CN106097328A - A kind of image missing values restoration methods based on non-rigid track base - Google Patents

Abstract

The present invention relates to a kind of image missing values restoration methods based on non-rigid track base, can be with self-defining feature first with track base, assume that non-rigid is made up of r primitive, the image moment rank of matrix formed is 3r+1, again it is carried out singular value decomposition, obtain projection matrix, the character of recycling projection matrix recovers row vector and the missing values of column vector, the missing values these recovered again replaces the missing values of image, successive ignition, until missing values position can correctly be recovered, the advantage of the method is the character that not only make use of column vector during recovering missing values, also use the character of row vector simultaneously, and to all of image and characteristic point all fair plays.

Description

A kind of image missing values restoration methods based on non-rigid track base

Technical field

The invention belongs to computer vision research technical field, especially for non-rigid carry out based on track base lack Mistake value restoration methods.

Background technology

In computer vision, being tracked the characteristic point extracted is one of the committed step of three-dimensional reconstruction.But by In light with the reason such as block, the characteristic point of a part can be caused to lose, this is one of the problem that have to consider of three-dimensional reconstruction. In order to recover missing values, Jacobs is at paper " motion structure having missing point recovers linear fit algorithm " (D Jacobs.Linear fitting with missing data for structure-from-motion[J].Computer Vision and Image Understanding, 2008,82 (1): 57-81.) in, it is the characteristic of 4 according to image array order, The method utilizing submatrix recovers missing values, but is susceptible to the impact of submatrix.Factorization method a big advantage It is can to process any number of image simultaneously, and the every piece image of fair play.Tang W K et al. utilizes factorization This advantage, at paper " a kind of projective reconstruction method of column vector " (Tang W K, Hung Y S.A column-space approach to projective reconstruction[J].Computer Vision and Image Understanding, 2006,101 (3): 166-176.) missing values is recovered, but only take into account the character of column vector. Non-rigid is due to reasons such as motion and deformation, and its degree of freedom increases, in the feelings not having any priori about deformation of body Under condition, the non-rigid containing missing values is carried out reconstruction and remains a difficult problem.Seeking the extra constraint about non-rigid With in terms of the hypothesis of deformation of body, related researcher has done a lot of research.Wherein, Akhter et al. is at nerve information (Akhter I, Sheikh Y, Khan S.Nonrigid Structure from Motion on reason system of IMS conference Trajectory Space[C].Conference on Neural Information Processing Systems, December.2008:41-48.) movement locus proposing non-rigid has the characteristic of lower-dimensional subspace.Gotardo et al. based on The thought of track base, (P F U Gotardo, A M Martinez.Non-in computer vision and pattern recognition meeting rigid structure from motion with complementary rank-3spaces.IEEE Conference On Computer Vision and Pattern Recognition [C], 2011:3065-3072.) utilize column vector matching Image containing missing values is rebuild by algorithm, but only uses the character of column vector.And it is based on forward projection model, When distance between camera to object is much larger than the depth of field of object, error is bigger.

Summary of the invention

All images are treated and based on existing for forward projection model for missing values restoration methods unfairness in prior art Shortcoming, the invention provides that a kind of error is little, each image of being impartial to and utilize column vector under perspective projection model Constraint and row vector constraint solving depth factor so realize missing values recover method.

To achieve these goals, the technical solution adopted in the present invention is to comprise the steps of:

(1) extract in image sequence every piece image can reflect the characteristic point data of movement locus, its image neat Form of degree n n is expressed asWhereinRepresent that the i-th width image contains disappearance Value,Representing the missing values of image, k represents the number of missing values, i=1 ..., n, j=1 ..., l, n and l are respectively image Number and feature are counted；

(2) solve the depth factor of image according to characteristic point data and recover missing values, particularly as follows:

(2.1) assume that camera model is pin-hole model, set up between each characteristic point and the space coordinates of each image Relation, establishes the homogeneous trajectory coordinates of three-dimensional of all characteristic points；

(2.2) the i-th width image jth missing values is set equal to the average of visible point, the most all missing values in the i-th width image All it is represented byE represents and counts seen from the i-th width image, and depth factor is initialized as 1, sets up and initializes Image sequence imaging model；

(2.3) set up image sequence matrix according to initialized image sequence imaging model, and it carried out SVD decomposition, Obtain orthogonal matrix；

(2.4) orthogonal matrix obtained according to decomposition, solves 3r+1 row and front 3r+1 row institute before in image sequence matrix Corresponding projection matrix, r is primitive number；

(2.5) projection matrix corresponding to front 3r+1 row utilizing gained orthogonal matrix calculates the depth factor of each column and right The row missing values answered；

(2.6) each column depth factor obtained by step (2.5) and corresponding row missing values are substituted into step (2.1), weight Multiple step (2.3) and the operation of step (2.5), solve the projection matrix of the front 3r+1 row of image array again, and to often going It is iterated analyzing, again solves the depth factor and corresponding row missing values often gone；

(2.7) step (2.6) is solved the depth factor often gone obtained and corresponding row missing values substitutes into step respectively (2.1), in, step (2.3)～(2.6) is repeated, until the 3r+2 eigenvalue v of gained image sequence matrix_3r+2Value be less than Equal to 10^-5, iteration ends, complete the recovery of missing point in image, the image missing values after being restored.

In above-mentioned steps (2.1), the jth characteristic point of the i-th width image is represented by formula

${\mathrm{\λ}}_{i,j}{\stackrel{\‾}{m}}_{i,j}=P_i{t}_{i,j}---\left(1\right)$

Wherein: λ_i,jIt is the depth factor of the jth characteristic point of the i-th width image,For step (1) The characteristic point homogeneous coordinates extracted, P_iIt is the projection matrix of 3 × 4, t_i,j=[x_i,j y_i,j z_i,j 1]^TIt is the of the i-th width image The three dimensions homogeneous coordinates of j characteristic point, x_i,j、y_i,jAnd z_i,jIt is the three-dimensional track x of jth characteristic point respectively_j、y_jAnd z_jIn The value of the i-th row,a_xd,j, a_yd,j, a_zd,jFor coefficient, σ_d=(σ_1,d … σ_n,d)^TIt is mark base vector, i=1 ..., the amount of images that n, n are extracted by step (1), j=1 ..., l, l are step (1) institute Having the characteristic point sum of image, r is primitive number；

The three-dimensional homogeneous trajectory coordinates expression formula of all characteristic points can be drawn by formula (1)

Above-mentioned steps (2.2) is specifically: by initialized image sequence imaging model matrix M_3n×lExpress, according to step Suddenly (2.1) understand,Then M_3n×l=P_3n×4nS_4n×l。

Above-mentioned steps (2.3) is specifically: brings formula (2) into step (2.2) and can obtain M_3n×l=P_3n×4nC_4n×(3r+1) D_(3r+1)×l, matrix M_3n×lOrder be 3r+1, i.e. to M_3n×lCarry out SVD decomposition, then have

M_3n×l=U_3n×3nV_3n×lΛ_l×l (3)

In formula: U_3n×3nAnd Λ_l×lFor orthogonal matrix, V_3n×l=diag (v₁ … v_3r+1 … v_q) it is diagonal matrix, q=min (3n, l), U is U_3n×3nFront 3r+1 row, Λ is Λ_l×lFront 3r+1 row.

Above-mentioned steps (2.4) is specifically: obtained the throwing of the front 3r+1 row U of all images in image sequence matrix by formula (3) Shadow matrixProjection matrix T with front 3r+1 row Λ_l×l, whereinT_l×l=I-Λ^TΔ, I is unit square Battle array.

Above-mentioned steps (2.5) is specifically: due to M_3n×lIn the orthogonal complement space of subspace that generates at U of either rank vector On projection be all 0, utilize the projection matrix of step (2.4) gainedI.e.

$T_{3n\×3n}^c\left[\begin{array}{c}{\mathrm{\λ}}_{1,j}{m}_{1,j}\\ .\\ .\\ .\\ {\mathrm{\λ}}_{i,j}{\stackrel{\‾}{m}}_{i,j}\\ .\\ .\\ .\\ {\mathrm{\λ}}_{n,j}{m}_{n,j}\end{array}\right]=0_{3n\×1}---\left(4\right)$

Wherein j=1 ..., l, λ_1,j,...,λ_n,jRepresent the depth factor of jth row.

The computing formula of the iterative analysis involved by above-mentioned steps (2.6) is:

${\mathrm{\λ}}_i{\stackrel{\‾}{m}}_i{T}_{l\×l}=0_{3\×l}---\left(5\right)$

Wherein i=1 ..., n, λ_iRepresent the depth factor of the i-th width image.

The present invention utilizes the track base can be with self-defining feature, it is assumed that non-rigid is made up of r primitive, the image formed Rank of matrix is 3r+1, then it is carried out singular value decomposition, obtains projection matrix, the character of recycling projection matrix recover row to Amount and the missing values of column vector, then the missing values of the missing values replacement image that these are recovered, successive ignition, until can be the most extensive Multiple missing values position, it is an advantage of the invention that the character that not only make use of column vector during recovering missing values, also simultaneously Make use of the character of row vector, and to all of image and characteristic point all fair plays.

Compared with prior art, the having the beneficial effects that of this method:

1) present invention utilizes the motion of each point of non-rigid to be continuous print, and the motion of each point is mutually restriction, Think movement locus a little constitute a lower-dimensional subspace.Assume that subspace, by the basis set one-tenth of small echo track, decreases not Know number, improve the robustness of arithmetic speed and algorithm.

2) present invention carries out row vector analysis on the basis of based on column vector analysis again, thus solve depth factor and Recovering missing values, it is ensured that error is less, result is more accurate.

3) in missing values recovery process, it is assumed that camera is pin-hole model, more tallies with the actual situation, the scope of application is more Extensively.

4), when asking the position of missing values, make use of the information of all images and characteristic point, and all of image and feature Point is treated coequally, does not rely on some image and characteristic point for counsel.

Accompanying drawing explanation

Fig. 1 is that in embodiment 1, missing values is the restoration result figure of 11.

Fig. 2 is that in embodiment 1, missing values is the restoration result figure of 13.

Fig. 3 is that in embodiment 1, missing values is the restoration result figure of 21.

Fig. 4 is that in embodiment 1, missing values is the restoration result figure of 31.

Detailed description of the invention

In conjunction with drawings and Examples, the present invention will be described, but the present invention is not limited only to following enforcement situation.

Embodiment 1

The one section of dinosaur sport video utilizing Cameron University's laboratory to record herein, converts thereof into image sequence, hands Work is extracted some characteristic points and is tested.

Carry out missing values recovery according to the method for the present invention, specifically comprise the following steps that

(1) extract in image sequence every piece image can react the characteristic point data of movement locus, its homogeneous form Can be expressed asWhereinRepresent that the i-th width image contains missing values,Representing the missing values of image, k represents the number of missing values, i=1 ..., n, j=1 ..., l, n and l are respectively picture number Mesh and feature are counted；

(2) solve the depth factor of image according to characteristic point data and recover missing values, particularly as follows:

(2.1) assume that camera model is pin-hole model, set up between the characteristic point of image and the space coordinates of characteristic point Relation, establishes the homogeneous trajectory coordinates of three-dimensional of all characteristic points；

The jth characteristic point of the i-th width image is represented by formula

${\mathrm{\λ}}_{i,j}{\stackrel{\‾}{m}}_{i,j}=P_i{t}_{i,j}---\left(1\right)$

Wherein: λ_i,jIt is the depth factor of the jth characteristic point of the i-th width image,For step (1) The homogeneous coordinates of the characteristic point extracted, P_iIt is the projection matrix of 3 × 4, t_i,j=[x_i,j y_i,j z_i,j 1]^TIt it is the i-th width image The three dimensions homogeneous coordinates of jth characteristic point, x_i,j、y_i,jAnd z_i,jIt is the three-dimensional track x of jth characteristic point respectively_j、y_jAnd z_j In the value of the i-th row,a_xd,j, a_yd,j, a_zd,jFor coefficient, σ_d=(σ_1,d … σ_n,d)^TIt is mark base vector, i=1 ..., the amount of images that n, n are extracted by step (1), j=1 ..., l, l are step (1) the characteristic point sum of all images, r is primitive number,

The three-dimensional homogeneous trajectory coordinates expression formula of all characteristic points can be drawn by formula (1)

In the present embodiment, amount of images n is 15 width, and characteristic point sum l is 49, Fig. 1, Fig. 2, Fig. 3, and the primitive number of Fig. 4 Mesh r is respectively 4,4,2,2, mark base vector σ_dRepresent with discrete cosine transform, a_xd,j, a_yd,j, a_zd,jIt is the numerical value randomly generated.

(2.2) the i-th width image jth missing values is set equal to the average of visible point, the most all missing values in the i-th width image All it is represented byE represents and counts seen from the i-th width image, and depth factor is initialized as 1, sets up and initializes Image sequence imaging model, by image sequence imaging model matrix M_3n×lExpress, then:

From step (2.1),Then M_3n×l=P_3n×4nS_4n×l。

(2.3) set up image sequence matrix according to image sequence imaging model, and it is carried out SVD decomposition, obtain orthogonal Matrix, specifically:

Formula (2) is brought in the image sequence imaging model of step (2.2), M_3n×l=P_3n×4nC_4n×(3r+1) D_(3r+1)×l, it is known that matrix M_3n×lOrder be 3r+1, to M_3n×lCarry out SVD decomposition, then have

M_3n×l=U_3n×3nV_3n×lΛ_l×l (3)

In formula: U_3n×3nAnd Λ_l×lFor orthogonal matrix, V_3n×l=diag (v₁ … v_3r+1 … v_q) it is diagonal matrix, q=min (3n, l), U is U_3n×3nFront 3r+1 row, Λ is Λ_l×lFront 3r+1 row.

(2.4) decompose the orthogonal matrix obtained according to SVD, solve 3r+1 row and front 3r+1 row before in image sequence matrix Corresponding projection matrixAnd T_l×l, r is primitive number, whereinT_l×l=I-Λ^TΔ, I is single Bit matrix.

(2.5) projection matrix corresponding to front 3r+1 row U of gained orthogonal matrix is utilizedCalculate the degree of depth of each column The factor and corresponding row missing values, specifically:

Due to M_3n×lIn either rank vector front 3r+1 row U generate subspace the orthogonal complement space on projection be all 0, utilize the projection matrix of step (2.4) gainedThe most available:

$T_{3n\×3n}^c\left[\begin{array}{c}{\mathrm{\λ}}_{1,j}{m}_{1,j}\\ .\\ .\\ .\\ {\mathrm{\λ}}_{i,j}{\stackrel{\‾}{m}}_{i,j}\\ .\\ .\\ .\\ {\mathrm{\λ}}_{n,j}{m}_{n,j}\end{array}\right]=0_{3n\×1}---\left(4\right)$

Wherein j=1 ..., l, λ_1,j,...,λ_n,jRepresent the depth factor of jth row.

AssumeObtain

${\mathrm{\λ}}_{1,j}(b_{\mathrm{\ω},1}{g}_{1,j}+b_{\mathrm{\ω},2}{h}_{1,j}+b_{\mathrm{\ω},3})+\mathrm{...}+{\mathrm{\λ}}_{i,j}(b_{\mathrm{\ω},3i-2}{\stackrel{\‾}{g}}_{i,j}+b_{\mathrm{\ω},3i-1}{\stackrel{\‾}{h}}_{i,j}+b_{\mathrm{\ω},3i})+\mathrm{...}+{\mathrm{\λ}}_{n,j}(b_{\mathrm{\ω},3(p+i)-5}{\stackrel{\‾}{g}}_{p+i-1,j}+b_{\mathrm{\ω},3(p+i)-4}{\stackrel{\‾}{h}}_{p+i-1,j}+b_{\mathrm{\ω},3(p+i)-3})=0$

ω=1 in formula ..., 3n, 0≤p≤n, p are positive integer, i.e. for M_3n×lIn jth row, 3n can be arranged just Journey, has n unknown λ_1,j,...,λ_n,jWith 2p missing valuesWithTherefore can solve The λ of each column_1,j,...,λ_n,jWith corresponding row missing valuesWith

In the present embodiment, amount of images n is 15 width, and characteristic point sum l is 49, the characteristic point of extraction such as Fig. 1, Fig. 2, figure Shown in O shape icon in 3, Fig. 4, missing values is respectively 11,13,21,31, such as Fig. 1, and extensive in Fig. 2, Fig. 3, Fig. 4 Shown in the * shape icon appeared again.

(2.6) each column depth factor obtained by step (2.5) and corresponding row missing values are substituted into step (2.1), weight Multiple step (2.3) and the operation of step (2.5), solve the projection matrix of the front 3r+1 row of image array again, and to often going It is iterated analyzing, again solves the depth factor often gone and corresponding row missing values, specifically:

The computing formula of iterative analysis is:

${\mathrm{\λ}}_i{\stackrel{\‾}{m}}_i{T}_{l\×l}=0_{3\×l}---\left(5\right)$

Wherein i=1 ..., n, λ_iRepresent the depth factor of the i-th width image.

Assume [f₁ f₂ … f_l]^TFor T_l×lEither rank, formula (5) can be expressed as

${\mathrm{\λ}}_{i,1}\left[\begin{array}{c}{g}_{i,1}\\ h_{i,1}\\ 1\end{array}\right]f_1+\mathrm{...}+{\mathrm{\λ}}_{i,j}\left[\begin{array}{c}{\stackrel{\‾}{g}}_{i,1}\\ {\stackrel{\‾}{h}}_{i,1}\\ 1\end{array}\right]f_j+\mathrm{...}+{\mathrm{\λ}}_{i,j+k-1}\left[\begin{array}{c}{\stackrel{\‾}{g}}_{i,j+k-1}\\ {\stackrel{\‾}{h}}_{i,j+k-1}\\ 1\end{array}\right]f_{j+k-1}+\mathrm{...}+{\mathrm{\λ}}_{i,l}\left[\begin{array}{c}{g}_{i,l}\\ h_{i,l}\\ 1\end{array}\right]f_l=0_{3\×1}$

0≤k≤l in formula, k is positive integer, i.e. for M_3n×lIn the i-th width image, 3l equation can be arranged, not have l not The λ known_i,1,...,λ_i,lWith 2k missing valuesWithTherefore can solve often row λ_i,1,...,λ_i,lWith corresponding row missing valuesWith

(2.7) depth factor often gone and the corresponding row missing values that step (2.6) are retrieved substitute into step respectively (2.1), in, step (2.3)～(2.6) is repeated, until the 3r+2 eigenvalue v of gained image sequence matrix_3r+2Value be less than Equal to 10^-5, iteration ends, complete the recovery of missing point in image, the image missing values after being restored.

In order to preferably show the effect of the present invention, it is table 1 by the data compilation in Fig. 1, Fig. 2, Fig. 3 and Fig. 4:

Table 1 is the result of embodiment 1

From table 1 it follows that the inventive method is in the problem recovering image missing values, effectiveness comparison is good, particularly Miss rate is more than 50% when, remain to accurately recover.

The image missing values restoration methods based on non-rigid track base of the present invention is applicable not only to dinosaur movement locus, also Be suitable to other all scenarios that can reflect movement locus, such as Yoga, jog.The content not described in detail in above content It is the content that those skilled in the art are known.

Claims (7)

1. an image missing values restoration methods based on non-rigid track base, it is characterised in that comprise the steps of:

(1) extract in image sequence every piece image can reflect the characteristic point data of movement locus, the homogeneous shape of its image Formula is expressed asWhereinRepresent that the i-th width image contains missing values,Representing the missing values of image, k represents the number of missing values, i=1 ..., n, j=1 ..., l, n and l are respectively picture number Mesh and feature are counted；

(2) solve the depth factor of image according to characteristic point data and recover missing values, particularly as follows:

(2.1) assume that camera model is pin-hole model, set up the relation between each characteristic point and the space coordinates of each image, Establish the homogeneous trajectory coordinates of three-dimensional of all characteristic points；

(2.2) the i-th width image jth missing values is set equal to the average of visible point, the most all missing values in the i-th width image It is expressed asE represents and counts seen from the i-th width image, depth factor is initialized as 1, sets up initialized figure As imaging sequences model；

(2.3) set up image sequence matrix according to initialized image sequence imaging model, and it is carried out SVD decomposition, obtain Orthogonal matrix；

(2.4) orthogonal matrix obtained according to decomposition, solves 3r+1 row and front 3r+1 row before in image sequence matrix corresponding Projection matrix, r is primitive number；

(2.5) projection matrix corresponding to front 3r+1 row utilizing gained orthogonal matrix calculates depth factor and the correspondence of each column Row missing values；

(2.6) each column depth factor obtained by step (2.5) and corresponding row missing values are substituted into step (2.1), repeat step Suddenly (2.3) and the operation of step (2.5), again solves the projection matrix of the front 3r+1 row of image array, and carries out often row Iterative analysis, solves the depth factor and corresponding row missing values often gone again；

(2.7) step (2.6) is solved the depth factor often gone obtained and corresponding row missing values substitutes into step (2.1) respectively In, repeat step (2.3)～(2.6), until the 3r+2 eigenvalue v of gained image sequence matrix_3r+2Value be less than or equal to 10^-5, iteration ends, complete the recovery of missing point in image, the image missing values after being restored.

Image missing values restoration methods based on non-rigid track base the most according to claim 1, it is characterised in that: described In step (2.1), the jth characteristic point of the i-th width image is represented by formula

${\mathrm{\λ}}_{i,j}{\stackrel{\‾}{m}}_{i,j}=P_i{t}_{i,j}---\left(1\right)$

Wherein: λ_i,jIt is the depth factor of the jth characteristic point of the i-th width image,Carried by step (1) The characteristic point homogeneous coordinates taken, P_iIt is the projection matrix of 3 × 4, t_i,j=[x_i,j y_i,j z_i,j 1]^TIt it is the jth of the i-th width image The three dimensions homogeneous coordinates of characteristic point, x_i,j、y_i,jAnd z_i,jIt is the three-dimensional track x of jth characteristic point respectively_j、y_jAnd z_jIn i-th The value of row,a_xd,j, a_yd,j, a_zd,jFor coefficient, σ_d=(σ_1,d … σ_n,d)^TIt is mark base vector, i=1 ..., the amount of images that n, n are extracted by step (1), j=1 ..., l, l are step (1) institute Having the characteristic point sum of image, r is primitive number；

The three-dimensional homogeneous trajectory coordinates expression formula of all characteristic points can be drawn by formula (1)

Image missing values restoration methods based on non-rigid track base the most according to claim 1, it is characterised in that: described Step (2.2) is specifically: by initialized image sequence imaging model matrix M_3n×lExpress, understand according to step (2.1),Then M_3n×l=P_3n×4nS_4n×l。

Image missing values restoration methods based on non-rigid track base the most according to claim 1, it is characterised in that: described Step (2.3) is specifically: brings formula (2) into step (2.2) and can obtain M_3n×l=P_3n×4nC_4n×(3r+1)D_(3r+1)×l, matrix M_3n×l Order be 3r+1, i.e. to M_3n×lCarry out SVD decomposition, then have

M_3n×l=U_3n×3nV_3n×lΛ_l×l (3)

In formula: U_3n×3nAnd Λ_l×lFor orthogonal matrix, V_3n×l=diag (v₁ … v_3r+1 … v_q) it is diagonal matrix, q=min (3n, L), U is U_3n×3nFront 3r+1 row, Λ is Λ_l×lFront 3r+1 row.

Image missing values restoration methods based on non-rigid track base the most according to claim 1, it is characterised in that: described Step (2.4) is specifically: obtained the projection matrix of the front 3r+1 row U of all images in image sequence matrix by formula (3)With The projection matrix T of front 3r+1 row Λ_l×l, whereinT_l×l=I-Λ^TΔ, I is unit matrix.

Image missing values restoration methods based on non-rigid track base the most according to claim 1, it is characterised in that: described Step (2.5) is specifically: due to M_3n×lIn either rank vector U generate subspace the orthogonal complement space on projection be all 0, utilize the projection matrix of step (2.4) gainedI.e.

$T_{3n\×3n}^c\left[\begin{array}{c}{\mathrm{\λ}}_{1,j}{m}_{1,j}\\ .\\ .\\ .\\ {\mathrm{\λ}}_{i,j}{\stackrel{\‾}{m}}_{i,j}\\ .\\ .\\ .\\ {\mathrm{\λ}}_{n,j}{m}_{n,j}\end{array}\right]=0_{3n\×1}---\left(4\right)$

Wherein j=1 ..., l, λ_1,j,...,λ_n,jRepresent the depth factor of jth row.

Image missing values restoration methods based on non-rigid track base the most according to claim 1, it is characterised in that: described The computing formula of the iterative analysis involved by step (2.6) is:

${\mathrm{\λ}}_i{\stackrel{\‾}{m}}_i{T}_{l\×l}=0_{3\×l}---\left(5\right)$

Wherein i=1 ..., n, λ_iRepresent the depth factor of the i-th width image.