cluster. Once all objects are assigned, the cluster center for each cluster is recalculated based on the objects existing in the cluster. This process will be repeated until some termination condition is met. The termination condition may be that no (or minimum number) objects are reassigned to different clusters, no (or minimum number) cluster centers are changed again, and the sum of squared errors is locally minimal.

The similarity metric is further illustrated according to one embodiment of the present invention, assuming that a given dataset is X ═ X_mWhere m is 1,2, …, total, and the samples in X describe the attribute a with d numbers₁,A₂,…,A_dAnd d description attributes are all continuity attributes. Data sample x_i＝(x_i1,x_i2,…,x_id)，x_j＝(x_j1,x_j2,…,x_jd). Wherein x_i1,x_i2,…,x_idAnd x_j1,x_j2,…,x_jdAre respectively a sample x_iAnd x_jCorresponding d description attributes A₁,A₂,…,A_dThe specific value of (a). Sample x_iAnd x_jThe similarity between them is usually determined by the distance d (x) between them_i,x_j) To show that the smaller the distance, the more similar the sample is, and the smaller the difference is; conversely, the more dissimilar, the greater the degree of difference.

While the present invention has been described in considerable detail and with particular reference to a few illustrative embodiments thereof, it is not intended to be limited to any such details or embodiments or any particular embodiments, but it is to be construed as effectively covering the intended scope of the invention by providing a broad, potential interpretation of such claims in view of the prior art with reference to the appended claims. Furthermore, the foregoing describes the invention in terms of embodiments foreseen by the inventor for which an enabling description was available, notwithstanding that insubstantial modifications of the invention, not presently foreseen, may nonetheless represent equivalent modifications thereto.

Claims (9)

1. A three-dimensional model identification and retrieval method based on thermonuclear is characterized by comprising the following steps:

s100) extracting the size-independent thermonuclear characteristics of the three-dimensional model, and expressing the thermonuclear characteristics of the three-dimensional model as a function HKS (x) in a time domain:

<mrow> <mi>H</mi> <mi>K</mi> <mi>S</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>:</mo> <msup> <mi>R</mi> <mo>+</mo> </msup> <mo>&amp;RightArrow;</mo> <mi>R</mi> <mo>,</mo> <mi>H</mi> <mi>K</mi> <mi>S</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>k</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&amp;infin;</mi> </munderover> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>&amp;lambda;</mi> <mi>i</mi> </msub> <mi>t</mi> </mrow> </msup> <msup> <msub> <mi>&amp;phi;</mi> <mi>i</mi> </msub> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow>

wherein λ_iAnd phi_i(x) The ith eigenvalue and the characteristic function of the Laplace-Beltrami operator of the shape;

s200) improving the thermonuclear characteristics to enable the thermonuclear characteristics to have scale transformation invariance, so as to obtain improved thermonuclear characteristics NSI-HKS (x);

s300) selecting amplitudes of second to sixth low-frequency components of NSI-HKS (x) as local features for sampling;

s400) carrying out K-means clustering processing on the sampled thermonuclear characteristics to obtain a cluster to which each data object belongs;

s500) measuring the similarity of the three-dimensional model, and identifying and retrieving the three-dimensional model.

2. The thermonuclear-based three-dimensional model identification and retrieval method of claim 1, wherein said step of extracting size-independent thermonuclear features of the three-dimensional model comprises:

s110) calculating Laplace-beltrami operator delta according to a formula_X＝A^-1W, wherein A and W are an area normalization matrix and a cosine weight matrix respectively;

s120) carrying out characteristic decomposition on the Laplace-beltrami operator to obtain lambda_iIs phi_iRespectively, the ith eigenvalue and eigenvector.

3. The thermonuclear-based three-dimensional model identification and retrieval method of claim 1, wherein the step of improving thermonuclear characteristics comprises:

s210) for each point x on the model, α with a time t^τThe thermal characteristics are sampled, and the discrete function is shown as formula (1):

h_τ＝h(x,α^τ) (1)

the scaling β of the S220 model is converted into a time shift S of 2log_αβ and amplitude scaling β²As shown in formula (2):

h′_τ＝β²h_τ+s(2)；

s230) h takes a logarithmic form, and then takes a discrete derivative to eliminate the constant β²As shown in formula (3):

<mrow> <msubsup> <mover> <mi>h</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>&amp;tau;</mi> <mo>&amp;prime;</mo> </msubsup> <mo>=</mo> <msub> <mover> <mi>h</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mi>&amp;tau;</mi> <mo>+</mo> <mi>s</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

wherein,

namely, it is

<mrow> <mover> <mi>h</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>&amp;alpha;</mi> <mi>&amp;tau;</mi> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&amp;infin;</mi> </munderover> <msub> <mi>&amp;lambda;</mi> <mi>i</mi> </msub> <msup> <mi>&amp;alpha;</mi> <mi>&amp;tau;</mi> </msup> <msup> <mi>log&amp;alpha;e</mi> <mrow> <mo>-</mo> <msub> <mi>&amp;lambda;</mi> <mi>i</mi> </msub> <msup> <mi>&amp;alpha;</mi> <mi>&amp;tau;</mi> </msup> </mrow> </msup> <msubsup> <mi>&amp;phi;</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>&amp;infin;</mi> </munderover> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msub> <mi>&amp;lambda;</mi> <mi>i</mi> </msub> <msup> <mi>&amp;alpha;</mi> <mi>&amp;tau;</mi> </msup> </mrow> </msup> <msubsup> <mi>&amp;phi;</mi> <mi>i</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>;</mo> </mrow>

S240) pairsPerforming discrete-time Fourier transform, as shown in equation (4):

H′(ω)＝H(ω)e^2πωs(4)

wherein H and H' are each independentlyAndis the Fourier transform of (a), omega ∈ [0,2 π ∈]

S250) then eliminate e by taking the modulus^2πωsThat is, | H' (ω) | H (ω) |, and | H (ω) | is denoted as NSI-hks (x).

4. The thermonuclear-based three-dimensional model identification and retrieval method of claim 3, wherein τ is selected as τ e [ τ ] in the feature function NSI-HKS (x)_min,τ_max]The amplitudes of the second to sixth low frequency components of NSI-HKS (x) are sampled as local features.

5. The thermonuclear-based three-dimensional model identification and retrieval method of claim 1, wherein the K-means clustering process comprises the steps of:

s410) randomly selecting K objects as initial clustering centers;

s420) calculating the distance between each object and each seed cluster center, and allocating each object to the cluster center closest to the object, wherein the cluster center and the objects allocated to the cluster center represent a cluster;

s430) after all the objects are distributed, the distribution center of each cluster is recalculated according to the existing objects in the cluster;

s440) continuously repeating the steps S420 and S430 until the data members of each cluster are not changed;

s450) obtaining the cluster to which each data object belongs.

6. The thermonuclear-based three-dimensional model identification and retrieval method of claim 5, wherein K is 60 in the K-means clustering process.

7. The thermonuclear-based three-dimensional model identification and retrieval method of claim 1, wherein the similarity measure comprises the method of Minkowski distance comparison.

8. The thermonuclear-based three-dimensional model identification and retrieval method of claim 7, wherein the Minkowski distance formula is:

<mrow> <mi>d</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>d</mi> </munderover> <mo>|</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mi>k</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msup> <mo>|</mo> <mi>q</mi> </msup> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mi>q</mi> </mrow> </msup> </mrow>

9. the thermonuclear-based three-dimensional model identification and retrieval method of claim 7, wherein the similarity measure step comprises:

s510) comparing the distance between the first point of the model A and all the points of the model B, and selecting one group with the minimum distance for matching;

s520) comparing the Minicosky distance between the second point of the model A and all the points of the model B (except the matched points), and selecting one group with the minimum distance for matching till 60 points are all matched;

s530) calculating the average value of the distances among all matched points, wherein the smaller the average value is, the more similar the samples are, and the smaller the difference is; the larger the average, the more dissimilar the samples, and the greater the degree of difference.