# Two-Dimensional Hermite Filters Simplify the Description of High-Order Statistics of Natural Images

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Two-Dimensional Hermite Functions: Definition and Properties

#### 2.2. Two-Dimensional Hermite Functions: Explicit Expressions

#### 2.3. Natural Images

#### 2.4. Analysis

## 3. Results

#### 3.1. Statistics of Rank Two TDH Filter Coefficients for Natural Images

#### 3.2. Statistics of Higher-Rank TDH Filter Coefficients for Natural Images

#### 3.3. Statistics TDH Filter Coefficients for Altered Images

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

TDH | Two-dimensional Hermite |

## References

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**Figure 1.**Two-dimensional Hermite (TDH) functions of rank 0–7 in (

**A**) polar form and (

**B**) Cartesian form. The pseudocolor scale (red positive, blue negative) is chosen separately for each function to cover the entire range. Modified with permission from Figure 1 in [29], Victor et al., J. Neurophysiol. 95, 375-400. American Physiological Society. 2006.

**Figure 2.**Cartesian TDH functions are a linear combination of polar TDH functions. Examples are shown for rank 2 (

**left**) and rank 3 (

**right**). For rank 2, the coefficients are $a=\sqrt{2}/2,b=1,c=-\sqrt{2}/2$. For rank 3, the coefficients are $a=1/2,b=\sqrt{3}/2,$ $c=-\sqrt{3}/2,d=-1/2$.

**Figure 3.**The seven filter sizes used to calculate image statistics, compared to the size of natural images used in this study (1536 × 1024).

**Figure 4.**Generalized steerability of the rank two TDH filters. Each unit-magnitude filter corresponds to a point on the surface of a sphere. The polar and Cartesian basis functions form two sets of orthogonal coordinate axes. Filters with a red frame are polar TDH filters; filters with a blue frame are Cartesian TDH filters; one filter is in both sets as indicated by its two frames. Filters without a frame are intermediate filters; they can be constructed from a linear combination of either polar or Cartesian filters.

**Figure 5.**Skewness and kurtosis for natural images filtered by rank two TDH filters across seven spatial scales. Each sphere represents the filter space of unit-length rank two TDH filters (oriented as shown in Figure 4). Skewness and kurtosis are averaged across all filtered images and plotted as a function of direction in the filter space. The pseudocolor scales for each skewness and kurtosis map are set to range from blue (minimum) to red (maximum). The minimum and maximum skewness and kurtosis values are shown under each sphere.

**Figure 6.**Variance, skewness and kurtosis for (

**A**) natural images filtered by polar TDH filters of rank 0–7 (spatial scale four) and (

**B**) modified TDH filters in which the polynomial component is replaced by its sign. Error bars are three standard errors of measurement (SEM).

**Figure 7.**Skewness and kurtosis of natural images filtered by 1000 random TDH filters of rank 2–7, at scale four. The abscissa is the projection of each random TDH filter onto the polar TDH filter shown at the lower right of each plot, which is the target-like filter for even ranks and the filter with a single, horizontal inversion axis for odd ranks. The filters placed along the abscissa are examples of filters whose projections onto the rightmost polar filter are 0, 0.25, 0.5 and 0.75. They illustrate the diversity of filters with a given value of the projection; the examples shown for the skewness and kurtosis columns at corresponding points along the abscissa are interchangeable.

**Figure 8.**At each spatial scale, the skewness of TDH-filtered images is characterized by two values: ${\gamma}_{3,target}$ (

**A**) for even ranks and ${\gamma}_{3,horiz}$ for odd ranks (

**B**); and kurtosis is characterized by ${\gamma}_{4,target}$ (

**C**) for even ranks and ${\gamma}_{4,non-target}$ for all ranks (

**D**).

**Figure 9.**Variance, skewness and kurtosis for (

**A**) natural images filtered by polar TDH filters of rank 0–7 (spatial scale four) after local mean subtraction; (

**B**) as in (A), but natural images are whitened prior to analysis. Error bars are three SEM.

**Figure 10.**Skewness and kurtosis TDH filters of rank 0–7 (spatial scale four) processed by pointwise nonlinearities prior to analysis. (

**A**) Logarithmic transformation; (

**B**) histogram equalization; (

**C**) Gaussian luminance distribution. Error bars are three SEM.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Hu, Q.; Victor, J.D.
Two-Dimensional Hermite Filters Simplify the Description of High-Order Statistics of Natural Images. *Symmetry* **2016**, *8*, 98.
https://doi.org/10.3390/sym8090098

**AMA Style**

Hu Q, Victor JD.
Two-Dimensional Hermite Filters Simplify the Description of High-Order Statistics of Natural Images. *Symmetry*. 2016; 8(9):98.
https://doi.org/10.3390/sym8090098

**Chicago/Turabian Style**

Hu, Qin, and Jonathan D. Victor.
2016. "Two-Dimensional Hermite Filters Simplify the Description of High-Order Statistics of Natural Images" *Symmetry* 8, no. 9: 98.
https://doi.org/10.3390/sym8090098