## On Advances in Statistical Modeling of Natural Images (2003)

Venue: | Journal of Mathematical Imaging and Vision |

Citations: | 112 - 6 self |

### BibTeX

@ARTICLE{Srivastava03onadvances,

author = {A. Srivastava and A. B. Lee and E. P. Simoncelli and S-c. Zhu},

title = {On Advances in Statistical Modeling of Natural Images},

journal = {Journal of Mathematical Imaging and Vision},

year = {2003},

volume = {18},

pages = {17--33}

}

### Years of Citing Articles

### OpenURL

### Abstract

Statistical analysis of images reveals two interesting properties: (i) invariance of image statistics to scaling of images, and (ii) non-Gaussian behavior of image statistics, i.e. high kurtosis, heavy tails, and sharp central cusps. In this paper we review some recent results in statistical modeling of natural images that attempt to explain these patterns. Two categories of results are considered: (i) studies of probability models of images or image decompositions (such as Fourier or wavelet decompositions), and (ii) discoveries of underlying image manifolds while restricting to natural images. Applications of these models in areas such as texture analysis, image classification, compression, and denoising are also considered.

### Citations

4018 |
Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images
- Geman, Geman
- 1984
(Show Context)
Citation Context ...(Winkler, 1995), states that full conditionals completely specify the joint density function (under a positivity assumption) and enabled the analysis of images using a Gibbs sampler. Geman and Geman (=-=Geman and Geman, 1984-=-) utilized the equivalence of MRFs and Gibbs distributions to sample from these distributions. Kersten (Kersten, 1987) worked on computing the conditional entropies of the pixel values, given the neig... |

2631 | A theory for multiresolution signal decomposition: The wavelet representation
- Mallat
- 1989
(Show Context)
Citation Context ...cusps at the median and large correlations across different scales. To our knowledge, Field (Field, 1987) was the earliest to highlight the highly kurtotic shapes of wavelet filter responses. Mallat (=-=Mallat, 1989-=-) pointed out that coefficients of multiscale, orthonormal wavelet decompositions of images could be described by generalized Laplacian density (given later in Section 3.2). This non-Gaussian behavior... |

1716 | Nonlinear Dimensionality Reduction by Locally Linear Embedding - Roweis, LK - 2000 |

1673 | Independent Component Analysis - Hyvärinen, Karhunen, et al. - 2001 |

1635 | Eigenfaces vs. Fisherfaces: recognition using class specific linear projection
- Belhumeur, Hespanha, et al.
- 1997
(Show Context)
Citation Context ...5 0 0 0.5 0.5 1 1s12 Srivastava et al. 1997; van Hateren, 1998). Other criteria lead to similar formulations of the subspace basis such as sparsity (Olshausen and Field, 1997), Fisher discrimination (=-=Belhumeur et al., 1997-=-), and non-negative factorization (Lee and Seung, 1999). The use of sparseness is often motivated by the scale invariance of natural images. Approximating the image manifold by a flat subspace is clea... |

1436 |
Independent component analysis, a new concept
- Comon
- 1994
(Show Context)
Citation Context ...to many multiscale decompositions. Similar results were obtained by minimizing the independence of coefficients, under linear decompositions, leading to independent component analysis (Cardoso, 1989; =-=Comon, 1994-=-; Hyvarinen et al., 2001). These observations justify widespread use of orthonormal wavelets in general image analysis applications. Use of Gabor wavelets is also motivated by the fact that the recept... |

1313 |
Embedded image coding using zerotree of wavelet coefcients
- Shapiro
- 1993
(Show Context)
Citation Context ...ependent (Wegmann and Zetzsche, 1990). Shapiro developed a heuristic method for taking advantage of joint dependencies between wavelet coefficients that revolutionized the field of image compression (=-=Shapiro, 1993-=-). Simoncelli and colleagues studied and modeled the dependency between responsessNatural Image Statistics 5 to pairs of bandpass filters and found that the amplitudes are strongly correlated, even wh... |

1239 |
Spatial Interaction and the Statistical Analysis of Lattice Systems (with Discussion
- Besag
- 1974
(Show Context)
Citation Context ...the neighborhood of a pixel is small, and furthermore, stationarity holds if the same conditional density is used at all pixels. Ising and Potts model are the simplest examples of this family. Besag (=-=Besag, 1974-=-; Besag, 1986) expressed the joint density of image pixels as a product of conditional densities, and ignored the normalizer to obtain a pseudo-likelihood formulation. Clifford-Hammersely theorem, see... |

1216 |
Image analysis and mathematical morphology
- Serra
- 1982
(Show Context)
Citation Context ...after contrast normalization at four different scales (N =1, 2, 4, 8). the finite image domain Ω ⊂ IR 2 is completely covered. The dead leaves model dates back to Matheron (Matheron, 1975) and Ser=-=ra (Serra, 1982) in mathema-=-tical morphology. They showed that the probability of any compact set K ⊂ Ω belonging to the same leaf and not being occluded by other leaves equals the ratio E[ν(X0⊖ ˘ K)] E[ν(X0⊕ ˘ K)] .... |

1131 |
Vision: A Computational Investigation into the Human Representation and Processing of Visual Information
- Marr
- 1982
(Show Context)
Citation Context ...elet basis has become an attractive tool. In particular, it is common to use Gabor wavelets (Gabor, 1946) for decomposing the observed images simultaneously in space and frequency. In addition, Marr (=-=Marr, 1982-=-) suggested using the Laplacian of Gaussian filter to model early vision. If one considers images as realizations on a finite, uniform grid in IR 2 , the image space becomes finitedimensional, and one... |

1090 | The Laplacian pyramid as a compact image code
- Burt, Adelson
- 1983
(Show Context)
Citation Context ..., which is based on a block-by-block frequency decomposition. In early 80s, it was recognized that multi-scale wavelet-style representations offer more flexibility and better compression performance (=-=Burt and Adelson, 1983-=-; Vetterli, 1984). Early coders were based on marginal models of coefficients, but this changed abruptly with the development of the first contextual coder by Shapiro (Shapiro, 1993). This coder, and ... |

1083 |
Learning the parts of objects by non-negative matrix factorization
- Lee, Seung
- 1999
(Show Context)
Citation Context ... 1998). Other criteria lead to similar formulations of the subspace basis such as sparsity (Olshausen and Field, 1997), Fisher discrimination (Belhumeur et al., 1997), and non-negative factorization (=-=Lee and Seung, 1999-=-). The use of sparseness is often motivated by the scale invariance of natural images. Approximating the image manifold by a flat subspace is clearly limiting in general situations. In (Zetzsche and R... |

991 |
Emergence of simple-cell receptive field properties by learning a sparse code for natural images
- Field, Olshausen
- 1996
(Show Context)
Citation Context ... cusp at zero and long exponential tails. This suggests a role for linear decompositions that maximize the kurtosis or some another measure of non-Gaussianity. Such efforts (Bell and Sejnowski, 1997; =-=Olshausen and Field, 1996-=-; van Hateren, 1998) have resulted in bases that are spatially oriented with (spatial) frequency bandwidths being roughly one octave, similar to many multiscale decompositions. Similar results were ob... |

987 | On the statistical analysis of dirty pictures
- Besag
- 1986
(Show Context)
Citation Context ...ood of a pixel is small, and furthermore, stationarity holds if the same conditional density is used at all pixels. Ising and Potts model are the simplest examples of this family. Besag (Besag, 1974; =-=Besag, 1986-=-) expressed the joint density of image pixels as a product of conditional densities, and ignored the normalizer to obtain a pseudo-likelihood formulation. Clifford-Hammersely theorem, see for example ... |

940 |
Theory of communication
- Gabor
- 1946
(Show Context)
Citation Context ...s. For instance, in order to capture the locality of objects in images, decomposition of images using a wavelet basis has become an attractive tool. In particular, it is common to use Gabor wavelets (=-=Gabor, 1946-=-) for decomposing the observed images simultaneously in space and frequency. In addition, Marr (Marr, 1982) suggested using the Laplacian of Gaussian filter to model early vision. If one considers ima... |

895 | De-noising by soft-thresholding
- Donoho
- 1995
(Show Context)
Citation Context ... to decompose image into bands of spatial frequency and to threshold the coefficients after some nonlinear transformation, proposed first by Bayer et al. (Bayer and Powell, 1986) and later by Donoho (=-=Donoho, 1995-=-). The nonlinear transformation is used essentially to shrink all wavelet coefficients towards zero. This shrinking is based on thresholding which can implemented as a hard threshold or a soft thresho... |

655 | Relations between the statistics of natural images and the response properties of cortical cells
- Field
- 1987
(Show Context)
Citation Context ...tistics 3 property, called the power law spectrum for images, was first observed by television engineers in the 50’s (Kretzmer, 1952; Deriugin, 1956) and discovered for natural images in late 80s by=-= (Field, 1987)-=- and (Burton and Moorhead, 1987). As summarized in (Mumford and Gidas, 2001), the value of η changes with the image types but is usually a small number. Although Fourier analysis is central to classi... |

641 | Sparse coding with an overcomplete basis set: A strategy by V1
- Olshausen, Field
- 1997
(Show Context)
Citation Context ... results (Comon, 1994; Bell and Sejnowski, −0.5 −0.5 0 0 0.5 0.5 1 1s12 Srivastava et al. 1997; van Hateren, 1998). Other criteria lead to similar formulations of the subspace basis such as sparsi=-=ty (Olshausen and Field, 1997-=-), Fisher discrimination (Belhumeur et al., 1997), and non-negative factorization (Lee and Seung, 1999). The use of sparseness is often motivated by the scale invariance of natural images. Approximati... |

592 |
Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters
- DAUGMAN
- 1975
(Show Context)
Citation Context ...n of one-dimension filters, which leads to non-oriented (mixed diagonal) subbands. Alternate representations that provide better orientation decomposition, and thus higher kurtosis response, include (=-=Daugman, 1985-=-; Watson, 987a; Simoncelli et al., 1992; Donoho and Flesia, 2001). 2.4. Non-Gaussianity of Joint Statistics In addition to the non-Gaussian behavior of marginal statistics, a number of authors have st... |

589 |
Application of the Karhunen-Loeve procedure for the characterization of human faces
- Kirby, Sirovich
- 1990
(Show Context)
Citation Context ...cipal) subspace of the data matrix, and is easily computed using eigen decomposition or singular value decomposition. A common application of PCA is in recognition of people from their facial images (=-=Kirby and Sirovich, 1990-=-) or study of natural images (Hancock et al., 1992). Instead, if the goal is to minimize the statistical correlation between the projected components, or to make them as independent as possible, then ... |

509 | The independent components of natural scenes are edge filters
- Bell, Sejnowskia
- 1997
(Show Context)
Citation Context ...ith high kurtosis, a sharp cusp at zero and long exponential tails. This suggests a role for linear decompositions that maximize the kurtosis or some another measure of non-Gaussianity. Such efforts (=-=Bell and Sejnowski, 1997-=-; Olshausen and Field, 1996; van Hateren, 1998) have resulted in bases that are spatially oriented with (spatial) frequency bandwidths being roughly one octave, similar to many multiscale decompositio... |

478 |
Random Sets and Integral Geometry
- Matheron
- 1975
(Show Context)
Citation Context ...es between adjacent pixels after contrast normalization at four different scales (N =1, 2, 4, 8). the finite image domain Ω ⊂ IR 2 is completely covered. The dead leaves model dates back to Mather=-=on (Matheron, 1975) and-=- Serra (Serra, 1982) in mathematical morphology. They showed that the probability of any compact set K ⊂ Ω belonging to the same leaf and not being occluded by other leaves equals the ratio E[ν(X... |

462 | Shiftable Multi-scale Transforms
- Simoncelli, Freeman, et al.
- 1992
(Show Context)
Citation Context ...hich leads to non-oriented (mixed diagonal) subbands. Alternate representations that provide better orientation decomposition, and thus higher kurtosis response, include (Daugman, 1985; Watson, 987a; =-=Simoncelli et al., 1992-=-; Donoho and Flesia, 2001). 2.4. Non-Gaussianity of Joint Statistics In addition to the non-Gaussian behavior of marginal statistics, a number of authors have studied joint statistics of filter respon... |

400 | Pyramid-based texture analysis/synthesis - HEEGER, BERGEN - 1995 |

297 | Independent component filters of natural images compared with simple cells in primary visual cortex - Hateren, Schaaf - 1998 |

231 |
Image Analysis, Random Fields and Dynamic Monte Carlo Methods
- Winkler
- 1995
(Show Context)
Citation Context ...shapes in contour surfaces of histograms. 3. Emerging Statistical Models on Image Space Earliest, and still widely used, probability models for images were based on Markov random field models (MRFs) (=-=Winkler, 1995-=-). An image field is as a collection of random variables, each denoting a pixel value on a uniformly spaced grid in the image plane. In MRFs, the conditional probability of a pixel value given the rem... |

225 |
General Pattern Theory
- Grenander
- 1993
(Show Context)
Citation Context ...and 2D images, are related by a projection (orthographic or perspective) map, manifolds formed by images can be generated by projecting the manifolds formed using 3D representations. As described in (=-=Grenander, 1993-=-; Miller and Younes, 2002), occurrences of physical objects in 3D system are modeled as the orbits of groups acting on the configurations of objects in the scenes. These orbits are projected into the ... |

209 | Noise Removal via Bayesian Wavelet Coring
- Simoncelli, Adelson
- 1996
(Show Context)
Citation Context ... be described by generalized Laplacian density (given later in Section 3.2). This non-Gaussian behavior of images has also been studied and modeled by Ruderman (Ruderman, 1994), Simoncelli & Adelson (=-=Simoncelli and Adelson, 1996-=-), Moulin & Liu (Moulin and Liu, 1999), and Wainwright (Wainwright and Simoncelli, 2000). Recent work of Thomson (Thomson, 2001) studies the statistics of natural images using phase-only second spectr... |

208 | Statistics of Natural Images and Models - Huang, Mumford - 1999 |

200 | Deriving intrinsic images from image sequences - Weiss - 2001 |

184 | Analysis of multiresolution image denoising schemes using generalized{Gaussian and complexity priors
- Moulin, Liu
- 1999
(Show Context)
Citation Context ...y (given later in Section 3.2). This non-Gaussian behavior of images has also been studied and modeled by Ruderman (Ruderman, 1994), Simoncelli & Adelson (Simoncelli and Adelson, 1996), Moulin & Liu (=-=Moulin and Liu, 1999-=-), and Wainwright (Wainwright and Simoncelli, 2000). Recent work of Thomson (Thomson, 2001) studies the statistics of natural images using phase-only second spectrum, a fourth order statistic, and dem... |

164 |
The statistics of natural images
- Ruderman
- 1994
(Show Context)
Citation Context ...Bialek, 1994) showed its invariance to scaling. Independently, Zhu et al. (Zhu and Mumford, 1997) showed a broader invariance by studying the histograms of wavelet decompositions of images. Ruderman (=-=Ruderman, 1994-=-; Ruderman, 1997) also provided evidence of scale invariance in natural images and proposed a physical model for explaining them. Turiel et al. (Turiel and Parga, 2000) investigated the multi-fractal ... |

158 |
Scale mixtures of normal distribution
- Andrews, Mallows
- 1974
(Show Context)
Citation Context ...elsen, 1977) as specific normal variance-mean mixtures that result when the mixing variable u is of certain class. For µ = β = 0 and ∆ = 1, the resulting family is also called Gaussian scale mixtu=-=re (Andrews and Mallows, 1974-=-) and has seensNatural Image Statistics 9 applications in financial mathematics (Bollerslev et al., 1994) and speech processing (Brehm and Stammler, 1987). Furthermore, if u is a scaled Gamma density ... |

155 | Prior learning and gibbs reaction-diffusion
- Zhu, Mumford
- 1997
(Show Context)
Citation Context ...ure of images. By studying the histograms of the pixel contrasts (log(I(x)/I0)) at many scales, Ruderman et al (Ruderman and Bialek, 1994) showed its invariance to scaling. Independently, Zhu et al. (=-=Zhu and Mumford, 1997-=-) showed a broader invariance by studying the histograms of wavelet decompositions of images. Ruderman (Ruderman, 1994; Ruderman, 1997) also provided evidence of scale invariance in natural images and... |

141 | Statistical Models for Images: Compression, Restoration and Synthesis
- Simoncelli
- 1997
(Show Context)
Citation Context ...ied and modeled the dependency between responsessNatural Image Statistics 5 to pairs of bandpass filters and found that the amplitudes are strongly correlated, even when the signed responses are not (=-=Simoncelli, 1997-=-; Buccigrossi and Simoncelli, 1999). This is illustrated in Figure 1, which shows conditional histograms for several pairs of coefficients. Note that unlike second-order correlations, these dependenci... |

118 |
Exponential decreasing distributions for the logarithm of particle size
- Barndorff-Nielsen
- 1977
(Show Context)
Citation Context ...he conditional density function of X given u is normal with mean µ + uβ and variance u∆, and u is called the mixing variable. Generalized hyperbolic distributions were introduced by BarndorffNiels=-=en (Barndorff-Nielsen, 1977) as-=- specific normal variance-mean mixtures that result when the mixing variable u is of certain class. For µ = β = 0 and ∆ = 1, the resulting family is also called Gaussian scale mixture (Andrews and... |

108 | Group actions, homeomorphisms, and matching: a general framework - Miller, Younes - 2001 |

94 | Source separation using higher order moments
- Cardoso
- 1989
(Show Context)
Citation Context ...ctave, similar to many multiscale decompositions. Similar results were obtained by minimizing the independence of coefficients, under linear decompositions, leading to independent component analysis (=-=Cardoso, 1989-=-; Comon, 1994; Hyvarinen et al., 2001). These observations justify widespread use of orthonormal wavelets in general image analysis applications. Use of Gabor wavelets is also motivated by the fact th... |

93 | Bayesian denoising of visual images in the wavelet domain
- Simoncelli
(Show Context)
Citation Context ...ween responsessNatural Image Statistics 5 to pairs of bandpass filters and found that the amplitudes are strongly correlated, even when the signed responses are not (Simoncelli, 1997; Buccigrossi and =-=Simoncelli, 1999-=-). This is illustrated in Figure 1, which shows conditional histograms for several pairs of coefficients. Note that unlike second-order correlations, these dependencies can not be eliminated with a li... |

89 | Color and spatial structure in natural scenes - Burton, Moorehead - 1987 |

89 | The nonlinear statistics of high-contrast patches in natural images - Lee, Pedersen, et al. |

77 |
Multidimensional sub-band coding: some theory and algorithms
- Vetterli
- 1984
(Show Context)
Citation Context ...ock-by-block frequency decomposition. In early 80s, it was recognized that multi-scale wavelet-style representations offer more flexibility and better compression performance (Burt and Adelson, 1983; =-=Vetterli, 1984-=-). Early coders were based on marginal models of coefficients, but this changed abruptly with the development of the first contextual coder by Shapiro (Shapiro, 1993). This coder, and many that follow... |

71 | The principal components of natural images
- Hancock, Baddeley, et al.
- 1992
(Show Context)
Citation Context ...uted using eigen decomposition or singular value decomposition. A common application of PCA is in recognition of people from their facial images (Kirby and Sirovich, 1990) or study of natural images (=-=Hancock et al., 1992-=-). Instead, if the goal is to minimize the statistical correlation between the projected components, or to make them as independent as possible, then the independent component basis results (Comon, 19... |

61 | Computational modeling of visual texture segregation
- Bergen, Landy
- 1991
(Show Context)
Citation Context ... models has revolutionized the area of texture analysis. In 1980 Faugeras et al. (Faugeras and Pratt, 1980) suggested using the marginals of filtered images for texture representations. Bergen et al (=-=Bergen and Landy, 1991-=-), Chubb et al. (Chubb et al., 1994), and Heeger et al. (Heeger and Bergen, 1995) also advocated the use of histograms. Zhu et al. (Zhu et al., 1997) showed that marginal distributions of filtered ima... |

57 | What are textons
- Zhu, Guo, et al.
- 2005
(Show Context)
Citation Context ...urther study. Another idea is to combine local basis elements, such as wavelet bases, into higher level structures that provide a better representation of the image manifold. For instance, Zhu et al (=-=Zhu et al., 2002-=-) have combined placements of transformed basis elements to form structures called textons in order to better characterize images and their manifolds. Shown in the left panel of Figure 5 is an example... |

51 | Efficient context-based entropy coding for lossy wavelet image compression
- Chrysafis, Ortega
- 1997
(Show Context)
Citation Context ...rginal models of coefficients, but this changed abruptly with the development of the first contextual coder by Shapiro (Shapiro, 1993). This coder, and many that followed (Rinaldo and Calvagno, 1995; =-=Chrysafis and Ortega, 1997-=-), took heuristic advantage of the joint statistical properties of wavelets. Some subsequent coders have been based more explicitly on such models (LoPresto et al., 1997; Buccigrossi and Simoncelli, 1... |

47 | Hilbert-Schmidt Lower Bounds for Estimators on Matrix Lie Groups for ATR
- Grenander, Miller, et al.
- 1998
(Show Context)
Citation Context ...sance variables s ∈ S. Here, the likelihood is computed via the nuisance integral P (I|Hi) = � S P (I,s|Hi)γ(ds). Nuisance variable estimation on a group S and estimation error bounds are derived=-= in (Grenander et al., 1998), -=-while the hypothesis testing for α and recognition error bounds are derived in (Grenander et al., 2000). − Metrics for Image Comparison: A broader goal in image analysis is to quantify differences ... |

46 | Probability models for clutter in natural images - Grenander, Srivastava |

45 | The earth mover’s distance under transformation sets
- Cohen, Guibas
- 1999
(Show Context)
Citation Context ... ˜d(f(x|p1,c1),f(x|p2,c2)), where ˜ d is a metric on the space of univariate densities. Several forms have been proposed for ˜ d including geodesic length (Riemannian metric), Earth Mover’s dista=-=nce (Cohen and Guibas, 1999), K-=-ullback-Leibler divergence, Renyi’s α-divergence (Hero et al., 2001), Jensen-Renyi divergence (He et al., 2002; Hamza et al., 2001), χ 2 -distance, and the L p norm for p =1, 2,...,. The choice of... |

45 | Alpha divergence for classification, indexing and retrieval - Hero, Ma, et al. - 2001 |