Final Report Application of Image Restoration Technique in Flow Scalar Imaging Experiment Guanghua Wang Center for Aeromechanics Research Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin Austin, Texas 78712-1085 Abstract Scalar imaging techniques are widely used in fluid mechanics, but the effects of imaging system blur on the measured scalar gradients are often inadequately considered. Depending on the flow condition and imaging system used, the blurring can cause unacceptable errors in gradient related measurements, which are much larger than those for the scalar itself. Planar Laser-Induced Fluorescence (PLIF) images of turbulent jet fluid concentration were corrected for blur based on the Richardson-Lucy Expectation Maximization (R-L-EM) image restoration algorithm. This algorithm relies on the shot-noise limited nature of PLIF images and the measured Point Spread Function (PSF). The restored PLIF images show much higher peak dissipations and thinner fine-scale structures in the images, particularly when the structures are clustered.1 of 8 1. Introduction The spatial resolution of the optical system is very important for flow imaging experiments and it depends on many factors [1], e.g. pixel size of the detector array, depth of the collection optics and magnification. In many scalar imaging experiments, the resolution is usually quoted in terms of the area that each pixel images in the flow. For an ideal optical system or for an optical system used at high f# (f# = f/D, where f is the focal length and D is the diameter of the lens) and magnification is close to the design condition, resolution is nearly diffraction limited. However for low light level flow imaging experiments, such as Raman, Rayleigh and Planar Laser Induced Fluorescence (PLIF) imaging, fast (low f#) optics are commonly used. For these experiments, pixel size is not the only factor that limits the resolution. The imaging system blur should also be considered, since the image is the convolution of the Point Spread Function (PSF) with the irradiance distribution of the object. The smallest objects that can be resolved are related to the size and shape of the PSF. PSF also tends to progressively blur increasingly smaller structures. This is essentially a result of the system’s inability to transfer contrast variations in the object to the image. Depending on the flow condition and imaging system used, the PSF could be of the same order as the characteristic length scale of the scalar structures in the flow field, which may cause unacceptable error in the scalar measurement. To represent the TRUE scalar structure statistics, e.g. thickness, dissipation and Probability Density Function (PDF), it is necessary to restore the flow scalar experimental images. This project will introduce image restoration technique in flow scalar imaging applications and focus on how scalar measurements are affected. PLIF images of turbulent jet fluid concentration were corrected for blur based on the Richardson-Lucy Expectation Maximization (R-L-EM) image restoration algorithm. The shot-noise limited nature of PLIF images and the measured PSF are used to get reliable restoration. 2. Background The general model [3-9] for a linear degradation caused by blurring and additive noise is ),(),(),(),( yxnyxoyxhyxi +∗= (1)2 of 8 where ),( yxi is the blurred and noisy “observed image” corresponding to the observation of the “true image” ),( yxo , ),( yxh is the blurring function or PSF of the imaging system, ∗ is the convolution operator, ),( yxn denotes the additive noise such as the electronic or quantization noise involved in obtaining the image. In Fourier domain, the degradation is ),(),(),(),( vuNvuOvuHvuI +⋅= (2) where ),( vuI , ),( vuH , ),( vuO and ),( vuN are the continuous Fourier transforms of ),( yxi , ),( yxh , ),( yxo and ),( yxn respectively and u and v are the spatial frequencies. The purpose of restoration is to determine ),( yxo knowing ),( yxi and ),( yxh . This inverse problem has led to a large amount of work. Main difficulties are coming from the additive noise [3-9] and the PSF [7-9]. The modeling of blurring can be divided in two parts: blurring function and noise modeling. Some ideal PSF models are Gaussian, out-of-focus and linear motion blur [3]. In astronomy, data extracted from clear stars in observed image is used to fit a synthetic PSF function by weighted least square method [7, 8]. The PSF measurement techniques are also discussed in [1]. The diversity of algorithms [3-9] developed nowadays reflects different ways of recovering a “best” estimate of the “true image”. Wiener and regularized filters are better for known PSF and additive noise [3, 4]. Some iterative restoration techniques [10], e.g. Expectation Maximization (EM) algorithms, work better for known PSF and unknown additive noise. Blind image restoration algorithms [5, 6] are more proper for unknown PSF and additive noise. Flow imaging experiments have a lot in common with astronomy observations. They both involve low light level imaging. In both cases images are degraded by imaging optical system and suffer from signal dependent noise (Poisson noise), CCD camera read-out noise and quantization noise. These physical similarities suggest that a better starting point in applying image restoration techniques in flow scalar image restoration is to consider those successful ones in astronomy. 3. Richardson-Lucy Expectation Maximization (R-L-EM) algorithm The Richardson-Lucy algorithm ([11, 12]), also called the expectation maximization (EM) method, is an iterative technique used heavily for the restoration of astronomical images in the presence of Poisson3 of 8 noise [7-9]. It attempts to maximize the likelihood of the restored image by using the Expectation Maximization (EM) algorithm. The EM approach constructs the conditional probability density [7, 10] )()()|()|( ipopoipiop = (3) where )(ip and )(op are the probabilities of the observed image i and the true image o respectively. Here )|( oip is the probability distribution of observed image i if o were the true image. The Maximum Likelihood (ML) solution maximizes the density)|( oip overo )|(maxarg oipooML= (4) where “argmax” means “the value that maximizes the function”. For true image o with Poisson noise !)()|()(,ioheoipiohyx∗Π=∗− (5) The maximum can be computed