CN107609638A - A kind of method based on line decoder and interpolation sampling optimization convolutional neural networks - Google Patents
A kind of method based on line decoder and interpolation sampling optimization convolutional neural networks Download PDFInfo
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Abstract
The invention belongs to field of image recognition, and in particular to a kind of method based on line decoder and interpolation sampling optimization convolutional neural networks.The convolutional neural networks that the present invention is built, including input layer, convolutional layer, pond layer, fully connected network network layers and output layer, trained first with convolution linear encoder and obtain weights, and in this, as the initial value of convolutional neural networks, then the characteristic pattern that convolutional layer obtains is subjected to multiple interpolation sampling pool respectively, by propagated forward and reversely regulation, the weights of the partial gradient of each neuron and each layer of convolutional layer convolution kernel in each layer are finally obtained.By carrying out contrast experiment with existing method, test result indicates that:When the convolutional neural networks built using the inventive method are classified to image, there is convergence rate faster, the advantages of accuracy is higher.
Description
Technical field
The invention belongs to field of image recognition, and in particular to one kind is based on line decoder and interpolation sampling optimization convolution god
Method through network.Image recognition, refer to handle image using computer, analyzed and understood, to identify various differences
The target of pattern and the technology to picture.It is the basis of the practical techniques such as stereoscopic vision, motion analysis, data fusion.
Background technology
Line decoder is a kind of special neutral net, similar with own coding neutral net, and it attempts to approach a perseverance
Deng function, so that exporting it includes 3 layers of neuron, it is input layer, hidden layer and output layer respectively.For hidden layer, god
S types (or tanh) excitation function is still used through member, but for output layer, output end uses identical excitation function as excitation
Function.Because the output area of S type excitation functions is [0,1], when output layer uses excitation function it is necessary to input limitation or
Scaling, is located in [0,1] scope, some data sets, such as MINST, very easily can zoom to output in [0,1],
But be difficult to meet the requirement to input, it can not solved the above problems simply very much during the identical excitation function of not excessive use.One
The self-encoding encoder that individual S types or tanh hidden layers and linear convergent rate layer are formed, referred to as line decoder.
One of the typical framework of modern computer vision convolutional neural networks are widely used in image procossing in recent years
Field.Since the proposition of convolutional neural networks, many researchers have carried out studying and proposing corrective measure to it.Convolution
The basic structure of neutral net is by input layer, convolutional layer, pond layer, full articulamentum and output layer.Wherein most crucial is and volume
The related convolution operation of lamination and the pondization related to pond layer operation.The present invention carries from the characteristics of convolutional neural networks
A kind of improved neural network model is gone out, experimental result represents, for the more traditional convolutional network of this method, has convergence speed
Degree faster, the advantages of accuracy is higher.
The content of the invention
In view of the deficiencies of the prior art, the present invention provides one kind based on line decoder and interpolation sampling optimization convolutional Neural
The method of network is improved the weights of convolutional neural networks and pond operation.
The technical solution adopted in the present invention is:Convolutional neural networks are initialized based on line decoder and inserted with multiple
It is worth the operation optimization convolutional neural networks structure of sampling, comprises the following steps:
Step 1, build convolutional neural networks, including input layer, S convolutional layer, S pond layer, fully connected network network layers and
Output layer, and set the number and size of convolution kernel in each convolutional layer;
Step 2, input layer, convolution kernel hidden layer are included according to the number of convolutional layer in the convolutional neural networks of structure, structure
With S convolution linear encoder of output layer, and training sample is randomly selected as first convolution linear encoder input layer
Input data, the output of previous convolution linear encoder can be respectively trained as the input of the latter convolution linear encoder
Obtain the weights of S convolution linear encoder;The implementation of wherein any one convolution linear encoder acquisition weights is as follows,
(1) for a single pass single input picture x, its corresponding k-th of convolution characteristic pattern is hk, wherein hkCan table
It is shown as,
hk(the x*w of=σ 1k+bk) (1)
Wherein σ 1 is activation primitive, and what * was represented is to carry out 2 dimension convolution operations, wkRefer to k-th of convolution kernel, i.e., k-th power
Value, bkFor biasing;
(2) for arbitrary input, the output y reconstructed by convolution linear encoder,
Wherein σ 2 is linear identical excitation function, and w' is the scrolling of the progress to convolution kernel, and b refers to the inclined of input
Put;
(3) loss function for the minimum value that need to optimize is represented with mean square deviation E, obtains the error of input and output, and root
According to the weights of the error transfer factor convolution linear encoder so that its error is minimum, obtains the weights of corresponding convolution linear encoder,
Wherein, n is training sample number;xiWhat is represented is that i-th of sample inputs, yiWhat is represented is that i-th of sample exports, E
What (θ) was represented is the error of input and output;
Step 3, the first of convolutional layer is corresponded in the convolutional neural networks constructed by weights as step 1 obtained using step 2
Initial value, realize to obtain corresponding characteristic pattern respectively by multiple interpolation sampling pool, then input convolution god of the picture to structure
Propagated forward is carried out through network, is exported, specific implementation is as follows,
Step 3.1, the initial value using the weights that step 2 obtains as convolutional neural networks;
Step 3.2, picture is then inputted to the convolutional neural networks of structure, by the S of S convolutional layeriIndividual characteristic pattern difference
Carry out multiple interpolation sampling pool, SiRepresent that i-th of convolutional layer corresponds to the number of convolution kernel, obtain the feature of corresponding pond layer
Figure, pulls into column vector feature, as the characteristic vector finally extracted by the characteristic pattern of last layer by row;
Step 3.3, input final characteristic vector as the input layer of full Connection Neural Network grader, connect afterwards
The hidden layer of one fully-connected network, the demarcation layer of the output layer of fully-connected network as fully-connected network;
Step 4, the label according to corresponding to being exported in step 3 and input picture is contrasted, to the convolutional Neural net of structure
Network is reversely adjusted, and the convolutional neural networks after being optimized, implementation is as follows,
Step 4.1, the partial gradient of each neuron in each layer is calculated with gradient descent method according to formula (8);
Calculating for convolution operation gradient, is obtained by following formula,
Wherein, x is input picture, and what E (θ) was represented is to input the error with output, wkRefer to k-th of convolution kernel, i.e., k-th
Weights, δ h and δ y are the partial gradient of hidden layer and output layer respectively, h'kRefer to partial gradient corresponding to k-th of characteristic pattern
Carry out the respective value obtained by scrolling;
Step 4.2, the weights of each layer of convolutional layer convolution kernel are updated,
Wherein, wkRefer to k-th of convolution kernel, bkFor biasing, η refers to learning rate.
Moreover, the implementation method that multiple interpolation samples in step 3 is as follows:
f_mapk=Multiple_interpolation_pooling (hk) (4)
Wherein, hkWhat is represented is k-th of characteristic pattern of convolutional layer, and Multiple_interpolation_pooling is represented
Multiple interpolation sampling function, f_mapkRepresent k-th of characteristic pattern of corresponding sample level;
Wherein, multiple interpolation sampling function is double by the use of the gray value of 16 points around point to be sampled as bicubic interpolation
Cubic interpolation formula is as follows:
If i+u, j+v are pixel coordinate to be asked, i, j are positive integer, and u, v are the decimal less than 1 more than zero, then pixel to be asked
The value f (i+u, j+v) of gray scale is,
F (i+u, j+v)=ABC (6)
Wherein A, B, C are matrix, and form is as follows:
A=[S (1+u) S (u) S (1-u) S (2-u)]
C=[S (1+v) S (v) S (1-v) S (2-v)]T
Wherein, f (i, j) represents the gray value of source images (i, j) place pixel.
Compared with prior art, the advantages of the present invention are:The present invention is first with convolution linear encoder
Training obtains weights initial value, and in this, as the initial value of convolutional neural networks, then enters the characteristic pattern that convolutional layer obtains
Row multiple interpolation sampling pool, by propagated forward and the reverse weights for adjusting, finally obtaining each layer of convolutional layer convolution kernel.It is logical
Cross and carry out contrast experiment with existing method, test result indicates that:Using the convolutional neural networks of the inventive method structure to figure
As when being classified, there is convergence rate faster, the advantages of accuracy is higher.
Brief description of the drawings
Fig. 1 is that convolution of embodiment of the present invention linear encoder initializes schematic diagram;
Fig. 2 is that multiple interpolation of the embodiment of the present invention adopts optimization sample convolutional neural networks schematic diagram.
Embodiment
Technical scheme is described further with reference to the accompanying drawings and examples.
A kind of method based on line decoder and interpolation sampling optimization convolutional neural networks provided by the invention carries out figure
As classification, comprise the following steps:
Step 1, build convolutional neural networks, including input layer, 2 convolutional layers, 2 pond layers, fully connected network network layers and
Output layer, and set the number and size of convolution kernel in each convolutional layer;
Step 2, input layer, convolution kernel hidden layer are included according to the number of convolutional layer in the convolutional neural networks of structure, structure
With 2 convolution linear encoders of output layer, and training sample is randomly selected as first convolution linear encoder input layer
Input data, the output of previous convolution linear encoder can be respectively trained as the input of the latter convolution linear encoder
Obtain the weights of 2 convolution linear encoders;
Step 2.1, the first half convolutional neural networks of 2 convolutional layers of band, the frame as convolution linear encoder are established
Frame structure;
Step 2.2, input number of the training sample of several 32*32 sizes as convolution linear encoder input layer is chosen
According to convolution kernel is the training weights of 6 5*5 sizes, then trains the convolution linear encoder, obtains first layer convolutional layer convolution
The weights of core;Similarly, the input as second convolution linear encoder is output it, obtains second layer convolutional layer convolution kernel 12
The training weights of individual 5*5 sizes, as shown in Figure 1.The acquisition of weights is described in detail below for first convolution linear encoder
Process:
(1) for a single pass single input picture x, its corresponding k-th of convolution characteristic pattern is hk, wherein hkCan table
It is shown as,
hk(the x*w of=σ 1k+bk) (1)
Wherein σ 1 is activation primitive, and what is taken in the embodiment of the present invention is Sigmoid functions, and what * was represented is to carry out 2 dimension convolution
Operation, wkRefer to k-th of convolution kernel, i.e. k-th of weights, bkFor biasing;
(2) for arbitrary input, the output y reconstructed by convolution linear encoder,
What wherein σ 2 took is linear identical excitation function, and w' is the scrolling of the progress to convolution kernel, and b refers to inputting
Biasing;
(3) loss function for the minimum value that need to optimize is represented with mean square deviation E:
Wherein, n is training sample number;xiWhat is represented is that i-th of sample inputs, yiWhat is represented is that i-th of sample exports, E
What (θ) was represented is the error of input and output, the weights of convolution linear encoder is adjusted according to this error so that its error
It is minimum.
Step 3, using input layer, convolutional layer, pond layer as the first half of convolutional neural networks, full articulamentum and output
Latter half of the layer as convolutional neural networks, as shown in Fig. 2 the convolution kernel initial value of wherein convolutional layer is obtained by step 2,
Pond layer is realized to obtain corresponding characteristic pattern by multiple interpolation sampling pool, then inputs convolutional Neural net of the picture to structure
Network carries out propagated forward, and the purpose of propagated forward is to be exported, and label corresponding to the output and input picture is carried out pair
Than finally obtaining an error, weights being adjusted according to this error;
Step 3.1, by obtain 6 5*5 and 12 5*5 weights be assigned to respectively first layer convolution kernel in convolutional layer and
Second layer convolution kernel, the initial value as convolutional neural networks;
Step 3.2, input picture is obtained into 6 characteristic patterns of first convolutional layer by 6 5*5 convolution kernel, respectively
Multiple interpolation sampling pool is carried out, obtains the characteristic pattern of 6 corresponding pond layers;Again second is obtained by 12 5*5 convolution kernel
12 characteristic patterns of individual convolutional layer, multiple interpolation sampling pool is carried out respectively, obtain the characteristic pattern of 12 corresponding pond layers;Will most
The characteristic pattern of later layer pulls into column vector feature by row, as the characteristic vector finally extracted, such as Fig. 2;
Multiple interpolation sampling can represent with the following method:
f_mapk=Multiple_interpolation_pooling (hk) (4)
Wherein, hkWhat is represented is k-th of characteristic pattern of convolutional layer, and Multiple_interpolation_pooling is represented
Multiple interpolation sampling function, f_mapkRepresent k-th of characteristic pattern of corresponding sample level.
Unlike maximum sampling and average sampling, the algorithm is made using the gray value of 16 points around point to be sampled
For bicubic interpolation, considering not only the gray scale of 4 direct neighbor points influences, and in view of gray-value variation rate between each adjoint point
Influence.Computing can obtain the amplification closer to high-definition picture with reducing effect three times.This algorithm, which needs to choose, to be inserted
Value basic function carrys out fitting data, and its most frequently used Interpolation-Radix-Function is:
Bicubic interpolation formula is as follows:
If i+u, j+v (i, j are positive integer, and u, v are the decimal less than 1 more than zero, similarly hereinafter) are pixel coordinate to be asked, then treat
The value f (i+u, j+v) for seeking pixel grey scale is,
F (i+u, j+v)=ABC (6)
Wherein A, B, C are matrix, and form is as follows:
A=[S (1+u) S (u) S (1-u) S (2-u)]
C=[S (1+v) S (v) S (1-v) S (2-v)]T
Wherein, f (i, j) represents the gray value of source images (i, j) place pixel.
Step 3.3, input final characteristic vector as the input layer of full Connection Neural Network grader, connect afterwards
The hidden layer of one fully-connected network, the demarcation layer of the output layer of fully-connected network as fully-connected network;
Step 4, the label according to corresponding to being exported in step 3 and input picture is contrasted, to the convolutional Neural net of structure
Network is reversely adjusted, the convolutional neural networks after being optimized;
Step 4.1, the partial gradient of each neuron in each layer is calculated with gradient descent method according to formula (8);
Calculating for convolution operation gradient, it can be obtained by following formula:
Wherein, δ hkRepresent the partial gradient of k-th of convolution kernel, y points of partial gradients for representing output layer of δ, h'kRefer to pair
Partial gradient corresponding to k-th of characteristic pattern carries out the respective value obtained by scrolling.
Step 4.2, the weights of each layer of convolutional layer convolution kernel are updated;
Last right value update is calculated by stochastic gradient descent method to come:
Wherein, η refers to learning rate, and 0.5 is taken in the present embodiment.
In order to further illustrate beneficial effects of the present invention, below to be tested in two datasets:MNIST standards
Data set, CIFAR-10 data sets, by its test on three different convolutional network frameworks, with average pond and maximum
Its result is analyzed in pond.
(1) MNIST standard data sets
The data set is made together by the Corinna Cortes of Google researchs department and the Yann LeCun of New York University
Complete, be mainly used in the experiment in the fields such as image procossing, computer vision, machine learning.MINST data sets specification and sample
Number is as shown in table 1;
The MINST data sets of form 1
For MNIST standard data sets, it is respectively adopted in the present embodiment using typical CNN network frames and lenet-5
Different methods are tested, and its result is as follows:
Interpretation of result:
It can see from above-mentioned experimental result, in same convolutional neural networks, different pond operations can be brought
Different effects, the classifying quality in wherein average pond is worst, and the effect in interpolation pond is best, and maximum pondization is placed in the middle.Boureau
Et al. mention and analyzing the characteristic value mentioned during influence caused by different pondization operations for different characteristics, maximum pond
Operation and the operation of uniform pondization can show different performances, and experimental result meets its description.The average that compares pond and maximum
It is worth for pond, while the advantage in interpolation pond is to improve accuracy rate, also accelerates the convergence of whole convolutional neural networks
Speed.
(2) CIFAR-10 data sets
The CIFAR-10 data sets are provided by Krizhevsky of Hinton team et al.;Including 60000 32x32 colours
Image, altogether 10 class:Aircraft, automobile, bird, cat, deer, dog, frog, horse, ship, truck have 6000 width images per class, wherein this
60000 width pictures include 50000 width training image samples and 10000 width test image samples, per shape, the face of class image
Very big difference be present in color, angle etc..
For CIFAR-10 data sets, the embodiment of the present invention is entered using typical CNN network frames using different methods
Test is gone, its result is as follows:
Interpretation of result:
Compared with before, when testing different data sets with same network frame, as a result differ larger.But with regard to it
For pondization operation, interpolation pondization is with respect to average pondization and maximum pond, or in occupation of obvious advantage:Net can be improved
The accuracy of the classification of network, moreover it is possible to increase the convergence of object function.
The present embodiment is trained by a number of picture to convolutional neural networks first, and training is led to again after finishing
Cross certain test sample to be tested, the quality of classifying quality is determined according to the picture ratio correctly classified.As a result show,
The image classification effect that the optimization method of this convolutional neural networks initialization and sampling is carried out is better than unmodified network.
It should be appreciated that the partial content that this specification does not illustrate belongs to existing more ripe technology category.
Specific embodiment described herein is only to spirit explanation for example of the invention.Technology belonging to the present invention is led
The technical staff in domain can be made various modifications or supplement to described specific embodiment or be replaced using similar mode
Generation, but without departing from the spiritual of the present invention or surmount scope defined in appended claims.
Claims (2)
- A kind of 1. method based on line decoder and interpolation sampling optimization convolutional neural networks, it is characterised in that including as follows Step:Step 1, convolutional neural networks, including input layer, S convolutional layer, S pond layer, fully connected network network layers and output are built Layer, and set the number and size of convolution kernel in each convolutional layer;Step 2, according to the number of convolutional layer in the convolutional neural networks of structure, structure includes input layer, convolution kernel hidden layer and defeated Go out S convolution linear encoder of layer, and randomly select input of the training sample as first convolution linear encoder input layer Data, the input exported as the latter convolution linear encoder of previous convolution linear encoder, are respectively trained and obtain S The weights of convolution linear encoder;The implementation of wherein any one convolution linear encoder acquisition weights is as follows,(1) for a single pass single input picture x, its corresponding k-th of convolution characteristic pattern is hk, wherein hkIt is represented by,hk(the x*w of=σ 1k+bk) (1)Wherein σ 1 is activation primitive, and what * was represented is to carry out 2 dimension convolution operations, wkRefer to k-th of convolution kernel, i.e. k-th of weights, bk For biasing;(2) for arbitrary input, the output y reconstructed by convolution linear encoder,<mrow> <mi>&gamma;</mi> <mo>=</mo> <mi>&sigma;</mi> <mn>2</mn> <mrow> <mo>(</mo> <munder> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>&Element;</mo> <mi>h</mi> </mrow> </munder> <msup> <mi>h</mi> <mi>k</mi> </msup> <mo>*</mo> <msup> <mi>w</mi> <mrow> <mo>,</mo> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>Wherein σ 2 is linear identical excitation function, and w' is the scrolling of the progress to convolution kernel, and b refers to the biasing of input;(3) loss function for the minimum value that need to optimize is represented with mean square deviation E, obtains the error of input and output, and according to this The weights of error transfer factor convolution linear encoder so that its error is minimum, obtains the weights of corresponding convolution linear encoder,<mrow> <mi>E</mi> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>Wherein, n is training sample number;xiWhat is represented is that i-th of sample inputs, yiWhat is represented is that i-th of sample exports, E (θ) What is represented is the error of input and output;Step 3, the initial value of convolutional layer is corresponded in the convolutional neural networks constructed by weights as step 1 obtained using step 2, Realize to obtain corresponding characteristic pattern by multiple interpolation sampling pool respectively, then input convolutional neural networks of the picture to structure Propagated forward is carried out, is exported, specific implementation is as follows,Step 3.1, the initial value using the weights that step 2 obtains as convolutional neural networks;Step 3.2, picture is then inputted to the convolutional neural networks of structure, by the S of S convolutional layeriIndividual characteristic pattern carries out more respectively Weight interpolation sampling pond, SiRepresent that i-th of convolutional layer corresponds to the number of convolution kernel, obtain the characteristic pattern of corresponding pond layer, will most The characteristic pattern of later layer pulls into column vector feature by row, as the characteristic vector finally extracted;Step 3.3, inputted final characteristic vector as the input layer of full Connection Neural Network grader, connect one afterwards The hidden layer of fully-connected network, the demarcation layer of the output layer of fully-connected network as fully-connected network;Step 4, the label according to corresponding to being exported in step 3 and input picture is contrasted, and the convolutional neural networks of structure are entered Row reversely regulation, the convolutional neural networks after being optimized, implementation is as follows,Step 4.1, the partial gradient of each neuron in each layer is calculated with gradient descent method according to formula (8);Calculating for convolution operation gradient, is obtained by following formula,<mrow> <mfrac> <mrow> <mo>&part;</mo> <mi>E</mi> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&part;</mo> <msup> <mi>W</mi> <mi>k</mi> </msup> </mrow> </mfrac> <mo>=</mo> <mi>x</mi> <mo>*</mo> <msup> <mi>&delta;h</mi> <mi>k</mi> </msup> <mo>+</mo> <msup> <mi>h</mi> <mrow> <mo>,</mo> <mi>k</mi> </mrow> </msup> <mo>*</mo> <mi>&delta;</mi> <mi>y</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>Wherein, x is input picture, and what E (θ) was represented is to input the error with output, wkRefer to k-th of convolution kernel, i.e. k-th of weights, δ h and δ y are the partial gradient of hidden layer and output layer respectively, h'kRefer to turning over partial gradient corresponding to k-th of characteristic pattern The respective value of volume operation gained;Step 4.2, the weights of each layer of convolutional layer convolution kernel are updated,<mrow> <msup> <mi>w</mi> <mi>k</mi> </msup> <mo>=</mo> <msup> <mi>w</mi> <mi>k</mi> </msup> <mo>-</mo> <mi>&eta;</mi> <mfrac> <mrow> <mo>&part;</mo> <mi>E</mi> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&part;</mo> <msup> <mi>W</mi> <mi>k</mi> </msup> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow><mrow> <msup> <mi>b</mi> <mi>k</mi> </msup> <mo>=</mo> <msup> <mi>b</mi> <mi>k</mi> </msup> <mo>-</mo> <mi>&eta;</mi> <mfrac> <mrow> <mo>&part;</mo> <mi>E</mi> <mrow> <mo>(</mo> <mi>&theta;</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&part;</mo> <msup> <mi>b</mi> <mi>k</mi> </msup> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>Wherein, wkRefer to k-th of convolution kernel, bkFor biasing, η refers to learning rate.
- 2. a kind of method based on line decoder and interpolation sampling optimization convolutional neural networks as claimed in claim 1, its It is characterised by:The implementation method that multiple interpolation samples in step 3 is as follows:f_mapk=Multiple_interpolation_pooling (hk) (4)Wherein, hkWhat is represented is k-th of characteristic pattern of convolutional layer, and Multiple_interpolation_pooling represents multiple Interpolation sampling function, f_mapkRepresent k-th of characteristic pattern of corresponding sample level;Wherein, the gray value of multiple interpolation sampling function by the use of 16 points around point to be sampled is used as bicubic interpolation, bicubic Interpolation formula is as follows:<mrow> <mi>S</mi> <mrow> <mo>(</mo> <mi>w</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mo>|</mo> <mi>w</mi> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>+</mo> <mo>|</mo> <mi>w</mi> <msup> <mo>|</mo> <mn>3</mn> </msup> <mo>,</mo> <mo>|</mo> <mi>w</mi> <mo>|</mo> <mo><</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>4</mn> <mo>-</mo> <mn>8</mn> <mo>|</mo> <mi>w</mi> <mo>|</mo> <mo>+</mo> <mn>5</mn> <mo>|</mo> <mi>w</mi> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>-</mo> <mo>|</mo> <mi>w</mi> <msup> <mo>|</mo> <mn>3</mn> </msup> <mo>,</mo> <mn>1</mn> <mo>&le;</mo> <mo>|</mo> <mi>w</mi> <mo>|</mo> <mo><</mo> <mn>2</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>0</mn> <mo>,</mo> <mo>|</mo> <mi>w</mi> <mo>|</mo> <mo>&GreaterEqual;</mo> <mn>2</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>If i+u, j+v are pixel coordinate to be asked, i, j are positive integer, and u, v are the decimal less than 1 more than zero, then pixel grey scale to be asked Value f (i+u, j+v) be,F (i+u, j+v)=ABC (6)Wherein A, B, C are matrix, and form is as follows:<mrow> <mtable> <mtr> <mtd> <mrow> <mi>A</mi> <mo>=</mo> <mo>&lsqb;</mo> <mtable> <mtr> <mtd> <mrow> <mi>S</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>S</mi> <mrow> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>S</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>S</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>-</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>B</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mi>j</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mi>j</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mi>j</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>S</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>S</mi> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>S</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>S</mi> <mrow> <mo>(</mo> <mn>2</mn> <mo>-</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>Wherein, f (i, j) represents the gray value of source images (i, j) place pixel.
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