CN111754426B - Automatic restoration method for mural shedding disease based on genetic algorithm - Google Patents
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Abstract
The invention relates to a mural shedding disease automatic repairing method based on a genetic algorithm, which comprises the following steps: acquiring a digital mural shedding image, and labeling the mural shedding disease image through mathematical morphology; and respectively repairing the structural information and the texture information of the falling and damaged area through a genetic algorithm. According to the method, the self color characteristics of the falling of the mural are fully utilized, the multi-scale morphological edge gradient detection is used for extracting edge information in the automatic labeling process of the falling diseases, and the target area of the image is highlighted through an image enhancement technology to obtain the falling edge; and then, respectively repairing according to the structural information and the texture information of the falling damaged area by utilizing the optimization performance of the genetic algorithm, and recovering the lost information in the mural image to make the mural image clearer and more natural, thereby maintaining the integral visual effect with certain similar precision to the original mural image and making the recovery effect more accurate and effective.
Description
Technical Field
The invention belongs to the technical field of digital image processing, and particularly relates to an automatic restoration method for mural peeling diseases based on a genetic algorithm.
Background
The mural culture is an indispensable corner of cultural heritage in China, and as a representative of murals, the Dunhuang murals comprise a plurality of cave murals such as Dunhuang Mogao caves, west Qianfo caves, anxi elm forest caves and the like, mainly relate to various categories such as Buddha image paintings, warp-changing paintings, human portraits, decorative paintings, landscape paintings, story paintings and the like, and the historical murals are as large as more than five ten thousand square meters. In the Dunhuang Mogao cave, more than sixteen countries, inertials, tang dynasties, fifth generations, yuan and other more than ten dynasties are stored, and the mural is rich in variety, rich in content and valuable in history, culture and artistic value; however, the mogao cave suffers from corresponding changes, and various diseases such as falling off, armor raising, cracks, and base of the ground stick appear on the mural under the influence of external environment and human factors, and the diseases cause irreversible damage to the mural.
Aiming at the above diseases, the traditional repairing method mainly adopts a manual repairing mode to repair the mural. The mural protection situation is severe due to the problems of shortage of professional technicians, single repairing means, long time consumption, low repairing efficiency and the like. With the advent of new technologies such as computer technology and image processing technology, mural digitization plays an increasingly important role in mural protection work, and is regarded as a development trend by national and cultural relic researchers. Through the digital collection of the ancient architecture murals, the murals are permanently stored through a modern digital technology, and meanwhile, the latest theory and method in the field of computer images are utilized to label and repair the murals, so that effective technical support is provided for the protection of the ancient architecture murals.
The final aim of the digital image restoration technology is to realize the restoration function of the ancient architecture murals, namely, the restoration of the murals is completed by using the advanced artificial intelligence technology. Firstly, a high-resolution camera is used for converting a continuous analog mural image signal into a discrete mural digital signal; and secondly, repairing and restoring the digital mural image. The existing digital image repairing method is generally used for repairing the shedding diseases by simultaneously repairing the structure and the texture, but the method has a very small repairing effect on the shedding diseases passing through the contour line, so that the shedding diseases passing through the contour line are difficult to repair, and a more appropriate repairing method needs to be explored.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, provides an automatic repairing method of a mural shedding disease based on a genetic algorithm, and solves the problem that the mural shedding disease cannot be effectively recovered.
The technical problem to be solved by the invention is realized by adopting the following technical scheme:
a mural shedding disease automatic repairing method based on a genetic algorithm comprises the following steps:
step 2, respectively repairing the structural information and the texture information of the falling and damaged area through a genetic algorithm; the specific method of the step comprises the following steps:
extracting marked mural structure information by adopting a Canny edge detection algorithm, and acquiring contour line related information and surrounding information; the contour line related information refers to each related contour line which is in contact with the damaged area; the surrounding information is the basic characteristics of the surrounding image in contact with the damaged area, including average brightness, contrast and curve smoothness;
secondly, constructing a genetic algorithm target function, searching matched contour lines, and recovering structural information in the damaged region of the image; the specific method of the step comprises the following steps:
(1) the following optimization objective assumptions are proposed: there is at least one pair of matched correlation contour lines; the matching associated profile curves must be possible to connect in the damaged area;
(2) designing a genetic algorithm objective function by adopting a curve fitting method, and converting an optimization objective into the objective function; the specific implementation method of the step is as follows:
for any contour, let y = f (x), its curvature at any point x is calculated as:
the objective function is expressed as:
wherein β isRepresenting a matching threshold parameter for balancing the optimization objective; m is the matching logarithm of the matching contour; d i Representing the matching degree of the ith pair of matching contour lines;
the fitness function of a chromosome is expressed as:
wherein N is ind Is the number of chromosomes, N x Is the sequence number of the chromosome x in the descending order; max represents the maximum value of fitness, and the higher the fitness of the chromosome, the greater the genetic probability, i.e., the greater the probability of selecting the retained chromosome gene;
filling texture information into all damaged areas to finish repairing the detached mural of the mural; the specific implementation method of the step is as follows:
(1) calculating the repair priority, and determining the repair block psi in the mural image to be repaired p Repair priority P (P):
P(p)＝max{nS(p)}
wherein S (p) represents the pixels of the undamaged area in the repair block area, and n represents the number of the pixels of the undamaged area in the repair block area;
after the repair priority P (P) is determined, the repair block with the maximum repair priority is calculated as the current repair-ready block and is marked as psi p ^：
Wherein Z represents the total number of pixels in the damaged area in the repair block, and P (P) is the repair priority of each pixel corresponding to the repair block;
(2) filling texture information to find and correct the block psi p The most similar repair block is marked as the best matching block psi q And B, completing restoration:
wherein D represents a damaged area in the repair block, D (ψ) p ^,ψ q ) Is defined as two blocks of pixels psi p ^ and psi q The sum of squares of color value errors between.
Further, the step 2 is followed by a step of repairing the paint layer of the actual wall painting shedding disease according to the digital image repairing result.
Further, the specific implementation method of step 1 includes the following steps:
the method comprises the steps of acquiring a digital mural falling image by a high-resolution digital camera, and processing and converting the mural falling image acquired in an RGB space into an HSV color space;
secondly, preprocessing the obtained image of the HSV color space by a median filtering method;
thirdly, acquiring the edge of the preprocessed mural image by adopting a multi-scale morphological edge gradient detection method;
fourthly, enhancing the edge scale of the image by adopting a high-cap transformation method;
fifthly, obtaining the falling edge by adopting a maximum class variance method adaptive threshold value segmentation technology;
sixthly, acquiring a mural falling edge mask by using a connected component marking algorithm, and internally filling the acquired falling edge mask.
Further, the mathematical expression of the step three multi-scale morphological edge gradient detection method is as follows:
wherein f (x, y) represents a gray image, b i (x, y) is a structural element, k is a scale parameter,
the morphological erosion operator is represented by a graphical representation,
a morphological dilation operator is represented.
The invention has the advantages and positive effects that:
1. according to the method, the self color characteristics of the falling of the mural are fully utilized, multi-scale morphological edge gradient detection is used for extracting edge information in the automatic marking process of the falling diseases, the target area of the image is highlighted through an image enhancement technology, and the falling edges are obtained; and then, introducing the GA algorithm into the ancient architecture mural image repairing process by utilizing the optimization performance of a Genetic Algorithm (GA), respectively repairing according to the structural information and the texture information of the falling damaged area, and recovering the lost information in the mural image to make the mural image clearer and more natural, thereby maintaining the integral visual effect with certain similar precision to the original mural image and making the recovery effect more accurate and effective.
2. The invention aims at the application of the method for preventing the wall painting from falling off. In the marking link of the falling-off disease, aiming at the mural falling-off disease with the surrounding structure information of the damaged area being a linear structure, a disease segmentation mode is adopted, the color characteristics of a mural example in an HSV color space are analyzed, median filtering is adopted to remove noise and enhance image contrast, the target edge of the mural example is extracted by using morphological gradient transformation and top-hat transformation in mathematical morphology, the obtained falling-off edge is internally filled, and finally the obtained falling-off area mask code and an original image are added for operation, so that the accurate marking function of the mural falling-off disease is realized.
3. The marked mural image is repaired by adopting an image repairing algorithm based on a genetic algorithm, the structural repairing problem is firstly converted into a genetic algorithm optimizing problem in the repairing process, then the connection of structural information of a damaged area is realized by utilizing the global optimizing capability of the genetic algorithm, and then the texture is filled, so that an ideal repairing effect is obtained, and the problem of repairing the falling-off disease through the contour line is reasonably and effectively solved.
Drawings
FIG. 1 is a flow chart of a repair method of the present invention;
FIG. 2 is a diagram of the process of dropping a label through a single contour line according to the present invention;
FIG. 3 is a schematic diagram of the structural information fitting process of the present invention;
FIG. 4 is a schematic diagram of a repair block and a matching block according to the present invention;
FIG. 5 is a diagram of the "dance drawing" repair result of the present invention via a single contour line;
FIG. 6 is a diagram of the "femto map" repair result from a single contour line of the present invention;
FIG. 7 is a graph of the meditation graph repair result with a single contour line;
FIG. 8 is a repair result chart of the invention through a multi-contour line for a color band diagram;
FIG. 9 is a drawing of a "calligraphical meeting" repair result of the present invention through a single contour line;
FIG. 10 is a diagram of the "crown uniform map" repair result of the present invention via a single contour line;
FIG. 11a is a PSNR index analysis chart of mural peeling repair evaluation indexes of the present invention and other algorithms;
FIG. 11b is an analysis chart of mural peeling repair assessment indicators (FSIM indicators) of the present invention and other algorithms;
FIG. 11c is a graph of mural peeling repair assessment index analysis (SR-SIM index) for the present invention and other algorithms;
FIG. 11d is a graph of the mural exfoliation repair assessment index analysis (VSI index) of the present invention and other algorithms.
Detailed Description
The present invention is further described in detail below with reference to the accompanying drawings.
The design idea of the invention is as follows: firstly, labeling a mural falling disease image through mathematical morphology, and extracting structure information of the labeled mural disease, wherein the structure part mainly extracts contour lines of damaged and surrounding areas of the image, the contour lines to be matched in the extraction process mainly take the contour lines connected with two ends of the damaged area of the image as main parts, and then curve fitting is carried out on the contour lines through a genetic algorithm to find out an optimal contour repairing scheme, so that the structure information repairing of the falling mural is completed; the whole image is subjected to region segmentation while the structural information is repaired by the falling-off mural, so that in the process of repairing the texture part, only the damaged region needs to be filled according to the region. The algorithm flow for the detached part of the mural is shown in fig. 1.
A mural shedding disease automatic repairing method based on a genetic algorithm comprises the following steps:
(1) Color space conversion
In consideration of the characteristic that the color characteristics of the drop-off area of the mural are outstanding, compared with the RGB space, the HSV color space has more visual and convenient color characteristics. Therefore, before the shedding disease is marked, color space conversion is firstly carried out on the image to be calibrated, and preprocessing conversion from an RGB space to an HSV space is completed.
(2) Median filtering
Median filtering is a commonly used non-linear smoothing filter. Compared with other filtering algorithms, the median filtering not only has a good filtering effect on impulse noise, but also has a protection effect on edge information, and can avoid fuzzification caused by filtering.
Let f (x, y) represent the gray value of any point in the image, N × N represent the size of the filter window, the point (i, j) in the filter window constitutes the region S, and the pixels of the point after median filtering can be represented as:
g(x,y)＝med{f(i-x,j-y)}((i,j)∈S) (1)
wherein med represents taking an average value, and (i, j) and (x, y) represent pixel points in the image respectively.
(3) Image edge acquisition
And acquiring an edge mask of a falling part under a complex background by utilizing multi-scale morphological edge gradient detection.
Wherein f (x, y) represents a gray image, b i (x, y) are structural elements, k is a scale parameter,
the morphological erosion operator is represented as a function of time,
a morphological dilation operator is represented.
(4) Edge scale enhancement
The edge scale enhancement can compress or stretch the dynamic range of the image gray level, and the effects of sharpening edge information, enhancing image contrast and the like are achieved. The edge detection based on mathematical morphology has better performance in the aspects of detecting edges and retaining image details. The top hat transformation is an important algorithm form of mathematical morphology, is the difference of the open operation results of the original image and the image, has the advantages of enhancing the image contrast, weakening background interference and the like, and can highlight the local characteristics of the image edge. Therefore, the high-hat transformation is carried out on the to-be-calibrated mural falling-off image after the edge of the image is obtained, so that the edge scale of the image can be effectively enhanced, and the subsequent image processing can be carried out. The mathematical expression is as follows:
wherein b is a structural element, f is an original image,
represents a morphological open operation.
(5) Obtaining the edge of the drop and filling the inside
And (4) performing self-adaptive threshold segmentation on the image processed in the steps to obtain a falling edge. And only preserving the mural falling edge mask by using a connected domain marking algorithm, and internally filling the obtained falling edge mask, thereby effectively extracting the falling part of the mural disease.
(6) Obtaining the edge of the drop and filling the inside
And performing addition operation on the falling part and the original mural image to obtain a mural falling disease labeling part. Taking a mural "dance graph" passing through a single contour line as an example, the specific algorithm labeling process is shown in fig. 2.
And 2, respectively repairing the structural information and the texture information of the falling and damaged area through a genetic algorithm. The specific implementation method of the step is as follows:
(1) Structural information extraction
The structure information extraction part adopts a Canny edge detection algorithm. The algorithm firstly uses a Gaussian function to carry out smooth filtering on the image and remove noise in the image. Let the two-dimensional gaussian function be:
where σ is the standard deviation of the gaussian distribution and is used to determine the degree of smoothing of the processed image.
By using Canny edge detection, all contour lines of the calibrated mural can be extracted, and after the extraction process is finished, the original structure information containing all contour lines is analyzed and divided into relevant information and irrelevant information.
Wherein the "related information" refers to each related contour line in contact with the damaged area. Because the damaged area is calibrated for a mural image to be repaired, calibration disease contour lines are necessarily existed, and these contour lines are not existed in the original image, i.e. they are "irrelevant information", besides, other contour lines which are not connected with the damaged area also belong to "irrelevant information", so that it has no need of retaining them. And after the subsequent processing of the irrelevant information is finished, obtaining the relevant outline line structure of the falling-off mural.
Besides obtaining the 'relevant information' of the falling-off mural, the basic features of the surrounding image contacting with the damaged area are also needed to be obtained, the basic features mainly comprise basic statistical parameters such as average brightness, contrast and curve smoothness, and finally the matching optimization of the structural information is completed. In the invention, in order to verify the feasibility of the algorithm, the feature parameters of the average brightness value d and the average contrast σ are selected to describe the features of the surrounding image connected with the damaged area.
Wherein the average luminance value d is represented as:
the average contrast σ is expressed as:
wherein L represents the total number of pixels in the relevant area; z is a radical of i Representing the gray value of the ith pixel in the relevant area; p (z) i ) Indicating the probability that this luminance value occurs within the extraction area.
(2) Structural information recovery
The invention adopts a curve fitting method to design a GA target function, and aims to search a matched contour line and restore structural information in a damaged area of an image.
Let y = f (x) for any one contour, its curvature at any point x is calculated as:
the objective function is expressed as:
where β is a match threshold parameter representing a target for equilibrium optimization; m is the matching logarithm of the matching contour; d i And indicating the matching degree of the ith pair of matching contour lines.
The fitness function of a chromosome is expressed as:
wherein N is ind Is the number of chromosomes, N x Is the sequence number of the chromosome x in the descending order; max represents the maximum fitness, and the higher the fitness of a chromosome, the greater the probability of inheritance, i.e., the probability of selecting a gene that retains the chromosome.
Therefore, we can get the best matching associated contour to recover the structural information of the broken image. In the partial process of repairing the best matching related contour line, the invention adopts a consistent and gradual point-by-point repairing strategy, namely, pixels on the connecting line are uniformly colored, and the fitting process is shown in figure 3.
(3) Texture split region filling
In the texture filling and zoning repair process, the repair process mainly comprises the following two steps:
1) Calculating repair priority
In the image restoration process, it is most important to determine the restoration block ψ in the mural image to be restored p The repair priority P (P) is calculated by the formula
P(p)＝max{nS(p)} (10)
Wherein S (p) represents the number of pixels in the repair block region, and n represents the number of pixels in the repair block region.
The more pixels of the undamaged part of the repair block in the current image to be repaired are available, the higher the repair priority of the repair block is, and the more the filling order is.
Repair block psi p The number of the repair blocks is determined by the number of the pixels in the damaged area, and after the repair priority P (P) of the repair block corresponding to each pixel is calculated, the repair block with the maximum repair priority is selected as the current quasi-repair block and is marked as psi p A, its expression is
Wherein Z represents the total number of pixels in the damaged area in the repair block, and P (P) is the repair priority of the repair block corresponding to each pixel.
2) Filling in texture information
After calculating the quasi-repair block psi having the maximum repair priority p After ^ it extracts texture information from the unbroken area, such as the S area in FIGS. 5-8 (b). For each quasi-repair block, the repair process searches the unbroken area by the following calculation formula to find the quasi-repair block ψ p The most similar repair block, called best match block, is noted as psi q ^：
Wherein D represents a damaged area in the repair block, D (ψ) p ^,ψ q ) Is defined as two blocks of pixels psi p ^ and psi q The sum of squares of color value errors between.
The filling process is shown in fig. 4. The repair process completes the copying of texture information from the unbroken area to the target area. And after finding out the best matching block of a certain quasi-repair block, filling the damaged area in the quasi-repair block with the new pixel value of the best matching block, then updating the priority of the unfilled part of pixels, and calculating and repairing the repair block with the highest priority by the same method until texture information is filled in all the damaged areas, so that the repair result is finished.
In order to verify the effectiveness of the method, two types of mural peeling diseases passing through a single contour line and a plurality of contour lines are selected as cases, the method is applied to carry out data restoration effect analysis, and meanwhile, the data restoration effect is compared and analyzed with the existing common mural restoration TV algorithm, the Criminsi algorithm and the PConv algorithm.
The invention selects the 2 types of shedding diseases which commonly exist in murals and pass through a single contour line and a plurality of contour lines to carry out a disease repair experiment, and selects 6 murals as mural shedding test patterns.
Shedding disease through a single contour line: the example includes the falling-off disease in the 82 nd Grotto's music dance picture, the 85 th Grotto's flying picture and the 9 th Grotto's meditation picture in late Tang period. The repair results using different repair algorithms are shown in fig. 5-7.
Shedding disease through multiple contour lines: the case includes the falling-off disease of the 146 th Groover pattern in the Mogao of the fifth generation period, the 112 th Groove book conference pattern in the Mogao of the Zhongdong period, and the 285 th Groove clothes pattern in the Mogao of the West Wei period. The repair results using different repair algorithms are shown in fig. 8-10.
The invention selects a plurality of evaluation indexes to evaluate the quality of the data restoration effect, and comprises the following steps: peak Signal to Noise Ratio (PSNR), feature Similarity (FSIM), spectral information Similarity (SR-SIM), and Visual Significance Index (VSI) were used as Similarity evaluation indexes in this section to evaluate image quality.
Wherein, the peak signal-to-noise ratio (PSNR) reflects the change degree of distortion data, measures the similarity between images before and after restoration, is the most widely applied evaluation index in image restoration research, and the calculation formulas are respectively as follows:
in the formula: i is 1 ,I 2 The two images to be compared respectively refer to an original image and a repaired image; m and n respectively represent the sizes of the two images; MSE (Mean Square Error) represents the Mean Square Error;
by utilizing the repairing method, the marked mural drops are respectively repaired, and each repairing effect evaluation index is calculated. The method of the present invention is compared with the TV algorithm, criminsi algorithm, and PConv algorithm repair effects, as shown in fig. 11a, 11b, 11c, and 11 d.
The abscissa in fig. 11a, 11b, 11c, and 11d is the similarity evaluation index. The analysis here takes as an example a falling wall painting "dance map" through a single contour line. The PSNR value of the GA optimization algorithm is 43.6152dB, and the PSNR values of the comparison TV algorithm, the Criminsi algorithm and the PConv algorithm are 41.0315dB, 41.7552dB and 38.6420dB respectively, so that the PSNR value of the GA optimization algorithm provided by the invention is improved by 16.54% to the maximum, the FSIM value is increased by 4.31% to the maximum, the SR-SIM value is increased by 2.08% to the maximum, and the VSI value is increased by 1.07% to the maximum. Therefore, various evaluation indexes of the GA optimized mural repair algorithm provided by the invention are obviously superior to those of the conventional image repair algorithm. The algorithm carries out curve fitting on the structural information by using the GA algorithm, recovers the structural information, and then adopts a texture filling mode to carry out regional parallel repair, thereby effectively improving the repair precision.
It should be emphasized that the embodiments described herein are illustrative rather than restrictive, and thus the present invention is not limited to the embodiments described in the detailed description, but also includes other embodiments that can be derived from the technical solutions of the present invention by those skilled in the art.
Claims (4)
1. A mural shedding disease automatic repairing method based on genetic algorithm is characterized in that: the method comprises the following steps:
step 1, collecting a digital mural falling image, and labeling the mural falling disease image through mathematical morphology;
respectively repairing the structural information and the texture information of the falling and damaged area through a genetic algorithm; the specific method of the step comprises the following steps:
extracting marked mural structure information by adopting a Canny edge detection algorithm, and acquiring contour line related information and surrounding information; the contour line related information refers to each related contour line which is in contact with the damaged area; the surrounding information is the basic characteristics of the surrounding image in contact with the damaged area, including average brightness, contrast and curve smoothness;
secondly, constructing a genetic algorithm target function, searching matched contour lines, and recovering structural information in the damaged region of the image; the specific method of the step comprises the following steps:
(1) the following optimization objective assumptions are proposed: there is at least one pair of matched correlation contour lines; the matching associated profile curves must be possible to connect in the damaged area;
(2) designing a genetic algorithm objective function by adopting a curve fitting method, and converting an optimization objective into the objective function; the specific implementation method of the step is as follows:
let y = f (x) for any one contour, its curvature at any point x is calculated as:
the objective function is expressed as:
where β is a match threshold parameter representing a target for equilibrium optimization; m is the matching logarithm of the matching contour; d i Representing the matching degree of the ith pair of matching contour lines;
the fitness function of a chromosome is expressed as:
wherein N is ind Is the number of chromosomes, N x Is the sequence number of the chromosome x in the descending order; max represents the maximum value of fitness, and the higher the fitness of the chromosome is, the greater the genetic probability is, namely the probability of selecting and retaining the chromosome gene is;
filling texture information into all damaged areas to finish repairing the detached mural of the mural; the specific implementation method of the step is as follows:
(1) calculating the repair priority, and determining the repair block psi in the mural image to be repaired p Repair priority P (P):
P(p)＝max{nS(p)}
wherein S (p) represents the pixels of the undamaged area in the repair block area, and n represents the number of the pixels of the undamaged area in the repair block area;
after the repair priority P (P) is determined, a repair block with the maximum repair priority is calculated as a current quasi-repair block and is marked as psi p^ ：
Wherein Z represents the total number of pixels in the damaged area in the repair block, and P (P) is the repair priority of each pixel corresponding to the repair block;
(2) filling texture information to find and correct the block psi p^ The most similar repair block, denoted as the best match block psi q^ And completing the repair:
wherein D represents a damaged area in the repair block, D (ψ) p^ ,ψ q ) Is defined as two blocks of pixels psi p^ And psi q The sum of the squared errors of the color values in between.
2. The automatic restoration method for the mural shedding disease based on the genetic algorithm according to claim 1, characterized in that: and step 2, repairing the pigment layer with the actual wall painting shedding disease according to the digital image repairing result.
3. The automatic restoration method for the mural shedding disease based on the genetic algorithm according to claim 1 or 2, characterized in that: the specific implementation method of the step 1 comprises the following steps:
the method comprises the steps of acquiring a digital mural falling image by a high-resolution digital camera, and processing and converting the mural falling image acquired in an RGB space into an HSV color space;
secondly, preprocessing the obtained image of the HSV color space by a median filtering method;
thirdly, acquiring the edge of the preprocessed mural image by adopting a multi-scale morphological edge gradient detection method;
fourthly, enhancing the edge scale of the image by adopting a high-cap transformation method;
fifthly, obtaining the falling edge by adopting a maximum class variance method adaptive threshold value segmentation technology;
sixthly, acquiring a mural falling edge mask by adopting a connected domain marking algorithm, and internally filling the acquired falling edge mask.
4. The method for automatically repairing the mural peeling disease based on the genetic algorithm according to claim 3, characterized in that: the mathematical expression of the step three multi-scale morphological edge gradient detection method is as follows:
wherein f (x, y) represents a gray image, b i (x, y) is a structural element, k is a scale parameter,
the morphological erosion operator is represented as a function of time,
representing a morphological dilation operator.
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