CN106530360A - Complementary color wavelet color image processing method - Google Patents

Abstract

The invention belongs to the field of color picture signal processing technology, and particularly to a complementary color wavelet color image processing method. According to the complementary color wavelet color image processing method, a sets of wavelets which are applied on an R channel, a G channel and a B channel of an image and have phases of 0, 2pi/3 and 4pi/3 are designed according to a color relation which is defined by a complementary color theory in visual physiology; designed complementary color decomposition processing is performed on an image for obtaining four complementary color relation coefficient sub-bands of red-green, green-pinkish-red, blue-yellow and black-white; and each complementary color relation coefficient sub-band comprises characteristic coefficients in eight directions of npi/8, n=0,1,....,7. After spatial domain analysis is performed on the characteristic coefficients, the characteristic coefficients are applied on directional color filtering and reinforcing of a color image. After statistical modeling is performed on the characteristic coefficients, the characteristic coefficients are applied on color image channel statistical modeling and pattern searching. Compared with the prior art, the complementary color wavelet color image processing method has advantages of realizing color filtering and reinforcing which are difficultly realized through a traditional method, and supplying more accurate image wavelet statistical modeling, thereby greatly improving color image pattern searching success rate.

Description

Complementary color wavelet color image processing method

Technical Field

The invention belongs to the technical field of color image signal processing, and particularly relates to a color image processing method.

Background

Over the past decades, mainstream image processing methods have been directed to grayscale images. One typically processes a color image by treating the three color channels of the image separately as grayscale images and adding the processing results of the channels as a result of the color image processing. Gray scale map processing methods such methods for processing individual color channels ignore relationships between individual color channels^[1][2]Easily cause the problems of edge distortion, color distortion, etc^[3][4][5]。

In the field of modern image processing, the wavelet method has an irreplaceable position due to the advantages of time-frequency multi-scale analysis, low computational complexity, flexible post-processing and the like. However, most wavelet image processing methods are also based on grayscale images. When applied to color image processing, different color channels often employ identical wavelet filters. In this case, the high-frequency wavelet coefficients due to different channels have a strong strengthThe local color information is difficult to extract^[6]. Other disadvantages of using the same wavelet filter for different channels are also not negligible. Such as lack of phase information, directional selectivity, and shift invariant properties, resulting in randomness of the coefficient relationship between color channels^[7]This randomness may cause hue distortion^[2]And color distortion^[5]. When color processing between channels is to be performed, such as statistical modeling, a large number of complex samplers are required to counteract the strong correlation and coefficient randomness^[8]。

In order to take full advantage of the color relationship, several approaches have been tried. Two of these examples are based on quaternions (Quaterninonalgebra)^[9][10]And single actor (Monogenic)^{[5][11][12][13]}The method of (1). They take the three color components of a pixel point as a whole pure four-element vector. I.e. the RGB channels of the image are mapped as a three-dimensional (3D) vector onto three mutually perpendicular axes of the color cube. Such mutually perpendicular color axis definitions have become widely accepted in the field of image processing^{[2][9][14][15]}. The first and only "color wavelet" to be found in the literature is based on such a definition of the vertical of color pixels^[5]Although projecting colors onto axes that are perpendicular to each other fills the gap in the color relationship, there is no support by visual theory, even more problematic is that such a color description may be more confusing^[9][14]If one were to project such a relationship into a three-dimensional color cube, a misinterpretation would result of red × being green-blueSome simple applications of cube processing, e.g. color edge extraction^[14]And become very complicated.

Therefore, three-dimensional color cubes are not the best way to represent color, but are just an expedient when many color image processing does not find tools. The field of color image processing lacks new processing methods for establishing a connection between color channels^[5]。

It is well known that the visual system of the human eye is a set of tristimulus systems that can be described by the relationship of RGB color rings and complementary colors^[16]Complementary colors are the earliest defined color relationship, which can be traced back to the definition of newton color rings. If white is obtained by mixing two colors, such a pair of colors is complementary. The RGB tristimulus system and the corresponding color circle proposed by the International Commission on illumination (CIE) in 1931 are one of the most widely used color systems. On an RGB color wheel, white can be obtained by mixing any two colors with pi phase difference. The RGB system is therefore particularly suitable for complementary color applications. Recent studies have shown that complementary colors have irreplaceable positions in the fields of color mixing, color constancy, color perception and the like^[16][17]. On the one hand, the field of color image processing is very lacking in new processing methods for establishing color channel relations. On the other hand, complementary colors, which have a color relationship that is important in human vision, have not been applied to color image processing tools such as wavelet analysis. Therefore, the present invention is expected to introduce the complementary color relationship into the wavelet color image processing to design a new color image processing method to fill the blank in the color image processing field.

Disclosure of Invention

The invention aims to provide a color image processing wavelet analysis method based on a human visual system complementary color theory so as to overcome the defects of the existing color image processing method.

The invention provides a color image processing wavelet analysis method, namely a complementary color wavelet color image processing method. The wavelet with corresponding phase relation is designed by adopting complementary color theory conforming to human visual physiological mechanism, and is applied to RGB channel of color image, thereby improving the effect of color image wavelet processing, and the specific steps are as follows:

(1) and designing a one-dimensional complementary color wavelet. According to the definition of complementary color theory, R, G, B three colors correspond to phases 0,2 pi/3 and 4 pi/3 on the color circle, so a group of one-dimensional wavelet filter groups psi with relative phases 0,2 pi/3 and 4 pi/3 is designed by adopting a Thiran all-pass filter and combining a wavelet design method of common factors⁰，ψ^2π/3And psi^4π/3It is called one-dimensional complementary color wavelet. The detailed process is as follows:

in the color ring of the three-color system, the R, G, B three axes are respectively located in the 0,2 pi/3 and 4 pi/3 directions on the color ring. Two colors of arbitrary pi phase difference form a pair of complementary colors. Of these, four pairs of complementary colors, namely red-cyan, green-magenta, blue-yellow and black-white, play an important role in human vision and color perception^[16]. The directions of 0,2 pi/3 and 4 pi/3 of the RGB base can be respectively expressed as 1 and e in a rectangular coordinate system^j2π/3，e^j4π/3. Any color value is [ r, g, b ]]The color tone (direction on the color wheel) of (1) can pass through its phase ∠ (r + g · e) on the color wheel^j2π/3+b·e^j4π/3) To obtain the compound. The intensity value can be obtained by summing the absolute values thereof, i.e. r + g + b | + | g · e^j2π/3|+|b·e^j4π/3L. In order to copy the excellent properties into the color image processing tool of the wavelet, the invention simulates e in a rectangular coordinate system^j2π/3To construct a family of wavelets with a phase difference of 2 pi/3. To this end, the present invention employs a common factor (common factor) algorithm to approximate such a phase delay^[19]。

Consider a Thiran all-pass filter as follows^[19][21]：

Wherein,

and,

wherein, (x)_nRepresents the ascending factor:

(x)_n:＝(x)(x+1)…(x+n-1)；

by D_τ(z), the following approximation can be obtained:

A_τ(z)≈z^-τand z is about 1.

When τ is 2/3, there is:

all-pass filters with phase differences of 2 pi/3 and 4 pi/3 can be obtained, based on which the following three low-pass filters can be constructed:

near z ═ 1, there are:

these three low-pass filters are used to filter out,andthere is a phase difference of approximately 2 pi/3.

Wherein the common factor F₍z₎Can be determined by spectral factorization methods^[19]。

Designed 2 pi/3 phase wavelet, i.e. psi⁰，ψ^2π/3And psi^4π/3Their sum, and their sum of squares are shown in figure 1. From this we can see that the designed wavelet combines 1, e in a rectangular coordinate system^j2π/3And e^j4π/3The property of good relation. Their energy is concentrated under the wavelet framework. The sum of which is 0, i.e.

ψ⁰+ψ^2π/3+ψ^4π/3＝0 (1)

(2) And designing the one-dimensional complementary color wavelet expansion into a two-dimensional single-channel complementary color wavelet. The method comprises the steps of respectively carrying out one-dimensional complementary color wavelet decomposition on two dimensions of horizontal and vertical dimensions, arranging and combining obtained phase components into two-dimensional components, screening non-0 two-dimensional components to obtain two-dimensional wavelet groups with approximate eight directions of n-k pi/8, k-1, 2 …,8 and three phases of theta-0, 2 pi/3 and 4 pi/3. The detailed process is as follows:

in order to make the two-dimensional complementary color wavelet have a phase difference of 2 pi/3, we know the three phase wavelets psi of the one-dimensional complementary color wavelet from equation (1)⁰、ψ^2π/3、ψ^4π/3Satisfy psi⁰+ψ^2π/3+ψ^4π/3＝0。

By employing low-pass and/or high-pass filter banks in two dimensions, respectively, the following three equations can be constructed:

where the subscripts h and v represent the horizontal and vertical dimensions, respectively. The superscript represents the relative phase used for the corresponding dimension. Note that phi need not be a concern⁰+φ^2π/3+φ^4π/3Because of phi on the left side of (2) to (4)⁰+ψ^2π/3+ψ^4π/3＝0。

Formulas (2) to (4) are developed, and there are:

rewriting equations (5) to (7) are the following uniform forms:

wherein,

w_hw_v＝φ_hψ_v,ψ_hφ_v,orψ_hψ_v(9)

and (8) can be further simplified into:

w₁+w₂+w₃+w₄+w₅+w₆+w₇+w₈+w₉＝0， (10)

wherein,

in order to obtain a two-dimensional complementary color wavelet 2 pi/3 phase relationship, the left side of the equation (10) is decomposed into three terms, wherein the sum of the three terms is 0, and the phase difference between the terms is 2 pi/3. In total, there are four component methods, respectively:

(a) w₁+w₅+w₉，w₂+w₆+w₇，w₃+w₄+w₈；

(di) w₁+w₆+w₈，w₂+w₄+w₉，w₃+w₅+w₇；

(III) w₁+w₂+w₃，w₄+w₅+w₆，w₇+w₈+w₉；

(IV) w₁+w₄+w₇，w₂+w₅+w₈，w₃+w₆+w₉。 (12)

As is clear from the formula (10), the sum of the above groups is 0. Therefore, two terms in each group of the formula (12) can determine the remaining one. Without loss of generality, the first two terms of each group in equation (12) are selected and labeled sequentially as u₁,…,u₈. Considering the overall constraint of equation (10), we can define the following partition transform:

u＝Tw, (13)

wherein

u＝(u₁u₂u₃u₄u₅u₆u₇u₈0)^T，

w＝(w₁w₂w₃w₄w₅w₆w₇w₈w₉)^T，

Where T denotes transposition.

Therefore, the element w in the formula (10) can be represented by the formula (13)_iThe combination of i-1, 2, …, and 9 is three terms, where each term has a relative phase of 2 pi/3. (13) T in the formula represents a total of four combinations, and four divisions in the formula (12) can be obtained, where each group can be represented as:

u_2i-1,u_2i,-u_2i-1-u_2i(14)

wherein i is 1,2,3,4

As described in equation (9), the three sub-bands are phi_hψ_v(LH)，ψ_hφ_v(HL) and psi_hψ_v(HH). When the partition transform (13) is applied to three subbands, respectively, we get:

u_φψ＝Tw_φψ，(LH) (15)

u_ψφ＝Tw_ψφ，(HL) (16)

u_ψψ＝Tw_ψψ，(HH) (17)

wherein:

w_φψ＝(w_1φψ,…,w_9φψ)^T， (18)

w_ψφ＝(w_1ψφ,…,w_9ψφ)^T， (19)

w_ψψ＝(w_1ψψ,…,w_9ψψ)^T。 (20)

in practical application, when filters with different phases are applied to the horizontal axis and the vertical axis as shown in formula (9), nine sets of Discrete Wavelets (DWTs) can be used for decomposition to obtain w₁…w₉Wherein each w_nWith three sub-bands w_nφψ、w_nψφAnd w_nψψ. Then, the decomposition coefficients of the same sub-band of the nine discrete wavelet DWT are respectively reduced to form expressions (18) to (20). Final u_φψ、u_ψφAnd u_ψψThis can be achieved by changing the formulae (15) to (17). Each of which contains four divisions of equation (14). But some groups are useless with zero values. We therefore consider only those groups that are non-zero.

In the LH sub-band because of psi⁰+ψ^2π/3+ψ^4π/30, there is:

similarly, there are u_6φψ0 and u_9φψ0. Only in the resulting LH sub-bandThe six terms (corresponding to the three groupings) are non-zero, and they are each u_1φψ、…、u_4φψ、u_7φψAnd u_8φψ。

In the HL sub-band, there are u as well_7ψφ＝u_8ψφ＝u_9φψ0. This means that the HL sub-band has only six terms (three groups), u_1ψφ、…、u_6ψφIs non-zero.

In the HH sub-band, it can be verified as well:

u_5ψψ＝u_6ψψ＝u_7ψψ＝u_8ψψ＝0。

that is, four terms (two groups), u, in the HH sub-band_1ψψ、…、u_4ψψIs non-zero.

There are sixteen total non-zero terms (eight subband groups) that are:

u_1φψ…u_4φψ、u_7φψ、u_8φψ、u_1ψφ…u_6ψφ、u_1ψψ…u_4ψψ。 (21)

therefore, we can define equation (21) as a two-dimensional complementary color wavelet sub-band.

Figure 2 shows eight wavelet sub-bands corresponding to two-dimensional complementary color wavelets. The first two lines are directly from sixteen complementary color wavelet subbands in equation (21) corresponding to the partition transform equation (13), and the third line is derived from equation (10).

(3) And (3) designing the two-dimensional single-channel complementary color wavelet expansion into a two-dimensional multi-channel complementary color wavelet. Namely 2 pi/3 phase difference wavelet of complementary color wavelet level j direction nAndare respectively represented asAndwavelet decomposition is carried out on each color channel of the color image to obtain a complementary color wavelet coefficient vector in the direction n of the hierarchy j Andwhere r, g and b are the channel vectors of the color image and x represents the convolution. Then through the defined intensity operator O^I(d)＝|d^R|+|d^G|+|d^BComplementary color operator O of black and white^C(d)＝d^R+d^G+d^BRed-green complementary color operator O^R(d)＝d^R-d^G-d^BGreen-magenta complementary color operator O^G(d)＝d^G-d^R-d^BBlue-yellow complementary color operator O^B(d)＝d^B-d^R-d^GThe characteristic coefficients of red-cyan, green-magenta, blue-yellow, black-white, etc. in each direction were obtained.

The detailed process is as follows:

the two-dimensional single-channel complementary color wavelet decomposes the gray-scale map into eight-directional sub-bands. Each directional subband contains two directly computed complementary color wavelet subbands and a 2 pi/3 phase difference subband derived from (10).

Renumbering wavelets of these subbands toWhere j is the decomposition level and θ, n correspond to the row and column phases in fig. 2, respectively, i.e. the wavelet groups with approximate direction n-k pi/8, k-1, 2 …,8 and relative phase θ -0, 2 pi/3, 4 pi/3.

Wavelet with 2 pi/3 phase difference in j direction n of complementary color wavelet hierarchyAndare respectively represented asAnd

the complementary color wavelet coefficient vector in the color image j level n direction can be expressed asAndwhere r, g and b are the channel vectors of the color image and x represents the convolution. We can define the two-dimensional complementary color operator as follows:

(1) intensity operator:

O^I(d)＝|d^R|+|d^G|+|d^B|, (22)

(2) chroma or black-white complementary color operator:

O^C(d)＝d^R+d^G+d^B(23)

(3) red-cyan complementary color operator:

O^R(d)＝d^R-d^G-d^B(24)

(4) green-magenta complementary color operator:

O^G(d)＝d^G-d^R-d^B(25)

(5) blue-yellow complementary color operator:

O^B(d)＝d^B-d^R-d^G(26)

the operators are developed from complementary color relations, and can be used for effectively describing the mutual relations among three channels of the color image RGB, so that the color image RGB channels can be used as a unified whole to achieve better processing effect, and the operators are particularly applied to complementary color related color image processing^[16]。

The method can be applied to directional color filtering and enhancement of color images through the spatial domain analysis of the characteristic coefficients, and obtains a better result compared with the prior similar method. The statistical modeling of the characteristic coefficients is applied to the statistical modeling and texture retrieval of the color image channel, and a better result can be obtained compared with the existing similar method.

The complementary color wavelet is based on the traditional wavelet, but has stronger directivity compared with the traditional wavelet, and the mutual relation of the color channels accords with the visual mechanism of human eyes, thereby not only realizing all functions which can be realized by the traditional wavelet, but also conveniently realizing the color image processing capability which is deficient by the traditional wavelet, such as the directional color filtering of a color image. Based on the operators of the complementary color wavelets, (22) to (26), multi-scale color detection, filtering and enhancement can be performed which are not possible or difficult to achieve with conventional wavelets. (22) Intensity operator O in^IAll wavelet coefficients are added up in absolute value and are therefore sensitive to changes in all image channels. The effect of this operator is similar to a conventional wavelet that adds the absolute values of different channel coefficients. Therefore, the operator O in (22)^IThe features which can be extracted by most of the classical wavelets can be extracted to a great extent. Chroma operators such as O in (23)^CAs shown. When the chroma of the color signal changes, the chroma operator O^CAnd will vary accordingly. (24) Operator O given in (26)^R、O^G、O^BThe three pairs of complementary color operators are corresponding to complementary colors of red-cyan, green-magenta and blue-yellow. They can easily extract information that color variations in a color image deviate from complementary color axes. By passingThe color change in each direction can be detected quickly by decomposing the complementary color wavelet and solving the operators (22) to (26).

The complementary color wavelet can establish a good statistical model among color image channels, and is very suitable for color image statistical applications such as color image texture retrieval. Color image texture retrieval is a popular field of image research where the most similar images are searched for in a database based on a known image. One of the most commonly used image texture retrieval methods is the wavelet statistical class supported by physiological studies^[8][27]. Studies have shown that the human eye is difficult to distinguish if the wavelet subband coefficients of both textures have the same edge density^[24]。

The grayscale discrete wavelet generally used for wavelet-based texture retrieval lacks translational invariance and directional selectivity, and therefore lacks color relationship information and much directional information required for feature extraction. The complementary color wavelet focuses on the characteristics of directivity and translation invariance related to colors, so that the retrieval success rate can be greatly improved after the complementary color wavelet replaces a discrete wavelet.

In the search task, wavelet coefficients are usually classified using a Generalized Gaussian Distribution (GGD) + KL divergence (KLD) framework. Where the generalized gaussian distribution is used to model the wavelet coefficients and the KL divergence is used to measure the similarity. The framework is recognized to have good performance under low overhead and high precision in the field of texture retrieval^[8][27]. But the traditional wavelet coefficient classified by the frame lacks directivity and color information, thereby greatly influencing the classification effect. The traditional wavelet coefficient in the GGD + KLD frame is replaced by the complementary color wavelet coefficient with good direction and color discrimination to carry out color texture retrieval, and experiments prove that the success rate of classification is greatly improved.

Drawings

Fig. 1 is a complementary color wavelet base, their sum, and their sum of squares.

Fig. 2 is a diagram of different sub-bands of complementary color wavelets.

Fig. 3 is an analysis (a) and synthesis (b) filter bank of complementary color wavelets.

Fig. 4 is a color image process applying complementary color wavelets. Wherein: (a) an original image. (b) The results of the level 2 vertical chroma operator filtering are used. (c) The results of the level 2 horizontal chroma operator filtering are used. (d) The results of the level 3 diagonal (3 pi/4) chroma operator filtering are used. (e) The result of the sum of the filter coefficients of the red-cyan complementary color operator in all directions of level 2 is used. (f) An original image. (g) The result of the sum of the filter coefficients of the chroma operators in all directions of level 1 is used. (h) The results of the level 4 diagonal (3 pi/4) chroma operator filtering are used. (i) And (5) adopting a result of filtering by using a level 4 diagonal (pi/4) chroma operator. (j) The result of the sum of the filter coefficients of the red-cyan complementary color operator in all directions of level 2 is used.

Fig. 5 is a sample of 10 color textures. From left to right, upper row: brick.01, building.09, Fabric.04, Fabric.09, flowers.05. And (3) lower row: leave.10, metal.00, misc.02, Sand.00, Wood.02.

Detailed Description

1. Design and structure of one-dimensional complementary color wavelet

In order to effectively apply the one-dimensional complementary color wavelet, three sets of real discrete wavelets are generally used to generate three phase components. Each discrete wavelet uses a set of low-pass and high-pass filters satisfying the conditions of Conjugate Quadrature Filter (CQF) and complete reconstruction (PR), and corresponding reconstruction filter bank^[23]. Low-pass decomposition filter obtained in the summary of the inventionThen obtaining corresponding decomposition high-pass filter through CQF conditionAnd obtaining a corresponding reconstruction low-pass filter and a corresponding reconstruction high-pass filter through PR conditions of the wavelets:

then, the wavelet sub-band coefficients are obtained through pyramid type fast decomposition. The one-dimensional complementary color wavelet analysis and synthesis filter bank and flow are shown in fig. 3.

2. Design and structure of two-dimensional single-channel complementary color wavelet

After obtaining the low-pass and high-pass filter banks and the corresponding reconstruction filter banks of the one-dimensional complementary color wavelet, according to the principle of the two-dimensional single-channel complementary color wavelet in the invention content, the steps of analyzing and integrating the two-dimensional single-channel complementary color wavelet can be realized by the following processes:

wavelet analysis:

(1) decomposition low-pass and high-pass filter bank for reading complementary color waveletAnd

(2) the discrete wavelet decomposition is performed in the horizontal direction and the vertical direction by one of three filter groups. There are a total of nine sets of discrete wavelet decompositions, corresponding to the nine combinations in equation (13).

(3) For each discrete wavelet DWT level (except for the low-pass residual level)

(a) According to the formula (18) fromNine LH sub-bands from nine groups of discrete wavelets to form w_φψTaking out nine HL subbands to form w according to equation (19)_ψφTaking out nine HH sub-bands to construct w according to equation (20)_ψψ。

(b) Obtaining u by dividing and converting expressions (15) to (17)_φψ、u_ψφAnd u_ψψ。

(c) According to (21) from u_φψ、u_ψφAnd u_ψψSixteen complementary color wavelet sub-bands are extracted.

Wavelet synthesis:

(1) for each complementary color wavelet sub-band (except for the low-pass residual level)

(a) Carrying the sixteen complementary color sub-bands back to u_φψ、u_ψφAnd u_ψψ. Sub-band u_5φψ、u_6φψ、u_7ψφ、u_8ψφ、u_5ψψ、u_6ψψ、u_7ψψAnd u_8ψψIs set to 0.

(b) Performing an inverse partition transform w ═ T^-1u, passing u_φψ、u_ψφAnd u_ψψObtain corresponding w_φψ、w_ψφAnd w_ψψWherein T is^-1Is an inverse matrix of T in the formula (13).

(c) From w_φψ、w_ψφAnd w_ψψThe same number of items in LH, HL and HH sub-bands are taken out to obtain nine groups of discrete wavelet DWT.

(2) Three sets of reconstruction filters are respectively adopted along the horizontal axis and the vertical axis according to the sequence in the formula (11) Andto reconstruct the wavelet.

3. Design, construction and application of two-dimensional multi-channel complementary color wavelet

The complementary color operators (23) to (26) defined in the summary of the invention can be written in the form of a matrix for each level and direction as follows:

(27) the formula is also called two-dimensional complementary color transformation/operator. Its corresponding inverse transform can be expressed in pseudo-inverse form as follows:

since the inverse transform (28) equation can fully recover the original sub-band coefficients, the (27) and (28) equations are a complete complementary color transform/operator pair.

And performing complementary color wavelet decomposition on the color image to be processed and solving a transformation operator, namely filtering and enhancing components of the complementary color operators in different levels and directions.

FIG. 4 illustrates the use of complementary color operators for directional color filtering of a color image in accordance with the present invention. Fig. 4(a) is an original image. FIGS. 4(b) and (c) are level 2 vertical and horizontal chroma operators, respectivelyAndFIG. 4(d) is a level 3 diagonal chroma operatorFIG. 4(e) is a red-cyan complement for all levels 2Sum of color operators, i.e.

From fig. 4(b) to (d), it can be seen that the chroma operator can suppress the black-white variation well, only highlighting the color variation. Comparing fig. 4(b) and (c), it can be seen that our filter has very good directivity. They react only to color changes in the direction of interest, namely the vertical chrominance change in fig. 4(b) and the horizontal chrominance change in fig. 4 (c). Comparing the level 2 results of fig. 4(b) (c) with the level 3 results of fig. 4(e), it can be seen that level 2 filtering is more responsive to finer variations (level 2 for 4-8 pixels in the wavelet level) and level 3 is more sensitive to coarser variations (level 3 for 8-16 pixels). The approximate width of the variation can be extracted from the responses of the different levels. Fig. 4(e) we choose the red-cyan complementary color operator to suppress the variation along the red-cyan axis. The change in the black-white axis, i.e. the change in brightness, is now highlighted over the change in the red color. From these results we can see that the complementary color wavelet can easily extract the variation of the color of interest along the direction of interest.

Fig. 4(f) to (j) are another example. Fig. 4(f) is an original image. Fig. 4(g) is the sum of all level 1 chroma operators, that is,FIGS. 4(h) and (i) are level 4 chroma operators respectively,andFIG. 4(j) is the sum of all level 2 Red-cyan complementary color operators, i.e.FIG. 4(g) extracts the edge of the color change by extracting the chroma operators in all directions of the finest level. FIG. 4(j) pressing on the red-cyan axisAll changes were made to highlight the black-white change. The two examples shown in fig. 4 are also used in the literature^[2]. By comparison, the method not only can reflect the change of the color, but also can clearly indicate the type, the direction and the size of the change.

A good statistical model can be established among color image channels by utilizing the complementary color wavelet, so that the method is very suitable for color image statistical applications such as color image texture retrieval. Image texture retrieval based on statistical method generally adopts generalized Gaussian distribution GGD and KL divergence measurement based on discrete wavelet^[27]. In the search task, we also use the framework of the generalized Gaussian distribution GGD + KL divergence to classify the wavelet coefficients. Wherein the generalized Gaussian distribution GGD is used to model wavelet coefficients and the KL divergence is used to measure similarity. The reason for adopting this framework is its acknowledged low overhead, high accuracy and good performance in the texture retrieval field^[8][27]。

To compare the inventive CCFL wavelet with a discrete wavelet, dual-tree complex wavelet (DT-CWT) in texture retrieval^[7]) We adopt the formula [27 ]]The same generalized Gaussian distribution GGD + KL divergence setting. The generalized Gaussian distribution GGD + KL divergence is completely kept unchanged, and only the input grayscale discrete wavelet sub-band coefficients are replaced by a grayscale dual-tree complex wavelet, an RGB channel dual-tree complex wavelet and a complementary color wavelet respectively. MIT VisionTexture (Vis-Tex) database^[28]The 40 texture images were retrieved. Our complementary color wavelet uses the complementary color transform given by equation (27). Table 1 shows the average retrieval (%) for the different wavelets. When all conditions were kept constant, replacing only the discrete wavelet with the complementary color wavelet of the present invention, the retrieval rate rose significantly from 75.63% to 86.35%. In contrast, the dual-tree complex wavelet with better directivity and its RGB channel version can only slightly increase the retrieval rate from 75.63% to 77.14% and 78.96%, respectively.

Table 1: average retrieval ratio of different wavelets (%)

Discrete wavelet	Dual-tree complex wavelet	RGB dual-tree complex wavelet	Complementary color wavelet
				75.63	77.14	78.96	86.35

In order to further explore the specific reasons for increasing the retrieval rate, 10 representative texture images were selected, the texture images are shown in fig. 5, and the retrieval rate (%) is shown in table 2.

Table 2: search Rate (%) for 10 specific textures

	Discrete wavelet	Dual-tree complex wavelet	RGB dual-tree complex wavelet	Complementary color wavelet
					Brick.01	76.17	85.55	89.06	100
Buildings.09	90.23	74.61	67.97	93.75
					Fabric.04	62.89	65.23	71.88	80.86
Fabric.09	83.59	84.38	83.20	96.88
					Flowers.05	54.69	59.38	64.06	91.41
Leaves.10	33.98	41.80	51.56	84.77
					Metal.00	73.83	80.47	81.64	98.44
Misc.02	76.95	88.28	91.02	99.61
					Sand.00	77.34	72.27	78.52	99.22
Wood.02	81.64	78.13	72.66	94.92

Textures such as build.09 and wood.02 contain many horizontal and vertical textures. Since the discrete wavelet and the complementary color wavelet have horizontal and vertical components, the dual-tree complex wavelet does not. The discrete wavelet and the complementary color wavelet should perform better than the dual-tree complex wavelet. For directional textures other than horizontal and vertical, such as fabric.04 and misc.02, the dual-tree complex wavelet and the complementary-color wavelet should perform better than the discrete wavelet. As can be seen from Table 2, these points are confirmed.

For textures with multi-directionality, such as fabric.09, flowers.05, metal.00, and san.00, the subbands of the complementary color wavelet are better suited to capture these directional features because of their richer directionality and more concentrated directional subband energy. This is also confirmed in table II.

When texture is more characterized by color variations than directional patterns, such as Brick.01, Fabric.04, and Wood.02, the complementary color wavelets CCWTs of the present invention can accurately retrieve them due to their good color characteristics. When the color and texture features are obvious, such as leave.10, the complementary color wavelet of the invention fully exerts the advantages of directional color filtering, and can greatly improve the retrieval accuracy.

Reference to the literature

[1]C.E.Moxey，S.J.Sangwine，and T.A.Ell，“Hypercomplex correlationtechniques for vector images，”Signal Processing，IEEE Transactions on，vol.51，no.7，pp.1941-1953，2003.

[2]Y.Xu,L.Yu,H.Xu,H.Zhang,and T.Nguyen,“Vector sparse representationof color image using quaternion matrix analysis,”Image Processing,IEEETransactions on,vol.24,no.4,pp.1315–1329,2015.

[3]J.Van De Weijer,T.Gevers,and J.-M.Geusebroek,“Edge and cornerdetection by photometric quasi-invariants,”Pattern Analysis and MachineIntelligence,IEEE Transactions on,vol.27,no.4,pp.625–630,2005.

[4]J.Van de Weijer,T.Gevers,and A.D.Bagdanov,“Boosting color saliencyin image feature detection,”Pattern Analysis and Machine Intelligence,IEEETransactions on,vol.28,no.1,pp.150–156,2006.

[5]R.Soulard,P.Carr′e,and C.Fernandez-Maloigne,“Vector extension ofmonogenic wavelets for geometric representation of color images,”ImageProcessing,IEEE Transactions on,vol.22,no.3,pp.1070–1083,2013.

[6]N.-X.Lian,V.Zagorodnov,and Y.-P.Tan,“Edge-preserving imagedenoising via optimal color space projection,”Image Processing,IEEETransactions on,vol.15,no.9,pp.2575–2587,2006.

[7]I.W.Selesnick,R.G.Baraniuk,and N.G.Kingsbury,“The dualtree complexwavelet transform,”Signal Processing Magazine,IEEE,vol.22,no.6,pp.123–151,2005.

[8]R.Kwitt,P.Meerwald,and A.Uhl,“Efficient texture image retrievalusing copulas in a bayesian framework,”Image Processing,IEEE Transactions on,vol.20,no.7,pp.2063–2077,2011.

[9]T.A.Ell and S.J.Sangwine,“Hypercomplex fourier transforms of colorimages,”Image Processing,IEEE Transactions on,vol.16,no.1,pp.22–35,2007.

[10]W.L.Chan,H.Choi,and R.G.Baraniuk,“Coherent multiscale imageprocessing using dual-tree quaternion wavelets,”Image Processing,IEEETransactions on,vol.17,no.7,pp.1069–1082,2008.

[11]R.Soulard and P.Carr′e,“Color monogenic wavelets for imageanalysis,”in Image Processing(ICIP),2011 18th IEEE International Conferenceon.IEEE,2011,pp.273–276.

[12]M.Felsberg and G.Sommer,“The monogenic signal,”Signal Processing,IEEE Transactions on,vol.49,no.12,pp.3136–3144,2001.

[13]G.Demarcq,L.Mascarilla,and P.Courtellemont,“The color monogenicsignal:A new framework for color image processing.application to coloroptical flow,”in Image Processing(ICIP),2009 16th IEEE InternationalConference on.IEEE,2009,pp.481–484.

[14]S.J.Sangwine,“Colour image edge detector based on quaternionconvolution,”Electronics Letters,vol.34,no.10,pp.969–971,1998.

[15]Q.Barth′elemy,A.Larue,and J.I.Mars,“Color sparse representationsfor image processing:Review,models,and prospects,”Image Processing,IEEETransactions on,vol.24,no.11,pp.3978–3989,2015.

[16]R.W.Pridmore,“Complementary colors theory of color vision:Physiology,color mixture,color constancy and color perception,”ColorResearch&Application,vol.36,no.6,pp.394–412,2011.

[17]R.W.Pridmore,“Complementary colors:the structure of wavelengthdiscrimination,uniform hue,spectral sensitivity,saturation,chromaticadaptation,and chromatic induction,”Color Research&Application,vol.34,no.3,pp.233–252,2009.

[18]N.Kingsbury,“Complex wavelets for shift invariant analysis andfiltering of signals,”Applied and computational harmonic analysis,vol.10,no.3,pp.234–253,2001.

[19]I.W.Selesnick,“The design of approximate hilbert transform pairsof wavelet bases,”Signal Processing,IEEE Transactions on,vol.50,no.5,pp.1144–1152,2002.

[20]R.Yu and H.Ozkaramanli,“Hilbert transform pairs of orthogonalwavelet bases:Necessary and sufficient conditions,”Signal Processing,IEEETransactions on,vol.53,no.12,pp.4723–4725,2005.

[21]J.-P.Thiran,“Recursive digital filters with maximally flat groupdelay,”Circuit Theory,IEEE Transactions on,vol.18,no.6,pp.659–664,1971.

[22]R.A.Gopinath,“The phaselet transform-an integral redundancynearly shift-invariant wavelet transform,”Signal Processing,IEEE Transactionson,vol.51,no.7,pp.1792–1805,2003.

[23]I.Daubechies et al.,Ten lectures on wavelets.SIAM,1992,vol.61.

[24]D.J.Heeger and J.R.Bergen,“Pyramid-based texture analysis/synthesis,”in Proceedings of the 22nd annual conference on Computer graphicsand interactive techniques.ACM,1995,pp.229–238.

[25]L.R.Williams and K.K.Thornber,“A comparison of measures fordetecting natural shapes in cluttered backgrounds,”International Journal ofComputer Vision,vol.34,no.2-3,pp.81–96,1999.

[26]L.Itti,C.Koch,and E.Niebur,“A model of saliency-based visualattention for rapid scene analysis,”IEEE Transactions on Pattern Analysis&Machine Intelligence,no.11,pp.1254–1259,1998.

[27]M.N.Do and M.Vetterli,“Wavelet-based texture retrieval usinggeneralized gaussian density and kullback-leibler distance,”Image Processing,IEEE Transactions on,vol.11,no.2,pp.146–158,2002.

[28]M.Vision and M.Group,“Vision texture,”online at http://vismod.media.mit.edu/vismod/imagery/VisionTexture/.。

Claims (5)

1. A complementary color wavelet color image processing method adopts complementary color theory conforming to human visual physiological mechanism to design wavelets with corresponding phase relation and applies the wavelets to RGB channels of a color image, thereby improving the effect of color image wavelet processing, and is characterized by comprising the following specific steps:

(1) designing a one-dimensional complementary color wavelet: according to the definition of complementary color theory, R, G, B three colors correspond to phases 0,2 pi/3 and 4 pi/3 on a color ring, and a group of wavelet design method with relative phases of 0,2 pi/3 and 4 pi respectively is designed by adopting a Thiran all-pass filter and combining common factorsOne-dimensional wavelet filter bank psi of/3⁰，ψ^2π/3And psi^4π/3Called one-dimensional complementary color wavelet;

(2) the one-dimensional complementary color wavelet expansion is designed into a two-dimensional single-channel complementary color wavelet: respectively carrying out one-dimensional complementary color wavelet decomposition on a horizontal dimension and a vertical dimension, arranging and combining the obtained phase components into two-dimensional components, screening non-0 two-dimensional components to obtain two-dimensional wavelet groups with approximate eight directions of n-k pi/8, k-1, 2-8 and three phases of theta-0, 2 pi/3 and 4 pi/3;

(3) the two-dimensional single-channel complementary color wavelet expansion is designed into a two-dimensional multi-channel complementary color wavelet: namely 2 pi/3 phase difference wavelet of complementary color wavelet level j direction nAndare respectively represented asAndwavelet decomposition is carried out on each color channel of the color image to obtain a complementary color wavelet coefficient vector in the direction n of the hierarchy jAndwhere r, g and b are the channel vectors of the color image and x represents the convolution; then through the defined intensity operator O^I(d)＝|d^R|+|d^G|+|d^BComplementary color operator O of black and white^C(d)＝d^R+d^G+d^BRed-green complementary color operator O^R(d)＝d^R-d^G-d^BIs greenMagenta complementary color operator O^G(d)＝d^G-d^R-d^BBlue-yellow complementary color operator O^B(d)＝d^B-d^R-d^GThe characteristic coefficients of red-cyan, green-magenta, blue-yellow, and black-white in each direction are obtained.

2. The method for processing a complementary color wavelet color image according to claim 1, wherein the process of step (1) is as follows:

in a three-color system color ring, R, G, B three axes are respectively positioned in the directions of 0,2 pi/3 and 4 pi/3 on the color ring; two colors with any pi phase difference form a pair of complementary colors; and the directions of 0,2 pi/3 and 4 pi/3 of the RGB base are respectively expressed as 1 and e in a rectangular coordinate system^j2π/3，e^j4π/3(ii) a Any color value is [ r, g, b ]]Of which the hue can pass through its phase ∠ (r + g · e) on the color circle^j2π/3+b·e^j4π/3) The intensity value can be obtained by the sum of the absolute values thereof, i.e. r + g + b | + | g · e^j2π/3|+|b·e^j4π/3L, |; the property is copied into a color image processing tool of a wavelet, and e in a rectangular coordinate system is imitated^j2π/3To construct a family of wavelets with a phase difference of 2 pi/3, for which a common factor algorithm is used to approximate such phase delays;

the following third all-pass filter was used:

wherein,

and,

wherein (x) n represents an ascending factor:

(x)_n:＝(x)(x+1)…(x+n-1)；

by D_τ(z), the following approximation is obtained:

A_τ(z)≈z-^τnear z ═ 1;

when τ is 2/3, there is:

all-pass filters with phase differences of 2 pi/3 and 4 pi/3 are obtained, based on which the following three low-pass filters are constructed:

near z ═ 1, there are:

these three low-pass filters are used to filter out,andhas a phase difference of approximately 2 pi/3;

wherein the common factor F (z) is determined by a spectral factorization method;

designed 2 pi/3 phase wavelet is phi⁰，ψ^2π/3And psi^4π/3The sum of them is 0:

ψ⁰+ψ^2π/3+ψ^4π/3＝0 (1)。

3. the method for processing a complementary color wavelet color image according to claim 2, wherein the flow of step (2) is:

in order to make two-dimensional complementary color wavelets have a phase difference of 2 pi/3, low-pass and/or high-pass filter banks are used in two dimensions, respectively, and the following three equations are constructed:

wherein, subscripts h and v represent horizontal and vertical dimensions, respectively, and superscripts represent relative phases used by the corresponding dimensions; because there is psi on the left side of the equations (2) to (4)⁰+ψ^2π/3+ψ^4π/30; formulas (2) to (4) are developed, and the following components are arranged:

w₁+w₂+w₃+w₄+w₅+w₆+w₇+w₈+w₉＝0， (10)

wherein,

wherein,

w_hw_v＝φ_hψ_v,ψ_hφ_v,orψ_hψ_v(9)

in order to obtain a two-dimensional complementary color wavelet 2 pi/3 phase relationship, the left side of the equation (10) is decomposed into three terms, wherein the sum of the three terms is 0, and the phase difference between the terms is 2 pi/3, and the four components are respectively adopted in total:

(a) w₁+w₅+w₉，w₂+w₆+w₇，w₃+w₄+w₈；

(di) w₁+w₆+w₈，w₂+w₄+w₉，w₃+w₅+w₇；

(III) w₁+w₂+w₃，w₄+w₅+w₆，w₇+w₈+w₉；

(IV) w₁+w₄+w₇，w₂+w₅+w₈，w₃+w₆+w₉(12)

As can be seen from the formula (10), the sum of each of the above-mentioned groups is 0, so that the two terms of each of the groups in the formula (12) can determine the remaining one term, and without loss of generality, the first two terms of each of the groups in the formula (12) are selected and the order thereof is marked as u₁,…,u₈(ii) a Considering the overall constraint of equation (10), the following partition transform:

u＝Tw, (13)

wherein

u＝(u₁u₂u₃u₄u₅u₆u₇u₈0)^T，

w＝(w₁w₂w₃w₄w₅w₆w₇w₈w₉)^T，

Wherein T represents transpose;

therefore, the element w in the formula (10) is represented by the formula (13)_iI is 1,2, …,9 is three terms, wherein each term has a relative phase of 2 pi/3; (13) t in the formula represents a total of four combinations, and four divisions in the formula (12) can be obtained, where each group can be represented as:

u_2i-1,u_2i,-u_2i-1-u_2i(14)

wherein i is 1,2,3,4

As described in equation (9), the three sub-bands are phi_hψ_v(LH)，ψ_hφ_v(HL) and psi_hψ_v(HH); applying the partition transform (13) to the three subbands, respectively, yields:

u_φψ＝Tw_φψ，(LH) (15)

u_ψφ＝Tw_ψφ，(HL) (16)

u_ψψ＝Tw_ψψ，(HH) (17)

wherein:

w_φψ＝(w_1φψ,…,w_9φψ)^T， (18)

w_ψφ＝(w_1ψφ,…,w_9ψφ)^T， (19)

w_ψψ＝(w_1ψψ,…,w_9ψψ)^T(20)

when filters having different phases are applied to the horizontal and vertical axes as shown in equation (9), nine sets of discrete wavelet decompositions are used to obtain w₁…w₉Wherein each w_nWith three sub-bands w_nφψ、w_nψφAnd w_nψψ(ii) a Then respectively integrating the decomposition coefficients of the same sub-bands of the nine groups of discrete wavelets DWT to form expressions (18) to (20); final u_φψ、u_ψφAnd u_ψψBy changing the formulae (15) to (17); wherein each term comprises four divisions of the formula (14); but some of the groups are zero-valued and useless, so only those groups that are non-zero are considered;

there are sixteen non-zero entries in total, eight subband groups, which are:

u_1φψ…u_4φψ、u_7φψ、u_8φψ、u_1ψφ…u_6ψφ、u_1ψψ…u_4ψψ(21)

the formula (21) is defined as a two-dimensional complementary color wavelet sub-band.

4. The method for processing a complementary color wavelet color image according to claim 3, wherein the flow of step (3) is:

the two-dimensional single-channel complementary color wavelet decomposes the gray scale image into eight-direction sub-bands, and each direction sub-band comprises two complementary color wavelet sub-bands obtained by direct calculation and a 2 pi/3 phase difference sub-band obtained by derivation according to the formula (10);

renumbering wavelets of these subbands toWhere j is the decomposition level and θ, n correspond to the row and column phases, respectively, i.e. a wavelet group with approximate direction n-k pi/8, k-1, 2 …,8 and relative phase θ -0, 2 pi/3, 4 pi/3;

wavelet with 2 pi/3 phase difference in j direction n of complementary color wavelet hierarchyAndare respectively represented asAnd

the complementary color wavelet coefficient vector in the color image j level n direction is expressed as Andwhere r, g and b are the channel vectors of the color image, representing the convolution; the two-dimensional complementary color operator is defined as follows:

(1) intensity operator:

O^I(d)＝|d^R|+|d^G|+|d^B|, (22)

(2) chroma or black-white complementary color operator:

O^C(d)＝d^R+d^G+d^B(23)

(3) red-cyan complementary color operator:

O^R(d)＝d^R-d^G-d^B(24)

(4) green-magenta complementary color operator:

O^G(d)＝d^G-d^R-d^B(25)

(5) blue-yellow complementary color operator:

O^B(d)＝d^B-d^R-d^G(26)

the operators are developed from complementary color relations and are used for describing the mutual relations among the three channels of the color image RGB, so that the color image RGB channels can be used as a unified whole to achieve a better processing effect.

5. The complementary color wavelet color image processing method according to claim 4, wherein directional color filtering and enhancement applied to a color image by spatial analysis of said feature coefficients; and the statistical modeling of the characteristic coefficient is applied to the statistical modeling and texture retrieval of the color image channel.