# Scale-space and edge detection using anisotropic diffusion

## Summary (2 min read)

### 1. INTRODUCTION HE importance of multiscale descriptions of images

- This paper is organized as follows: Section I1 critiques the standard scale space paradigm and presents an additional set of criteria for obtaining ''semantically meaningful" multiple scale descriptions.
- In Section I11 the authors show that by allowing the diffusion coefficient to vary, one can satisfy these criteria.
- In Section VI the authors compare their scheme with other edge detection schemes.

### A . The Maximum Principle

- The causality criterion requires that no new features are introduced in the image in passing from fine to coarse scales in the scale-space.
- Therefore the causality criterion can be established by showing that all maxima and minima in the scale-space belong to the original image.
- The diffusion equation (3) is a special case of a more general class of elliptic equations that satisfy a maximum principle.
- The principle states that all the maxima of the solution of the equation in space and time belong to the initial condition (the original image), and to the boundaries of the domain of interest provided that the conduction coefficient is positive.
- Where the adiabatic boundary case is also treated, and weaker hypotheses on the conduction coefficient are used.the authors.

### B. Edge Enhancement

- With conventional low-pass filtering and linear diffusion the price paid for eliminating the noise, and for performing scale space, is the blurring of edges.
- The authors will show here that if the conduction coefficient is chosen to be an appropriate function of the image gradient they can make the anisotropic diffusion enhance edges while runningfonvard in time, thus enjoying the stability of diffusions which is guaranteed by the maximum principle.
- Imentally one observes that the areas where 4r V. EXPERIMENTAL RESULTS.
- Less crude approximations of the gradient yielded perceptually similar results at the price of increased computational complexity.
- It is possible to verify that, whatever the choice of the approximation of the gradient, the discretized scheme still satisfies the maximum (and minimum) principle provided that the function g is bounded between 0 and 1 .

### VI. COMPARISON TO OTHER EDGE DETECTION SCHEMES

- This section is devoted to comparing the anisotropic diffusion scheme that the authors present in this paper with previous work on edge detection, image segmentation, and image restoration.
- The authors can thus summarize the advantages of the scheme they propose over linear fixed-neighborhood edge detectors.
- In schemes based on linear smoothing or fixedneighborhood processing the shape and size of the areas where smoothing occurs are constant throughout the image.
- Compare to Fig. 17 where anisotropic diffusion preserves edge junctions, shape, and position.

### B. Energy-Based Methods for Image Reconstruction and Segmentation

- The energy function ( 14) is the sum of two terms: the a priori term (the sum of the "clique" functions V containing the a priori knowledge about the image spacesee any one of [6], [16], [2] for a complete discussion), and a term depending on the data available (the sum of the functions W i ) .
- The gradient of the energy function, which may be computed from (16) differentiating with respect to the zi, is the vector of components therefore the gradient descent algorithm is where A is some "speed" factor.
- The data (the original image) are used as the initial condition.
- Perhaps the only exception is the GNC algorithm proposed by Blake and Zisserman [2] which does not guarantee to find the global optimum for generic images, but appears to be a good compromise between speed and accuracy.

### VII. CONCLUSION

- The authors have introduced a tool, anisotropic diffusion, that they believe will prove useful in many tasks of early vision.
- Implementations on massively parallel architectures like the connection machine would be almost trivial.
- Implementations using hybrid analog-digital networks also seem feasible.
- The authors have shown that the simplest version of anisotropic diffusion can be applied with success to multiscale image segmentation.
- As a preprocessing step it makes thinning and linking of the edges unnecessary, it preserves the edge junctions, and it does not require complicated comparison of images at different scales since shape and position are preserved at every single scale.

Did you find this useful? Give us your feedback

##### Citations

11,727 citations

### Cites background from "Scale-space and edge detection usin..."

...There are a large variety of approaches to achieve this goal, from adaptive Wi ener filtering [31], to implementing isotropic [50] and anisotropic [44] local diffusion processes , a topic which recently received renewed interest [19, 37, 56]....

[...]

9,599 citations

8,738 citations

### Cites background from "Scale-space and edge detection usin..."

...we prevent averaging across edges, while still averaging within smooth regions? Anisotropic diffusion [12, 14] is a popular answer: local image variation is measured at every point, and pixel values are averaged from neighborhoods whose size and shape depend on local variation....

[...]

...Many efforts have been devoted to reducing this undesired effect [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 17]....

[...]

6,804 citations

### Cites background or methods from "Scale-space and edge detection usin..."

...The idea of such filter goes back to Perona and Malik [11]....

[...]

...[11] P. Perona and J. Malik....

[...]

...This averaging may be performed locally: the Gaussian smoothing model (Gabor [7]), the anisotropic filtering (Perona-Malik [11], Alvarez et al. [1]) and the neighborhood filtering (Yaroslavsky [16], Smith et al. [14], Tomasi et al. [15]), by the calculus of variations: the Total Variation minimization (Rudin-Osher-Fatemi [13]), or in the frequency domain: the empirical Wiener filters (Yaroslavsky [16]) and wavelet thresholding methods (Coiffman-Donoho [5, 4])....

[...]

...We shall list formulas permitting to compute and analyze the method noise for several classical local smoothing filters: the Gaussian filtering [7], the anisotropic filtering [1, 11], the Total Variation minimization [13] and the neighborhood filtering [16]....

[...]

...This averaging may be performed locally: the Gaussian smoothing model (Gabor [7]), the anisotropic filtering (Perona-Malik [11], Alvarez et al....

[...]

^{1}, University of Iowa

^{2}, Harvard University

^{3}, Kitware

^{4}, General Electric

^{5}

4,786 citations

### Cites background from "Scale-space and edge detection usin..."

...Functionality includes arithmetic operations, Gaussian and anisotropic denoising filters [38] and intensity inhomogeneity bias field correction [39], among other tools....

[...]

##### References

28,073 citations

18,761 citations

5,516 citations

3,008 citations

2,641 citations