where the diagonal blocks are within-channel degradations and the off-diagonal blocks are between-channel degradations [29].
The image deconvolution problem is concerned with knowing the un-blurred image and PSF given the degraded image. Based on these equa-tions, the unblurred image can be easily determined when the models for the degraded image and PSF are known. However, this is not the case in actual applications. Images cannot be modeled in a straightforward manner thus, their features and properties are usually utilized. For some applications, probability models are created based on the imaging conditions and type of scenes [30]. On the other hand, blurring functions can be mathematically modeled. Exploiting the characteristics of these models can decrease the complexity of determining the unblurred image. The next section discusses some constraints and common types of blurring functions that have been studied in several deconvolution methods.
generate energy, i.e.,
X
∀(p,q)
hm(p, q) = 1. (2.10)
The rest of this section will discuss the commonly used blurs used in simulations and algorithm evaluations. Their mathematical models as well as their corresponding parameters will also be presented.
1. No blur
Images that do not exhibit any degradation have no blur. This means that the blurring function is a Dirac delta function. For discrete imag-ing systems, this is approximated by a unit impulse [25]. Thus,
hm(p, q) = δ(p, q) =
1, p=q = 0 0, otherwise
. (2.11)
This indicates that the PSF did not spread out the image pixel values.
An example of unblurred image is shown in Figure 2.1. The color image is in Figure 2.1(a) where the convolved PSF is in Figure 2.1(b).
The green channel for the image is extracted and shown in Figure 2.1(c) with its corresponding Fourier transform in Figure 2.1(d). It can be observed that in the frequency domain significant components are scattered within the spectrum.
2. Gaussian blur
This type is generally used to model a variety of devices such as cam-eras and optical scopes. In remote sensing and aerial imaging appli-cations, blurs are caused by several factors such as temperature and
(a) RGB image
-6 -4
-2 0
2 4
6
q -6
-4 -2 0 2 4 6
p 0 0.2 0.4 0.6 0.8 1
h
(b) PSF
(c) G channel (d) Fourier Transform of 2.1(c)
Figure 2.1: Image with no blur.
wind speed among others. For long-term atmospheric exposure, the Gaussian model is usually used. As a result, the term atmospheric turbulence blur is also used in some literatures. Mathematically, this can be defined by
hm(p, q) = Kexp
−p2+q2 2σ2
(2.12)
where K is a normalizing constant that ensures that equation (2.10) is satisfied and σ is the variance. An image degraded by Gaussian blur is shown in Figure 2.2. In this example, σ = 1.66 and the spread is based on 6σ. In the frequency domain, significant components are mostly concentrated in the lower frequencies.
3. Uniform/Linear Motion blur
This can be observed when the camera or object is moving faster than the camera’s exposure period. For spatially invariant cases, global translation is the most distinguishable effect on the degraded image.
This is frequently used with frequency based blur identification and deconvolution algorithms since this is characterized by frequency do-main zeros. It is generally modeled by
hm(p, q) =
1 L, p
p2+q2 < L2 and pcosθ=qsinθ
0, otherwise
(2.13)
where L is the motion length in pixels and θ is the angle between motion orientation and horizontal axis in degrees. The term horizontal
(a) RGB image
-6 -4
-2 0
2 4
6
q -6
-4 -2 0 2 4 6
p 0 0.01 0.02 0.03 0.04 0.05 0.06
h
(b) PSF
(c) G channel (d) Fourier Transform of 2.2(c)
Figure 2.2: Image with Gaussian blur.
motion (HM), which is more commonly encountered and used, results when θ = 0. This simplifies equation (2.13) into
hm(p, q) =
1
2L, −L≤p≤L and q= 0
0, otherwise
(2.14)
In this condition, the frequency domain zeros are located on lines perpendicular to the direction of blur with a spacing determined by L. Images blurred with motion in different directions are shown in Figures 2.3 through 2.6. The parameters are set as L = 7 and θ = {0o,45o,90o,135o}. The frequency domain plots show that the significant components are concentrated mostly on the direction per-pendicular to the motion.
4. Out-of-Focus blur (OOF)
This is typically observable as defocusing in imaging systems. It is caused by the finite size of the camera’s aperture that is assumed to be circular. Thus, a small disk, known as circle of confusion (COC), can be used to describe the image of a point source. A detailed discussion on the relationship between COC, object distance, and lens diameter can be found in [21]. Based on this, the discrete PSF approximation can be expressed as
hm(p, q) =
1 πr2, p
p2+q2 ≤r 0, otherwise
(2.15)
whereris the radius of the COC. The equation indicates that intensity
(a) RGB image
-6 -4
-2 0
2 4
6
q -6
-4 -2 0 2 4 6
p 0 0.05 0.1 0.15 0.2
h
(b) PSF
(c) G channel (d) Fourier Transform of 2.3(c)
Figure 2.3: Image with 0o motion blur.
(a) RGB image
-6 -4
-2 0
2 4
6
q -6
-4 -2 0 2 4 6
p 0 0.05 0.1 0.15 0.2
h
(b) PSF
(c) G channel (d) Fourier Transform of 2.4(c)
Figure 2.4: Image with 45o motion blur.
(a) RGB image
-6 -4
-2 0
2 4
6
q -6
-4 -2 0 2 4 6
p 0 0.05 0.1 0.15 0.2
h
(b) PSF
(c) G channel (d) Fourier Transform of 2.5(c)
Figure 2.5: Image with 90o motion blur.
(a) RGB image
-6 -4
-2 0
2 4
6
q -6
-4 -2 0 2 4 6
p 0 0.05 0.1 0.15 0.2
h
(b) PSF
(c) G channel (d) Fourier Transform of 2.6(c)
Figure 2.6: Image with 135o motion blur.
values must be constant and nonzero within COC and must be zero elsewhere. This blur also exhibits frequency domain zeros that can be observed as concentric circles about the origin and is nearly periodic depending on r. An image degraded with OOF is shown in Figure 2.7 with a COC diameter set to 7. In the frequency domain, the significant components are concentrically located. Slightly observable are the spectral zero patterns at higher frequencies.
5. Rectangular blur
This is also known as pillbox or uniform 2D blur. It is used in many simulations and sometimes utilized as a crude approximation of defo-cus blur. It also models sensor pixel integration especially in applica-tions dealing with superresolution restoration. It is modeled by
hm(p, q) =
1
s2, |p|< 2s and |q|< 2s 0, otherwise
(2.16)
where s is the size of the smoothing area. The image in Figure 2.8 is degraded with rectangular blur. Since this blur is a crude approx-imation of OOF, the characteristics of the image in the spatial and transform domain are closely similar to Figure 2.7.
(a) RGB image
-6 -4
-2 0
2 4
6
q -6
-4 -2 0 2 4 6
p 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
h
(b) PSF
(c) G channel (d) Fourier Transform of 2.7(c)
Figure 2.7: Image with OOF blur.
(a) RGB image
-6 -4
-2 0
2 4
6
q -6
-4 -2 0 2 4 6
p 0 0.005 0.01 0.015 0.02 0.025
h
(b) PSF
(c) G channel (d) Fourier Transform of 2.8(c)
Figure 2.8: Image with rectangular blur.