CN109785273A - The method that Steerable filter device is explained and extended based on cyclic coordinate descent method - Google Patents
The method that Steerable filter device is explained and extended based on cyclic coordinate descent method Download PDFInfo
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- CN109785273A CN109785273A CN201811533223.9A CN201811533223A CN109785273A CN 109785273 A CN109785273 A CN 109785273A CN 201811533223 A CN201811533223 A CN 201811533223A CN 109785273 A CN109785273 A CN 109785273A
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- filter device
- steerable filter
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Abstract
The present invention provides a kind of methods that Steerable filter device is explained and extended based on cyclic coordinate descent method, the following steps are included: step 1, it establishes Steerable filter device and small two and multiplies equivalence between the cyclic coordinate decline solver of objective function, and point out that the rolling filters solutions of Steerable filter device are derived during the minimum of objective function；Step 2, a general frame is found to define new class Steerable filter device, and new class Steerable filter device is developed in this frame；Step 3, mathematic(al) treatment is provided for the rolling filters solutions of guiding and be extended.
Description
Technical field
The present invention relates to a kind of computer vision and image processing techniques, especially a kind of to be based on cyclic coordinate descent method pair
The method that Steerable filter device is explained and extended.
Background technique
Most basic smooth tool should be edge perceptual filter in image procossing, can roughly be classified as two classes:
1) Explicit Filtering device converts the input into output using map operator, such as Gauss, bilateral and Steerable filter device, because they
It can clearly be showed with convolution operator；2) implicit filter, without map operator, filtering output is considered as
The minimum device of objective function.So contradiction is easily achieved there have been, Explicit Filtering device and to calculate cost lower, but design new
Explicit Filtering device it is just less simple, and design new implicit filter and can simplify to propose objective function and finding it
Solver is minimized, disadvantage is exactly to require a great deal of time.
The method for solving above-mentioned contradiction situation be exactly using the advantages of a kind of filter come overcome another filter lack
Point.Specifically, if between the iterative solution of the objective function of the map operator and implicit filter of Explicit Filtering device
Connection is established, then individually from the explicit or implicit insurmountable problem of angle, so that it may pass through two filtering of joint
Visual angle solves.There are two the benefits at this joint visual angle: 1) filtering operation of implicit filter and its usage of rolling filtering
It is all to be showed by iterator, therefore new filter can be defined by modified objective function, and it can also be from iteration
Their rolling filtering usage is disclosed during the minimum of solver；2) Explicit Filtering device has deepened the reason to implicit filter
Solution, to replace its original optimization to explain, helps because each minimum transmittance process is mapped by it with primary filtering intension
In intuitively understanding each iteration and its function in optimization.
Connection and stale is established between Explicit Filtering device and implicit filter.Enabling q, p, L and ∧ is respectively that N × 1 is tieed up
Output vector, constraint, N × N Laplacian Matrix and diagonal matrix.Forefathers have been proven that the output of Steerable filter device is similar to
Optimize a Jacobi iteration in (1).But there are also many shortcomings for its discovery, because of Steerable filter device and optimization (1)
Iterative solution device it is not stringent equal, therefore Jacobi algorithm cannot repeatedly replace the behavior of guiding image filter.
Summary of the invention
The purpose of the present invention is to provide one kind to be explained and be extended to Steerable filter device based on cyclic coordinate descent method
Method, comprising the following steps:
Step 1, the equivalence between Steerable filter device and the cyclic coordinate decline solver of least square objective function is established
Property, and point out that the rolling filters solutions of Steerable filter device can be derived during the minimum of objective function；
Step 2, a general frame is found to define new class Steerable filter device, and is developed in this frame new
Class Steerable filter device；
Step 3, mathematic(al) treatment is provided for the rolling filters solutions of guiding and be extended.
Compared with prior art, the present invention having the advantage that
1) we disclose the cyclic coordinate decline solver that Steerable filter device is equivalent to least square objective function, and refer to
The rolling filtering usage of Steerable filter device can be considered as the minimum process of objective function out.
2) we have found a general framework to define new class GF filter, and class GF filtering is developed in this frame
The new example of device.
3) we provide Fundamentals of Mathematics with the method for extending them for two kinds of rolling filters solutions of Steerable filter device.
The invention will be further described with reference to the accompanying drawings of the specification.
Detailed description of the invention
Fig. 1 is explanation reference figure of the invention.
Fig. 2 is Steerable filter device algorithm schematic diagram.
Fig. 3 is based on the Steerable filter device algorithm schematic diagram for rolling mutual junction structure filtering.
Specific embodiment
A method of Steerable filter device is explained and extended based on cyclic coordinate descent method, including establishes guiding filter
Equivalence between wave device and cyclic coordinate descent method, extension Steerable filter device roll filters solutions and explain and its extension, and totally four
A process.Table 1 is the Chinese meaning of specific embodiment Chinese and English abbreviation.
Table 1
In conjunction with Fig. 1, establish the equivalence between Steerable filter device and cyclic coordinate descent method the following steps are included:
Step 1, the definition of Steerable filter device (GF): initially, Steerable filter device be defined by following two points part it is more
Point filter frame:
(1) multiple spot is estimated: in the window ω centered on iiIn, according to linear transformation
Multiple spot, which is calculated, for each pixel i in image area estimates q'i, wherein I is guiding image, (ak,bk) be equation (2) minimum
Device (3), p are filtering inputs, and ε is a constant,WithIndicate filter window ωiThe average value and variance of middle x.
(2) it polymerize: due to k ∈ ωi, each pixel i has multiple estimation points, therefore filter result q is these estimations
The average value of point.
For clarity, the algorithm that above-mentioned Steerable filter device is given in Fig. 2 of Figure of description is realized.
Step 2, cyclic coordinate declines (CCD) algorithmic translation: the minimum device of multi-variable function F can be by once along one
A direction minimizes F to obtain.That is, in each iteration, CCD is according to equation (5) cyclically to each rope of problem
Draw i to be solved.
WhereinIt is vector with y.It therefore, can be from initialization x0Start and obtains sequence { x0,x1,x2..., meet F
(x0)≥F(x1)≥F(x2)≥…。
Step 3, using CCD optimization object function (6)
It can verify that (3) and (4) are the closed-form solutions of (6) by cyclically minimizing q and (a, b).
Step 1: enablingOptimize (7) using CCD
It obtains in (7)Closed-form solution it is as follows:
Step 2: fixedCCD is reused to minimize (9).
It is thereinIt obtainsAre as follows:
So far, except iteration index n except for an additional, equation (8) (10) and equation (3) (4) are of equal value.This means that
The first time cyclic coordinate decline iteration of Steerable filter device and equation (6) is of equal value, wherein original hypothesis q0=p.
Step 4, GF (q is allowedn, I, ε) and indicate that the input of Steerable filter device is qnWhen filtering output.
qn+1=GF (qn,I,ε) (11)
Roll the cyclic coordinate reduced minimum process that filters solutions (11) can be construed to (6).The reason is that, from n-th
What a cyclic coordinate decline iteration obtained has input qnSteerable filter device filter result be equal to qn+1, i.e., (n+1)th circulation
Coordinate declines iteration.
Extend Steerable filter device the following steps are included:
It, can be with modified objective function simultaneously due to the equivalence between Steerable filter device and first time cyclic coordinate decline iteration
New class GF filter is defined using CCD, and can derive rolling filters solutions from CCD iterative process, because of circulation
The iteration of coordinate decline minimizes process and discloses new rolling filters solutions.
This patent has expanded five class GF Steerable filter devices, shown in following steps:
Step 5, spatial simlanty ω spatial perception Steerable filter device (SGF): is added into GFki=exp (- | | k-i | |2/
σ), wherein σ is the constant for controlling spatial simlanty, can decline in solver from the cyclic coordinate of (12) and derive spatial impression
Know Steerable filter device.
Initialize q0The minimum device of=q, (12) can be calculated by being iterated to (13) (14), whereinWithIndicate window ωiIt is weighed relative to Gauss
Weight ωijxjWeighted mean and variance.It therefore, can be by spatial impression according to the equivalence of Steerable filter device and cyclic coordinate decline
Know that Steerable filter device is defined as (13) (14), wherein n=0 and q0=q.
Step 6, total variance Steerable filter device (TVGF): because the cost function (6) of Steerable filter device only considers guiding figure I
Constraint between output q, can not determine optimal output.So by the regular terms of total variance (TV)
It is attached in cost objective function (6) and constructs new objective function (15)
It enablesMake again
It is obtained minimizing device with CCD iterative calculation (16) (17).
It willOptimal solution be expressed as (18), it is identical as definition (7) of GF.
The minimum device of (17) unlike (9)It is equivalent to (19)
Wherein It is quick Fu
In leaf transformation (FFT),Then indicate complex conjugate.It is the Fourier transformation of delta function, | ωk| indicate window ωkIn
Sum of all pixels.
The first time cyclic coordinate that total variance Steerable filter device is defined as (18) (19) is declined into process, and rolling can be used
Filtered version (20) indicates that wherein q=TVGF (p, I, ε, λ) represents the filtering output q of total variance Steerable filter device.
qn+1=TVGF (qn,I,ε,λ) (20)
Step 7, guard Steerable filter device (CGF): the minimum by experiment discovery objective function (6) and (15) is can to neglect
Zero point slightly, shows that Steerable filter device and total variation Steerable filter device can consume image " energy in each iterative process
Amount ", so they are to dissipate.But preferably filter should be conservative, i.e. rolling filter result must converge to one
Untrivialo solution.Therefore cost objective function (21) are constructed, g indicates a reference picture here, for passing through data item (qi-gi)2
Derivation between constraint output q and g.
Initial state assumption q0=p reuses CCD iterative calculation (22) (23) to find the smallest point of objective function (21)
WhereinNotice (22)
Solution it is identical as (18), enableThe solution of (23) can be indicated again are as follows:
Similar to the definition of total variance Steerable filter device, the first time that conservative Steerable filter device is defined as (22) (23) changes
For path, q=CGF (p, I, g, ε, λ)=(1- α) GF (p, I, ε)+α g is then obtained.In addition, the circulation of objective function (21) is sat
Mark reduced minimum process can be restated as roll filtered version (25):
qn+1=CGF (qn,I,g,ε,λ) (25)
Step 8, inverse Steerable filter device (IGF) and inverse conservative Steerable filter device (ICGF): in the filtering side of Steerable filter device
In case, guiding figure is for calculating sharpening result.In turn, guiding figure G can also be estimated by sharpening result q, come with this anti-
Turn filter.Use objective function (26) and original hypothesis G0=I formulates inverse Steerable filter device (IGF).
CCD is applied to objective function (26), is iterated to calculate two sub-problems (27) (28),
Wherein Pass through the P in minimum (27)0(q,Gn, ε), and constructed with (29)Envelope
Closed solutions.
The Least-squares minimization for solving (28), obtains (30)
And inverse Steerable filter device will be defined as the first time cyclic coordinate descent path of (29) (30), wherein G0=I, and will
It exports G and is expressed as (31).
G=IGF (q, I, ε) (31)
Similarly, inverse conservative Steerable filter device (ICGF) G=IGF (q, I, ε) is defined as the first time CCD iteration of (32)
Path, here g control export the derivation between G and g.In practice, usually it is set to input guiding figure I.
The first step is rightIt is solved to obtain (33)
Second step, it is rightIt optimizes to obtain
(34)
It rolls filters solutions and explains and its extend and to comprise the steps of:
Step 9, roll mutual junction structure filtering (RMSF) scheme: GF assumes that the geometry of guiding figure and input weighs each other
It closes.But this is a strong assumption, and GF can be made to generate texture mapping pseudomorphism in filtering.Processing guiding figure input between structure not
A kind of consistent method is to estimate their mutual junction structures.So being optimized to (35) texture-free as a result, wherein to obtain
Optimization specifically can be by assuming that q0=p and G0=I, then four subproblems (36)-(39) Lai Shixian is iterated to calculate,
Wherein(36) (37) for estimate the linear coefficient used in GF and IGF, (38) (39) calculate GF and
The linear combination of IGF.
So far, the present invention successfully discloses RMSF explanation, therefore the above process is known as to be based on rolling mutual junction structure filter
The Steerable filter device (GF-RMSF) of wave.For clarity, the realization algorithm of GF-RMSF is given in Fig. 2 of Figure of description
Frame.
The present invention also uses (CGF (p, I, ε, λ), ICGF (p, I, ε, λ)) to replace based on the mutual junction structure filtering (CGF- of rolling
RMSF the filter in GF) re-assemblies based on the inverse of the mutual junction structure filtering of rolling (GF (p, I, ε), CGF (p, I, ε))
Conservative Steerable filter device (ICGF-RMSF), experiment show that the program has better filter result.
Step 10, flash of light/flashless filtering (RFNF) scheme is rolled: in order to improve flash of light/flashless image pair quality,
Forefathers synthesize a new images using Steerable filter device, which forms primary image B and from flash of light/flashless image to (If,
In) the detail pictures D that is calculated, and spectrum analysis is provided to illustrate why to roll using working as q0=In(41) when can be with
Generate the result of high quality.
Unlike, (41) are construed to the approximation of the cyclic coordinate decline solver of objective function (42),
Wherein Ie=GF (If,If,ε)+τ(If-GF(If,If, ε)), g is IeAlias, objective function is identical as (21).Cause
This, passes through orderThe minimum that (43) carry out function to achieve the objective is iterated to calculate again.
Enable α ≈ 0 andObtain (1- α) ≈ 1 and α Ie≈λ(If-GF(If,If,ε)).In addition, the q in (43)n+1It reduces
To the GF (q with (41) with same formk,If,ε)+λ(If-GF(If,If,ε)).This discovery illustrates the rolling filter of forefathers
Wave scheme is the approximation of (43) special circumstances.Therefore, (41) to (43) can be promoted.It is being transported importantly, compromise is extensive
Better result is produced in dynamic deblurring.
Claims (4)
1. a kind of method that Steerable filter device is explained and extended based on cyclic coordinate descent method, which is characterized in that including
Following steps:
Step 1, it establishes Steerable filter device and small two and multiplies equivalence between the cyclic coordinate decline solver of objective function, and refer to
The rolling filters solutions of Steerable filter device are derived during the minimum of objective function out；
Step 2, a general frame is found to define new class Steerable filter device, and is developed new class in this frame and led
To filter；
Step 3, mathematic(al) treatment is provided for the rolling filters solutions of guiding and be extended.
2. the method according to claim 1, wherein step 1 specifically comprises the following steps:
Step 101, the original definition of Steerable filter device is provided according to multiple spot estimation and polymerization；
Step 102, by establishing Steerable filter device to the objective function execution cyclic coordinate descent algorithm for realizing Steerable filter device
Equivalence between cyclic coordinate descent method.
3. the method according to claim 1, wherein step 2 is according to different scene demands, modified objective function
And minimum solution is carried out to it by cyclic coordinate descent method, Steerable filter device is extended to following five class:
(1) spatial perception Steerable filter device: spatial simlanty weight is added in Steerable filter device, and to objective function by following
Cyclic co-ordinate descent method carries out minimum solution, derives spatial perception Steerable filter device further according to path is minimized；
(2) total variance Steerable filter device: the regular terms of total variance is added in Steerable filter device, in conjunction with Fast Fourier Transform (FFT)
Cyclic coordinate reduced minimum is carried out to objective function and obtains total variance Steerable filter device；
(3) it guards Steerable filter device: being exported between input a reference picture is added in Steerable filter device to constrain filtering
The derivation of equation, obtain reusing cyclic coordinate descent method after new objective function and derive conservative Steerable filter device；
(4) inverse Steerable filter device: inversely estimating guiding figure by sharpening result, for inverting filter, obtains inverse guiding
Filter；
(5) inverse conservative Steerable filter device: being added a reference picture during inversely releasing guiding figure by sharpening result, should
" energy " that reference picture is used to constrain in image filtering dissipates.
4. the method according to claim 1, wherein step 3 specifically comprises the following steps:
Step 301, mathematic(al) treatment is carried out to the mutual junction structure filters solutions of rolling and be extended；
Step 302, mathematic(al) treatment is carried out to rolling flash of light/flashless filters solutions and be extended.
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Application publication date: 20190521 |