The application of directional derivative in the design of animation characters and background elements

Qing Zhang

Detalles de publicación y publicación

Applied Mathematics and Nonlinear Sciences

Detalles de la revista

Formato: Revista
eISSN: 2444-8656
Primera edición: 01 Jan 2016
Calendario de la edición: 2 veces al año
Idiomas: Inglés

As the main content of animation creation, animation characters are an important factor to attract the audience, which affects the creation quality of the whole animation work [1, 2]. The rise of the animation industry has gradually become clear, so the animation character design market has also been promoted, involving animation derivatives, games, advertisements, posters, mascots, etc., and the commercial value of animation characters has gradually increased [3, 4]. The characters in animation works, like the actors in film and television dramas, undertake the role of interpreting the content of the story, promoting the development of the plot, highlighting the character of the characters and forming the style of the film [5–7]. Excellent animated images can attract audiences of different ages. With time, audiences may gradually forget the plots of animation works, but animated characters with unique personality charm and modelling characteristics often leave a deep impression on audiences [8–10]. Excellent animation characters convey a correct outlook on life, world outlook and values by their pleasing modelling design, smart, humorous and optimistic personality characteristics [11, 12]. To a certain extent, it has the role of ideological inspiration for the audience [13–15]. There are many problems in animation creation. The application of directional derivatives and visual restoration can well solve the problems caused by animation characters during construction and can perform theoretical calculations and modelling more efficiently [16].

Video repair is the process of repairing the incomplete area of picture information, mainly to restore the information contained in the primitive itself. In today’s society where data technology is becoming more and more advanced, picture information has also become one of the important means of signal transmission for human beings. Video inpainting technology can be based on computer-aided geometric feature inpainting and texture-based video inpainting (completion) in two directions [17, 18]. Video enhancement is one of the key technologies in animation design and background element design. Due to the instability of the scene, inaccurate internal parameter values of the computer, equipment damage, etc., uneven lighting, colour distortion, and overlapping areas may appear in the creation caused by visual degradation. Under the higher requirements of human beings for visual quality, the contrast of the video occupies an important position. The greater the contrast, the better the visual effect of the video, and the higher the clarity of details [19, 20].

Compared with the above-mentioned simulation application of video texture and primitive repair, this paper will apply the theory of directional derivative to the design of animation characters and background elements. According to the information positioning of the directional derivative, the stable field model and equation of the local texture of the static video can be established, and the definition and information entropy can be obtained by calculating each known pixel point [21,22]. This can be now be compared with known algorithms, and then the realistic degree and effect of animation background elements under the directional derivative theory are evaluated [23,24]. In addition, the directional derivative will be calculated in the background based on the restoration time and the peak noise reduction ratio, and the accurate image restoration parameters will be obtained through research, which will confirm the feasibility of the directional derivative in the design of animation characters and background elements from the side [25, 26].

Introduction to directional derivatives

The knowledge of partial derivatives lets us know that the partial derivatives actually reflect the rate of change of the multivariate function at a point along the coordinate axis. Many problems in life involve solving the rate of change of a multivariate function at a point in other directions. Under this phenomenon, the concept of the directional derivative of a multivariate function at a point along a given direction is proposed.

The most widely admired is ‘Advanced Mathematics’ printed by Tongji University Press. The book takes the binary function $z = f (x, y)$ as an example, the set point is $P_{0} (x_{0}, y_{0})$ , the direction is $l = (\cos α, \cos β)$ , and α and β are the directions of the vector l horn.

When the moving length of the moving point P(x,y) along the l direction is ρ, the function increment is $f (P) - f (P_{0}) = f (x_{0} + ρ \cos α, y_{0} + ρ \cos β) - f (x_{0}, y_{0})$ , and the moving distance of the moving point is $ρ = | P_{0} P |$ . If the following limit exists, the directional derivative of $z = f (x, y)$ at the point $(x_{0}, y_{0})$ along the direction l is defined as this limit.

1 $${\left. {{{\partial f} \over {\partial l}}} \right|_{({x_0},{y_0})}} = \mathop {\lim }\limits_{\rho \to {0^ + }} {{f(P) - f({P_0})} \over {|{P_0}P|}} = \mathop {\lim }\limits_{\rho \to {0^ + }} {{f({x_0} + \rho \cos \alpha ,{y_0} + \rho \cos \beta ) - f({x_0},{y_0})} \over \rho }$$ ${\frac{\partial f}{\partial l}|}_{(x_{0}, y_{0})} = lim_{ρ \to 0^{+}} \frac{f (P) - f (P_{0})}{| P_{0} P |} = lim_{ρ \to 0^{+}} \frac{f (x_{0} + ρ \cos α, y_{0} + ρ \cos β) - f (x_{0}, y_{0})}{ρ}$

This article will study the connection between it and the design of animation elements according to the modified part.

The current rules and requirements for the design of animated characters and background elements

Animation character design is defined when the character designer analyses the character’s shape and character’s characteristics by refining the character information in the script and related materials, and then draws the character’s appearance, expression, siblings, movements, clothing, etc. The production activity lays the groundwork. In the production process of animation works, character design is an important link, and the character image directly affects the style of the work and the expression of the plot. It can transmit visual information to the audience for the first time and promote the development of the story. In animation works, there are usually several animation characters. When designing, we should fully consider the character’s characteristics of different characters and their effects on the development of the storyline, to produce animation characters with different characteristics and improve the audio-visual effect of animation works. Excellent animation character design should follow certain principles, which also play an important role in the development of animation derivatives and stand guarantee for commercial benefits.

The design of animation characters should have the following ideas:

3.1

Aesthetics

Visual effects are one of the important criteria to measure the quality of animation works. To enjoy the visual enjoyment of animation works, audiences need to have beautiful pictures, which include pleasing animation characters.

3.2

Simplicity

The concise and lively character image helps to deepen the audience’s memory, and is also conducive to the development of animation derivative products.

3.3

Unique

The character design of animation should have individuality, and it will have more appeal and vitality.

3.4

Harmony

The animation character design cannot be separated from the storyline and scene design, and the three must be consistent.

When designing character actions, first of all, there must be a clear purpose. The design is based on the overall style and character’s characteristics of animation works, follows the laws of motion, and uses the methods of elasticity, inertia and curve motion to make moderate exaggeration. Scientific and reasonable character pose design, which can be the key to shaping the character and unfolding the plot. Second, attention to details is imperative and needs to be paid. When designing a character’s action pose, a look, a smile, a frown, a widened nostril, clenched teeth, etc., are all detailed actions that express the character’s emotion and personality. The detailed action and posture designed based on the character’s character can make the character fresher, more infectious, and easy to resonate with the audience. Character designers should observe carefully in life, pay attention to accumulation, create creations according to the plot and character characteristics, and pursue better design effects.

The character pedigree of animation work is drawn after the design work of all the appearing characters is completed, and all characters are placed on the same horizon on the design drawing, reflecting the high degree of change and contrast between each character. For original artists and animators, character lineage is to define the relative proportions between different characters to ensure that when multiple characters appear in the same scene, the correct height ratio and body shape can be maintained between characters. This puts forward a stricter requirement for portraits, especially in terms of the clarity and fidelity of imaging, and in close-up shots, it is not allowed to have confusion in the structure and proportions between characters. Therefore, this requires original artists and animators to draw characters strictly under the character pedigree settings to ensure the accuracy and rigour of animation works.

There are many types of software for animation character creation. The software includes two categories: plane animation production software and 3D animation production software. For details, see Figure 1:

Classification of animation character creation software

Video enhancement and restoration application of directional derivatives in the design of animation characters and background elements

4.1

Video enhancement applications of directional derivatives

4.1.1

Video enhancement theory of directional derivatives

In recent years, researchers have extended the application field of directional derivatives into the design and processing of animation characters and scene elements so that the application research of directional derivatives has a new development direction and space. Mathematical variables are used to monitor and control the key parameters of the image in real time, to enhance the video of animations and scenes so that the edge of the image is easier for everyone to identify. At the same time, the texture information of the smooth area of the image is preserved. The G–L (Grümwald–Letnikov) fractional image enhancement method is used in the literature to solve the problem that the effect of integer order differentiation on image texture enhancement is not obvious, and the edge colour of RGB colour images will be distorted after differentiation. Combining the G–L (Grümwald–Letnikov) fractional differential with the directional derivative, the adjacent pixel information of the processed pixels can be fully utilised, and the edge information and texture detail information of the image can be significantly enhanced.

Caputo and G–L definitions are equivalent under the condition that function f(t) satisfies $f^{(k)} (a)$ , $k = 0, 1, 2, \dots, n - 1$ , and function f(t) has $m + 1$ order continuous derivative and m takes at least; otherwise, they are not equivalent. It can be seen from the above analysis that the Caputo definition is an improvement of the G–L definition alone, and the use of the Cauchy integral formula is not considered to be used due to the complex calculation process, poor timeliness and error-prone calculation in large-scale data sets. The G–L definition is mainly used to calculate the analytical solutions of some simpler functions; the Caputo definition is suitable for the analysis of the initial value problem of fractional differential equations, as well as the application in the engineering field. The G–L definition can approximate the function as the weighted summation of the displacement of the function and can be easily rewritten as the convolution operation of the function and the corresponding weight, so it is very suitable for application in the signal field. In this paper, the G–L definition of fractional calculus is mainly used and compared with the directional derivative to measure the ability of the directional derivative in video enhancement.

4.1.2

Derivation of video enhancement formula of direction derivative

4.1.2.1

Directional derivative

In the directional derivative of this part, redefine the directional function mentioned in the second part, then there is, let $z = f (x, y)$ be defined in the neighbourhood D of the point $M (x, y)$ , draw line l from point M. Let the angle of rotation from the positive x-axis to ray l be α, and let $M_{o} (x + Δ x, y + Δ y)$ be another point on l and $M_{0} \in D$ , as shown in Figure 2.

According to the correlation characteristics of directional derivatives, $z = f (x, y)$ can be derived at point $M (x, y)$ , then $z = f (x, y)$ at point M has directional derivatives along any direction l, inferred 2 $\frac{\partial f (x, y)}{\partial l} = \frac{\partial f (x, y)}{\partial x} \cos α + \frac{\partial f (x, y)}{\partial y} \sin α$

Similarly, if $f (x, y)$ has continuous k-order partial derivatives in D, the k-order directional derivative of the function $f (x, y)$ along the direction l at point M is defined as: 3 $${{{\partial ^k}f} \over {\partial {l^k}}} = {\left( {\cos \alpha {\partial \over {\partial x}} + \sin \alpha {\partial \over {\partial y}}} \right)^k}f(x,y)$$ $\frac{\partial^{k} f}{\partial l^{k}} = {(\cos α \frac{\partial}{\partial x} + \sin α \frac{\partial}{\partial y})}^{k} f (x, y)$

Among them, $k = 1, 2, \dots, n, n + 1$ .

4.1.2.2

Definition of G–L Fractional Calculus

The G–L fractional derivative is defined as: 4 $$_a^GD_t^vf(t) = \mathop {\lim }\limits_{h \to 0} {1 \over {{h^v}}}\mathop \sum \limits_{m = 0}^{\left[ {{{t - a} \over h}} \right]} {( - 1)^m} \times {{{\rm{\Gamma }}(v + 1)} \over {m!{\rm{\Gamma }}(v - m + 1)}} \times f(t - mh)$$ $_{a}^{G} D_{t}^{v} f (t) = lim_{h \to 0} \frac{1}{h^{v}} \sum_{m = 0}^{[\frac{t - a}{h}]} {(- 1)}^{m} \times \frac{Γ (v + 1)}{m! Γ (v - m + 1)} \times f (t - m h)$

Let’s analyse it here. The meaning of the left superscript in $aG D_{t}^{v}$ is G–L; v represents the differential degree of the function; a represents the lower boundary; t represents the upper boundary, that is, a is the initial value of t. Gamma function $Γ (n) = \int_{0}^{\infty} e^{- t} t^{n - 1} d t = (n - 1)!$ , h is the differential step size, $\left[ {{{t - a} \over h}} \right]$ $[\frac{t - a}{h}]$ symbol represents the integer in $\frac{t - a}{h}$ . The duration of f(t) refers to $t \in [t - a]$ . If it is evenly distributed according to the step size $h = 1$ , $m = {\left[ {{{t - a} \over h}} \right]^{h = 1}} = [t - a]$ $m = {[\frac{t - a}{h}]}^{h = 1} = [t - a]$ is obtained. The difference formula for deriving the ovrder differential of f(t) is: 5 $\frac{d^{v} f (t)}{d t^{n}} \approx f (t) + (- v) f (t - 1) + \frac{(- v) (- v + 1)}{2} f (t - 2) + \dots + \frac{Γ (- v + 1)}{m! Γ (- v + m + 1)} f (t - m)$

Let H be a variable function, which is defined to represent the change of function f(x,y) in direction l. To specify the amount of change of the function f(x,y) along a certain direction E at any point M(x,y), the fractional differential can be innovatively derived as: 6 $0G D_{t}^{v} f (x_{0}, y_{0}) = \sum_{n = 0}^{\infty} \frac{1}{n!} \frac{\partial^{n} f}{\partial l^{n}} 0 G D_{t}^{v} H^{n} + \frac{1}{(n + 1)!} \frac{\partial^{n + 1} f (x + θ Δ x, y + θ Δ x)}{\partial l^{n + 1}} 0 G D_{t}^{v} H^{n + 1}$ where $0 < θ < l$ , the above formula is called the fractional directional derivative along the direction l at point M(x,y).

4.1.3

Video enhancement effect and analysis of direction derivative

To make the pixels on the image boundary to be processed, it is necessary to calculate the processed image correspondingly in the computer-aided measurement system, and explore the enhancement index that can realise the image. Since the differential order of the fractional differential calculation can be adjusted, different fractional orders are used to process the image so that the better order of enhancement effect can be selected. To verify the superiority of the directional derivative, we compare the Laplacian method, the G–L fractional differential method and the directional derivative itself. Laplacian is the mainstream practical application method of image enhancement, but in most cases, the enhanced image is under micro-vision. It will still look rough, not only lost more detail information, but also has a lot of noise, the enhancement effect is not good. And the algorithm processing of the dark part of the image is not very effective, especially the texture part. However, the G–L fractional differential methods have enhanced it, and the enhancement amplitude is slightly smaller, and there is no sharpening phenomenon. However, the enhancement effect is still not obvious; the enhancement effect of the directional derivative is better, but there is a slight blurring phenomenon at the edge of the image; when the directional derivative is used as a parameter aid, the contrast of the overall grey level of the animated character and the background image is significantly increased, and the local details Obviously enlarged, the texture is clearly visible. The experimental results show that the directional derivative has the best enhancement effect when k = 2–4, and k 3 is selected in this paper. Therefore, in practical applications, the method in this paper can appropriately select the value of k according to the degree of image enhancement, to achieve the optimal situation of the image under the assistance of directional derivative parameterisation.

The quality of image enhancement in this paper is evaluated in a quantitative direction. Quantitative methods are used to evaluate the enhancement effect of the image to reflect the change of the grey value of the image and finally, conduct a comprehensive evaluation. Let x_in be the original image, x_out be the enhanced result image and $(M, N)$ be the size of the image, there are:

4.1.3.1

Clarity

7 $S = \frac{1}{M N} \sum_{i = 1}^{M} \sum_{j = 1}^{N} \sqrt{{(Δ I_{x} (i, j))}^{2} + {(Δ I_{y} (i, j))}^{2}}$

Among them, 8 $Δ I_{x} (i, j) = X_{o u t} (i, j) - X_{o u t} (i - 1, j)$ 9 $Δ I_{y} (i, j) = X_{o u t} (i, j) - X_{o u t} (i, j - 1)$

Clarity is the characteristic of the contrast unit of the tiny details of the video.

4.1.3.2

Information entropy

Let the video grayscale set be ${d_{0}, d_{1}, \dots, d_{i - 1}}$ , its corresponding probability is expressed as ${p_{0}, p_{1}, \dots,$ $p_{i - 1}}$ , and L is the grayscale set. The image histogram of the grayscale range ${0, 1, \dots, L - 1}$ , and its entropy H is as follows: 10 $H = - \sum_{i = 0}^{L - 1} p_{i} \ln (p_{i})$

The larger the H entropy value is, the more information factors the image has, so the image entropy value is a measure of the richness of the information contained in the image.

The differential order of the G–L fractional differential operator can be adjusted, and the image is processed accordingly, and a better order of enhancement effect is reserved. To verify the superiority of the method in this paper, we analyse and compare the Laplacian method, G–L fractional derivative method and directional derivative (k = 2) method, the test results are shown in Table 1.

Table 1

Information entropy and clarity of various enhancement methods

To enhance the image		Method
To enhance the image		Laplacian	G–L	Directional derivative (k=2)
Image 1	S	0.2305	6.5574	12.0354
Image 1	H	1.0455	7.5638	7.8928
Image 2	S	0.1788	5.4325	5.9807
Image 2	H	1.0022	6.8725	7.0901
Image 3	S	0.4482	10.1075	14.8263
Image 3	H	1.0228	6.5826	7.0304

It can be seen from Table 1 that the sharpness and information entropy of the enhanced image obtained by the Laplacian method are relatively low, resulting in a poor visual effect of the enhanced image; the G–L method has a small enhancement range, resulting in low image sharpness. The derivative (application of the method discussed in this paper) enhances the image with the highest clarity and the largest information entropy, indicating that this method has the best enhancement effect on the image. To sum up, the directional derivative proposed in this paper makes full use of the information of the surrounding pixels. It can be seen from both subjective evaluation and objective evaluation methods that the directional derivative has a significant effect on image enhancement. It is deduced from this that in the design that requires a higher quality of video processing technology for animation characters and background elements, the directional derivative can bring a better effect.

4.2

Video repair application of directional derivatives

4.2.1

Derivation of vision repair formula for direction derivatives

At present, the research of image inpainting technology is more popular in texture special effects and set feature video inpainting, which is mainly used for the reconstruction of small defect areas, and most of them use the calculation mode of the partial differential equation (PDE) solution. The edge information of the defect region is diffused from the region boundary to the boundary, and the diffusion information and diffusion direction are determined by the edge information. Its representative algorithms are: the BSCB model, and the curvature diffusion repair algorithm (CDD model). In addition to the different mathematical models used in the above algorithms, the main factors affecting the restoration effect are the selection of diffusion information and the control of diffusion direction. In general, these methods aim to make the overall effect of the restored image imperceptible to the human eye and focus on the overall visual effect, rather than on the reconstruction of the accurate value of a certain point, which also makes them difficult to detect. The clarity of edge and texture detail repair is not deliberately pursued. In addition, because this type of repair model requires repeated iterations and high-order partial derivative calculations, it will affect its repair efficiency. Through the image texture itself and the image texture formation mechanism. The video inpainting based on directional derivatives is proposed first.

4.2.2

Derivation of vision repair formula for direction derivatives

The directional derivative of a scalar field describes the magnitude of the rate at which the scalar function changes in a certain direction in the field. As shown in Figure 2, for the image scalar field with defects, let $I_{0} (r_{i})$ be the function describing the texture of the known area of the image, r_i be the coordinates of any point in the field, l be any ray from r_i in the field, and r be the adjacent l on r_i, and the distance between the two points is $Δ l$ , then the directional derivative of $I_{0} (r_{i})$ at point r_i along the l direction is defined as: 11 $${{\partial {I_0}({r_i})} \over {\partial l}} = \mathop {\lim }\limits_{{\rm{\Delta }}l \to 0} {{{I_0}(r) - {I_0}({r_i})} \over {{\rm{\Delta }}l}}$$ $\frac{\partial I_{0} (r_{i})}{\partial l} = lim_{Δ l \to 0} \frac{I_{0} (r) - I_{0} (r_{i})}{Δ l}$

Therefore, when $Δ l$ is extremely small, we have 12 $I_{0} (r) \approx Δ l^{*} \frac{\partial I_{0} (r_{i})}{\partial l} + I_{0} (r_{i})$

The ray drawn from point r_i along the normal direction of the equipotential line, in this direction, the scalar field $I_{0} (r_{i})$ has the maximum rate of change, which is the gradient of $I_{0} (r_{i})$ at point r_i.

Since the influence of points far away is too small to be ignored, when actually reconstructing a defective pixel point, the area with a distance not greater than $\sqrt{5}$ from the target reconstruction point is taken as the effective area, only considering the effects of points within the area. Definition: 13 $${I_0}({r_i}) = \left\{ {\matrix{ {\varphi ({r_i})} \hfill & {{r_i}\>{\rm{as}}\>{\rm{known}}} \hfill \cr 0 \hfill & {{r_i}\>{\rm{as}}\>{\rm{unknown}}} \hfill \cr } } \right.$$ $I_{0} (r_{i}) = {\begin{cases} φ (r_{i}) & r_{i} as known \\ 0 & r_{i} as unknown \end{cases}$

Let A represent the gradient of the pixel at point B, and C represent the directional derivative of point D in the direction of E. Figure 3 is a randomly selected point, and these quantities are marked. On the premise that F satisfies the appropriate order of magnitude (take the normalised value) available

Let $\nabla I_{0} (r_{i})$ represent the gradient of the pixel at the point r_i, and $\nabla^{\land} I_{0} (r_{i})$ represent the directional derivative of the point r_i in the direction of $(r, r_{i})$ . Figure 3 is a randomly selected point, and these quantities are marked. On the premise that $| r = r_{i} |$ satisfies the appropriate order of magnitude (take the normalised value), available: 14 $\hat{I} (r, r_{i}) = I_{0} (r) \approx I_{0} (r_{i}) + \nabla^{\land} I_{0} (r_{i}) \times | r - r_{i} |$

In the formula, $\hat{I} (r, r_{i})$ represents the estimated value of I(r) only under the influence of point r_i, and 15 $\nabla^{\land} I (r_{i}) = | \nabla I_{0} (r_{i}) | \cos θ$

In the formula, θ is the angle between the gradient $\nabla I_{0} (r_{i})$ and the vector $(r, r_{i})$ , available: 16 $$\hat I(r,{r_i}) = {I_0}({r_i}) + |{\nabla _0}({r_i})| \times |r - {r_i}| \times \cos \theta = {I_0}({r_i})\left( {1 + {{|{I_0}({r_i})| \times |r - {r_i}| \times \cos \theta } \over {{I_0}({r_i})}}} \right)$$ $\hat{I} (r, r_{i}) = I_{0} (r_{i}) + | \nabla_{0} (r_{i}) | \times | r - r_{i} | \times \cos θ = I_{0} (r_{i}) (1 + \frac{| I_{0} (r_{i}) | \times | r - r_{i} | \times \cos θ}{I_{0} (r_{i})})$

4.2.3

The effect and analysis of the visual restoration of the directional derivative

After the above parameters are calculated and processed mathematically, Table 1 lists the repair or reconstruction time and peak signal-to-noise ratio (PSNR) of the above three algorithms. It is convenient to compare the impact of image size on the repair or reconstruction time of several algorithms; at the same time, Image 1 and Image 4 are destroyed respectively, and the number of defective pixels in each image is quite different, so as to compare the impact of the size of the defect area on the repair and reconstruction time of each algorithm.

It can be seen from Table 2 that the repair time of the BSCB algorithm is mainly affected by the size of the image itself; the repair time of the CDD algorithm is greatly affected by the size of the image itself and the size of the defect area; while the repair time of the algorithm in this paper is almost completely determined by the size of the defect area. A horizontal comparison of the repair time in the table shows that the image scalar directional derivative repair efficiency is higher than the previous two algorithms. The larger the image size, the more obvious the advantage of this algorithm in terms of efficiency. From the perspective of PSNR, the scalar directional derivative algorithm is obviously better than the other ones. The algorithm in this paper and the BSCB and CDD methods have their own advantages for different types of images and different types of damage, but they are not as good as the image scalar directional derivative. To sum up, the image scalar directional derivative and the first two algorithms have better overall visual effects on image reconstruction or restoration; since the reconstruction model in this paper focuses on the reconstruction of each defective pixel point, in the reconstruction process, each defective point is reconstructed. All calculations are performed, so that the algorithm in this paper can reconstruct the edge and texture details more clearly and accurately; but because the calculation accuracy of the gradient and the influence function is not enough, the reconstruction accuracy of the edge details by the algorithm in this paper is not high enough, and the edge cannot be completely and accurately reconstructed. Since the reconstruction model in this paper avoids repeated iterations and calculation of high-order partial derivatives, it has high efficiency, especially for large-size images, the reconstruction time advantage of image scalar directional derivatives is obvious.

Table 2

Comparison of repair time and PSNR of several algorithms. PSNR, peak signal-to-noise ratio

Pattern	The total number of pixels of the image	Defect pixels	Repair time/s			PSNR/dB
Pattern	The total number of pixels of the image	Defect pixels	BSCB	CDD	directional derivative	BSCB	CDD	directional derivative
Image 1	128*128	366	5	38	2	36.8	38.2	46.9
Image 2	512*512	2405	58	287	6	45	47.3	36.8
Image 3	68610243	5933	389	1167	15	45.5	46.8	42.5

Conclusion

Through the specific application research of the directional derivative in video enhancement and video restoration, it can be known that its directional derivative is also suitable for the design of animation characters and background elements, and can be more prominent and better than the traditional algorithm. The directional derivative can improve the clarity of the video and the amount of information carried by a higher level, which is based on video enhancement; the directional derivative can also be repaired in the case of video damage, and the repair time and the PSNR should be high-quality and outstanding compared with the comparison object algorithm. Based on the above description, the conclusions can be summarised as follows:

In the aspect of video enhancement, traditional Laplacian, G–L fractional derivative and directional derivative can achieve better video visualisation effect, but in terms of quantitative analysis, whether it is the result of clarity and entropy, the research value of the directional derivative is the same. It is higher than the other two methods, indicating that the directional derivative has an excellent ability to enhance the image.

In terms of image restoration, when dealing with higher pixel images, the BSCB model, the curvature diffusion restoration algorithm CDD model, and the directional derivative model show that the restoration time and PSNR values of the two index values show that the directional derivative has the shortest restoration time among the three. The PSRN value is the smallest, and it maintains a good repair and shaping force for the texture details and edge construction of the video.

Directional derivatives have practical applications in both areas of visual enhancement and restoration in the design of animated characters and background elements. And play an important guiding role in the design of the rising national comics and the design of scene background elements. Especially in the function of texture filling and realistic characterisation and self-healing of edge details, the algorithm principle of directional derivative can bring higher efficiency.

Figuras y tablas

Information entropy and clarity of various enhancement methods

To enhance the image		Method
To enhance the image		Laplacian	G–L	Directional derivative (k=2)
Image 1	S	0.2305	6.5574	12.0354
Image 1	H	1.0455	7.5638	7.8928
Image 2	S	0.1788	5.4325	5.9807
Image 2	H	1.0022	6.8725	7.0901
Image 3	S	0.4482	10.1075	14.8263
Image 3	H	1.0228	6.5826	7.0304

Comparison of repair time and PSNR of several algorithms. PSNR, peak signal-to-noise ratio

Pattern	The total number of pixels of the image	Defect pixels	Repair time/s			PSNR/dB
Pattern	The total number of pixels of the image	Defect pixels	BSCB	CDD	directional derivative	BSCB	CDD	directional derivative
Image 1	128*128	366	5	38	2	36.8	38.2	46.9
Image 2	512*512	2405	58	287	6	45	47.3	36.8
Image 3	68610243	5933	389	1167	15	45.5	46.8	42.5

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Applied Mathematics and Nonlinear Sciences

The application of directional derivative in the design of animation characters and background elements

Qing Zhang
Qing Zhang

Publicado en línea: 23 Dec 2022

Volumen & Edición: AHEAD OF PRINT

Páginas: -

Recibido: 07 Jun 2022

Aceptado: 16 Aug 2022

Detalles de publicación y publicación

Applied Mathematics and Nonlinear Sciences

Vista previa del PDF

Artículo

Fig. 1

Fig. 2

Figuras y tablas

Fig. 1

Fig. 2

Information entropy and clarity of various enhancement methods

Comparison of repair time and PSNR of several algorithms. PSNR, peak signal-to-noise ratio

Referencias

Artículos Recientes

Planifique su conferencia remota con Sciendo

Applied Mathematics and Nonlinear Sciences

The application of directional derivative in the design of animation characters and background elements

Qing ZhangQing Zhang

Publicado en línea: 23 Dec 2022

Volumen & Edición: AHEAD OF PRINT

Páginas: -

Recibido: 07 Jun 2022

Aceptado: 16 Aug 2022

Detalles de publicación y publicación

Applied Mathematics and Nonlinear Sciences

Vista previa del PDF

Artículo

Fig. 1

Fig. 2

Figuras y tablas

Fig. 1

Fig. 2

Information entropy and clarity of various enhancement methods

Comparison of repair time and PSNR of several algorithms. PSNR, peak signal-to-noise ratio

Referencias

Artículos Recientes

Planifique su conferencia remota con Sciendo

Qing Zhang
Qing Zhang