## Highly sensitive size discrimination of sub-micron objects using optical Fourier processing based on two-dimensional Gabor filters

Optics Express, Vol. 17, Issue 14, pp. 12001-12012 (2009)

http://dx.doi.org/10.1364/OE.17.012001

Acrobat PDF (540 KB)

### Abstract

We use optical Gabor-like filtering implemented with a digital micromirror device to achieve nanoscale sensitivity to changes in the size of finite and periodic objects imaged at low resolution. The method consists of applying an optical Fourier filter bank consisting of Gabor-like filters of varying periods and extracting the optimum filter period that maximizes the filtered object signal. Using this optimum filter period as a measure of object size, we show sensitivity to a 7.5 nm change in the period of a chirped phase mask with period around 1µm. We also show 30nm sensitivity to change in the size of polystyrene spheres with diameters around 500nm. Unlike digital post-processing our optical processing method retains its sensitivity when implemented at low magnification in undersampled images. Furthermore, the optimum Gabor filter period found experimentally is linearly related to sphere diameter over the range 0.46µm-1µm and does not rely on a predictive scatter model such as Mie theory. The technique may have applications in high throughput optical analysis of subcellular morphology to study organelle function in living cells.

© 2009 OSA

## 1. Introduction

10. R. M. Pasternack, Z. Qian, J.-Y. Zheng, D. N. Metaxas, E. White, and N. N. Boustany, “Measurement of subcellular texture by optical Gabor-like filtering with a digital micromirror device,” Opt. Lett. **33(19)**, 2209–2211 (2008).
[CrossRef]

12. R. Mehtrotra, K. R. Namuduri, and N. Ranganathan, “Gabor filter-based edge detection,” Pattern Recognit. **25(12)**, 1479–1494 (1992).
[CrossRef]

13. J. G. Daugman, “Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters,” J. Opt. Soc. Am. **A 2(7)**, 1160–1169 (1985).
[CrossRef]

10. R. M. Pasternack, Z. Qian, J.-Y. Zheng, D. N. Metaxas, E. White, and N. N. Boustany, “Measurement of subcellular texture by optical Gabor-like filtering with a digital micromirror device,” Opt. Lett. **33(19)**, 2209–2211 (2008).
[CrossRef]

## 2. Methods

### 2.1 Optical Setup and Image Acquisition

10. R. M. Pasternack, Z. Qian, J.-Y. Zheng, D. N. Metaxas, E. White, and N. N. Boustany, “Measurement of subcellular texture by optical Gabor-like filtering with a digital micromirror device,” Opt. Lett. **33(19)**, 2209–2211 (2008).
[CrossRef]

*λ*

_{o}=632.8 nm) is passed through a spinning diffuser and coupled into a multimode fiber whose output is collimated and launched into the microscope’s condenser aligned in central Köhler illumination (NA<0.05). Image acquisition consisted of collecting on a 16-bit CCD (Roper Scientific Cascade 512B) a stack of spatially filtered dark-field images using a spatial filter bank generated by the DMD.

^{2}mirrors, which can be programmed to deflect the light towards or away from the CCD detector, thus allowing for binary on/off amplitude modulation of the field at each mirror. To analyze feature size for objects within the resolution of the microscope system, we programmed the DMD to display a set of two dimensional (2D) Gabor-like filters with varying period. As in [10

**33(19)**, 2209–2211 (2008).
[CrossRef]

**33(19)**, 2209–2211 (2008).
[CrossRef]

*λ*

_{o}=n/a. Positions of the orders of the grid in the Fourier plane are measured in mirrors from the zeroth order position (DC component) with the aid of a DMD mirror ruler which is simply a filter with passbands at regular intervals. The calibration gave on average 12.80 mirrors per order, giving 0.00781 cycles/µm/mirror. The maximum aperture (NA=0.75) for the 20X objective corresponds to 1.185 cycles/µm or a radius of 152 DMD mirrors.

### 2.2 Sample preparation

_{o}.

### 2.3 Analysis of the Optically Filtered Images

13. J. G. Daugman, “Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters,” J. Opt. Soc. Am. **A 2(7)**, 1160–1169 (1985).
[CrossRef]

_{1}and center located at frequencies (

*U*) in cycles/µm. In the object space, the period and orientation of the sinusoid are

_{h}, V_{h}_{s}. The filters are scaled such that the amplitude of the filter in the frequency domain remains constant. The Gabor filter response at each position in the object is given by the convolution of the Gabor filter with the object, and is directly obtained in our system by applying the Gabor-like filter (Fig. 1) on the DMD while collecting the image of the object on the CCD camera. To analyze all possible feature sizes that may be present in the object, a Gabor filter bank consisting of many individual Gabor filters with incrementally changing values of S is applied to the object. The size of the local object feature present at a given pixel in the resulting Gabor filtered images is analyzed by measuring the pixel response as a function of Gabor filter period. The Gabor filter period that maximizes the pixel response is then taken as a measure of the local feature size. The localization of the Gabor filter response in object space depends on the extent of the Gabor filter Gaussian envelope, σ

_{s}.

_{max}. The data are encoded into color-coded images indicating optimum Gabor frequency (or period) vs. pixel (Fig. 2(d)). At each pixel, the value of S

_{max}was then taken as a measure of local phase mask period or sphere diameter.

### 2.4 Digital Processing of Phase Mask Images

_{s}=S applied in image space. The period of the phase mask was measured at each pixel by optimizing the period of the Gabor filter to give maximum image pixel response. The optimization algorithm utilized an unconstrained nonlinear minimization (Nelder-Mead Simplex Method).

### 2.5 Digital Processing of Sphere Images

*t*) is the 2D translation, and σ and 2σ are the standard deviations of the two Gaussians, respectively. Then, at each of the spheres, the local sphere image was convolved with the 2D template, and the sphere size was measured by optimizing the 2D translation and the standard deviation of the 2D template. The optimized σ gave a relative measurement of the sphere size, which was converted to sphere size using the known magnification of the image in number of pixels per micron. This optimization algorithm also utilized an unconstrained nonlinear minimization (Nelder-Mead Simplex Method).

_{x}, t_{y}## 3. Results

### 3.1 Optical Processing of Periodic Phase Masks

_{s}=S. The image magnification for optical Gabor filtering was 0.275µm/pixel. The Gabor-filtered phase mask images are essentially featureless (Fig. 3(c)) due to the absence of the Gabor complex conjugate; however, the Gabor filtering results in images that are spatially confined as indicated by the preservation of the condenser field stop boundary. All images taken contain signal significantly above background levels due to the width of the Gabor filters; however, signal strength is maximized when the Gabor filter is centered over the diffraction order and decreases as the filter position is moved away. The pixel-by-pixel amplitude response as a function of applied Gabor filter frequency was fit using non-linear least-squares Gaussian fitting as explained previously (Fig. 2(c)), and encoded into color-coded images indicating mask order frequency vs. pixel (Fig. 4(a)). The Gaussian fit to the Gabor response at each pixel produced a correlation of greater than 80% in >99% of pixels. Analysis of the encoded images of the chirped phase mask produced a mask period vs. stage position relationship (Fig. 4(b)) in which the chirp mask period is linearly decreasing with relative position at a rate of 7.6nm/mm (correlation=0.98), 0.1 nm/mm higher than the provided manufacturer specifications but well within the error of the optical setup. Analysis of the encoded images for the unchirped phase mask produced a nearly constant period ranging from 1.074 +/- 0.002 to 1.077 +/-0.003 µm, and within 3 nm of the 1.075 µm period specified by the manufacturer (Fig. 4(b), gray squares).

### 3.2 Digital Gabor processing of Periodic Phase Masks

### 3.3 Optical Processing of Sphere Samples

_{s}=S/2. Each resulting set of optically filtered images (Figs. 5(c) and 5(h)) was analyzed pixel-by-pixel using the Gaussian fitting scheme previously described, and an encoded image of optimum Gabor period, S

_{max}, (Figs. 5(d) and 5(i)) was produced for each sample. This image was then scaled to the dark field intensity image (Figs. 5(e) and 5(j)).

_{max}, for each feature finally yielding a per-feature histogram depicting the relative number of features (spheres) at each value of S

_{max}for all three sphere samples (Fig. 6(b)). The statistics of these distributions are given in Table 1. For the spheres with diameter around 500nm, p values are shown for the comparison to the 494nm sphere size; for the 989nm spheres, the p value is shown for the comparison to the 1053nm sphere size. The differences between each sphere size are clearly observable. The distributions are tighter for the larger sphere sizes within the ~0.5µm range, and the standard deviations of the measurements for the 0.494 µm and 0.548 µm sphere sizes are within the manufacturer’s specifications (Fig. 6(b), Table 1). The relationship between Gabor period S

_{max}and sphere size is plotted in Fig. 6(c) and follows a linear fit with the intercept pegged to zero (correlation coefficient=0.99).

### 3.4 Digital post-processing of the Sphere Images

_{s}=S/2. For the DIC images, the orientation of the filter bank was parallel or perpendicular to the direction of maximal contrast. The generated filtered image transforms were then reverse-transformed and fit to a Gaussian pixel-by-pixel as was done for optical processing. After registration, the same pixels from the analog feature analysis are used in the digital analysis, yielding a per-feature histogram of filter responses for each sphere size. The measured filter responses for the DF data overlap and do not have differences in their distribution that are statistically significant (Fig. 7(a), Table 2).. The same is true for the DIC data ((Figs. 7(b) and 7(c), Table 2). The digital template-matching algorithm fared no better than digital Gabor filtering of the spheres, with no statistically significant difference in measured sphere size noticeable for DF (Fig. 7(d)) or DIC (Fig. 7(e)), with the exception of the 0.494 µm spheres which were actually measured larger than the other two sphere sizes in DIC (Fig. 7(e)).

## 4. Discussion

_{max}, giving maximum response as a measure of object size at each pixel. S

_{max}was obtained after fitting a Gaussian to the pixel response curve vs. Gabor filter frequency (e.g. Figure 2(c), 6(a)). We chose the Gaussian because it is capable of extracting a local maximum (if one exists) necessary for the analysis within the data set while making few other assumptions about the remainder of the data. It is positive for all possible values of frequency, as a real signal should be as well. However, it is important to note that the Gaussian function was only used here to retrieve objectively and reproducibly the spatial frequency position of the maximum response and for this purpose resulted in good fits with good correlation values. The nature of the actual response function will in general depend on the product of the Gabor filter with the object’s transform, and may not be known a priori for arbitrary samples. Better function choices could ultimately be made for specific objects where the shape of the scatter is known. This technique is similar to Optical Scatter Imaging [14

14. N. N. Boustany, S. C. Kuo, and N. V. Thakor, “Optical scatter imaging: subcellular morphometry in situ with Fourier filtering,” Opt. Lett. **26(14)**, 1063–1065 (2001).
[CrossRef]

14. N. N. Boustany, S. C. Kuo, and N. V. Thakor, “Optical scatter imaging: subcellular morphometry in situ with Fourier filtering,” Opt. Lett. **26(14)**, 1063–1065 (2001).
[CrossRef]

_{max}intrinsically describes the object size in the correct units of length, and the data in Fig. 6 show that the relationship between Smax and object size is linear for particles with diameter between 0.46µm and 1µm. The linear relationship between the sphere size and Gabor period is supported by the 0.99 correlation coefficient (Fig. 6(c)) although more sphere sizes would need to be tested in similar fashion to better define the dynamic range of this relationship. In particular, the sensitivity to the size of particles which are significantly below the resolution of the optical system will decrease due to signal truncation from the numerical aperture. This begins to be apparent in the broadening of the size distributions for the smaller sphere sizes in the histograms from Fig. 6(b). The slope of this relationship is also likely to depend on the shape of the finite objects being measured. Thus, although the parameter S

_{max}is highly sensitive to changes in object size, it may not measure the absolute object size accurately in a sample consisting of objects other than spheres. Nonetheless, a linear relationship is still expected to retrieve relative changes in object size accurately in all cases. A linear relationship between S

_{max}and object size imparts this technique with an advantage over spectroscopic methods, which require a scattering model of the spectrum in order to translate spectral changes into change in object size measured in the correct units of length [6]. The method presented here is therefore relevant for monitoring cell morphology in living samples, where the ability to detect subtle relative changes in structure without assumptions of a model may be of greater value than measuring absolute size accurately.

_{max}and sphere diameter. Changes in the refractive index ratio will however affect the amount of signal a great deal. Biological organelles have refractive index ratios m ~1.04, while for the polystyrene spheres in ployacrylamide gel m ~1.2. The decrease from m=1.2 to m=1.04 corresponds to an order of magnitude loss of signal due to the reduction in scattering cross section at the lower value of m. The signal-to-noise ratio (SNR) in a shot-noise limited detection system would drop by a factor of 3 or 4 (e.g. SNR in data of Fig. 6(a)), from which signal may still be recovered, albeit with lower sensitivity. This sensitivity may be recovered immediately, by using more powerful illumination, or by examining more than one pixel at a time to average the noise out. The current fluence at the sample is ~0.2mW/mm2 accounting for power losses at the diffuser and during coupling into the microscope. Thus the current laser power could be increased by a factor of 10 without damaging biological tissue, while pixel areas corresponding to 0.5µm x0.5µm (2×2 pixels at the current magnification) could be analyzed while remaining within the resolution of the Gabor filters used. Thus, for cases where the refractive index ratio is low (such as for cellular organelles in cytosol), accurate data collection is a signal to noise issue that can be solved either with a more powerful source of illumination, post-processing averaging, or both, and is not expected to be otherwise affected by index of refraction.

**33(19)**, 2209–2211 (2008).
[CrossRef]

## Acknowledgments

## References and links

1. | Z. Pincus and J. A. Theriot, “Comparison of quantitative methods for cell-shape analysis,” J. Microsc. |

2. | J. Angulo and S. Matou, “Application of mathematical morphology to the quantification of in vitro endothelial cell organization into tubular-like structures,” Cell Mol Biol (Noisy-le-grand) |

3. | A. Heifetz, J. J. Simpson, S.-C. Kong, A. Taflove, and V. Backman, “Subdiffraction optical resolution of a gold nanosphere located within the nanojet of a Mie-resonant dielectric microsphere,” Opt. Express |

4. | J. D. Wilson and T. H. Foster, “Mie theory interpretations of light scattering from intact cells,” Opt. Lett. |

5. | Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous Measurement of Angular and Spectral Properties of Light Scattering for Characterization of Tissue Microarchitecture and its Alteration in Early Precancer,” IEEE J. Sel. Top. Quantum Electron. |

6. | H. Fang, M. Ollero, E. Vitkin, L. M. Kimerer, P. B. Cipolloni, M. M. Zaman, S. D. Freedman, I. J. Bigio, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Noninvasive sizing of subcellular organelles with light scattering spectroscopy,” IEEE J. Sel. Top. Quantum Electron. |

7. | J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, “Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures,” J. Biomed. Opt. |

8. | J. D. Wilson, C. E. Bigelow, D. J. Calkins, and T. H. Foster, “Light scattering from intact cells reports oxidative-stress-induced mitochondrial swelling,” Biophys. J. |

9. | L. Xu, A. Taflove, and V. Backman, “Recent progress in exact and reduced-order modeling of light scatter properties of complex structures,” IEEE J. Sel. Top. Quantum Electron. |

10. | R. M. Pasternack, Z. Qian, J.-Y. Zheng, D. N. Metaxas, E. White, and N. N. Boustany, “Measurement of subcellular texture by optical Gabor-like filtering with a digital micromirror device,” Opt. Lett. |

11. | I. Fogel and D. Sagi, “Gabor filters as texture discriminator,” Biol. Cybern. |

12. | R. Mehtrotra, K. R. Namuduri, and N. Ranganathan, “Gabor filter-based edge detection,” Pattern Recognit. |

13. | J. G. Daugman, “Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters,” J. Opt. Soc. Am. |

14. | N. N. Boustany, S. C. Kuo, and N. V. Thakor, “Optical scatter imaging: subcellular morphometry in situ with Fourier filtering,” Opt. Lett. |

**OCIS Codes**

(070.1170) Fourier optics and signal processing : Analog optical signal processing

(070.6110) Fourier optics and signal processing : Spatial filtering

(120.5820) Instrumentation, measurement, and metrology : Scattering measurements

(180.0180) Microscopy : Microscopy

**ToC Category:**

Fourier Optics and Signal Processing

**History**

Original Manuscript: April 13, 2009

Revised Manuscript: May 28, 2009

Manuscript Accepted: June 1, 2009

Published: July 1, 2009

**Virtual Issues**

Vol. 4, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Robert M. Pasternack, Zhen Qian, Jing-Yi Zheng, Dimitris N. Metaxas, and Nada N. Boustany, "Highly sensitive size discrimination of sub-micron objects using optical Fourier processing based on two-dimensional Gabor filters," Opt. Express **17**, 12001-12012 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-12001

Sort: Year | Journal | Reset

### References

- Z. Pincus and J. A. Theriot, “Comparison of quantitative methods for cell-shape analysis,” J. Microsc. 227(2), 140–156 (2007). [CrossRef]
- J. Angulo and S. Matou, “Application of mathematical morphology to the quantification of in vitro endothelial cell organization into tubular-like structures,” Cell Mol Biol (Noisy-le-grand) 53(2), 22–35 (2007).
- A. Heifetz, J. J. Simpson, S.-C. Kong, A. Taflove, and V. Backman, “Subdiffraction optical resolution of a gold nanosphere located within the nanojet of a Mie-resonant dielectric microsphere,” Opt. Express 15(25), 17334–17342 (2007). [CrossRef]
- J. D. Wilson and T. H. Foster, “Mie theory interpretations of light scattering from intact cells,” Opt. Lett. 30(18), 2242–2244 (2005).
- Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, “Simultaneous Measurement of Angular and Spectral Properties of Light Scattering for Characterization of Tissue Microarchitecture and its Alteration in Early Precancer,” IEEE J. Sel. Top. Quantum Electron. 9(2), 243–256 (2003).
- H. Fang, M. Ollero, E. Vitkin, L. M. Kimerer, P. B. Cipolloni, M. M. Zaman, S. D. Freedman, I. J. Bigio, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Noninvasive sizing of subcellular organelles with light scattering spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 9(2), 267–276 (2003).
- J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, “Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures,” J. Biomed. Opt. 7(3), 378–387 (2002). [CrossRef]
- J. D. Wilson, C. E. Bigelow, D. J. Calkins, and T. H. Foster, “Light scattering from intact cells reports oxidative-stress-induced mitochondrial swelling,” Biophys. J. 88(4), 2929–2938 (2005). [CrossRef]
- L. Xu, A. Taflove, and V. Backman, “Recent progress in exact and reduced-order modeling of light scatter properties of complex structures,” IEEE J. Sel. Top. Quantum Electron. 11(4), 759–765 (2005).
- R. M. Pasternack, Z. Qian, J.-Y. Zheng, D. N. Metaxas, E. White, and N. N. Boustany, “Measurement of subcellular texture by optical Gabor-like filtering with a digital micromirror device,” Opt. Lett. 33(19), 2209–2211 (2008). [CrossRef]
- I. Fogel and D. Sagi, “Gabor filters as texture discriminator,” Biol. Cybern. 61(2), 103–113 (1989).
- R. Mehtrotra, K. R. Namuduri, and N. Ranganathan, “Gabor filter-based edge detection,” Pattern Recognit. 25(12), 1479–1494 (1992). [CrossRef]
- J. G. Daugman, “Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters,” J. Opt. Soc. Am. A 2(7), 1160–1169 (1985). [CrossRef]
- N. N. Boustany, S. C. Kuo, and N. V. Thakor, “Optical scatter imaging: subcellular morphometry in situ with Fourier filtering,” Opt. Lett. 26(14), 1063–1065 (2001). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.