Image Noise Covariance Can be Adjusted by a Noise Weighted Filtered Backprojection Algorithm

Abstract

It was believed that the noise covariance of an image reconstructed with the filtered backprojection (FBP) algorithm is anisotropic. This paper shows that the noise-weighted FBP algorithm is able to alter the noise covariance and make it approximately isotropic. For the noise-weighted FBP algorithm, this paper develops a closed-form expression for the noise variance image and a closed-form expression for the noise covariance image. Computer simulations are carried out to evaluate the noise covariance with and without the noise weighting in the FBP algorithm. Transmission and emission noise models are used in computer simulations. The noise weighted FBP algorithm has a parameter that emulates the iteration number in the iterative Landweber algorithm. It is observed that the noise covariance can be altered by this emulated iteration number when noise weighting is used. The noise weighting in the noise-weighted FBP algorithm is able to change the noise covariance and a proper selection of the emulated iteration number may be able to give an approximately isotropic image noise covariance function.

I. Introduction

IN tomography, images are reconstructed from sinogram data. Noise in the sinogram data will propagate into the reconstruction. Researchers have been studying noise propagation in tomography for more than 40 years. Initially mathematical expressions for the noise variance in the reconstructed image were derived for the filtered backprojection (FBP) algorithm [1, 2]. Later, computer simulations were carried out [3, 4]. It was reported that the choice of reconstruction kernel used in the reconstruction algorithm affects the magnitude and correlation of image noise [5]. People realized that in addition to image noise variance, noise covariance also affects the image quality for object detection tasks [6, 7]. Calculation of covariance image in fan-beam CT has found an application in the channelized Hotelling observer method [8].

Image reconstruction algorithms play an important roll in image noise behavior. It is more difficult to investigate the noise propagation for the iterative algorithms. Noise properties for the emission iterative ML-EM (maximum likelihood expectation maximization) algorithm have been investigated and compared with the FBP algorithm. In [9-11], recursive covariance expressions were derived as a function of the iteration number. The noise covariance of a reconstruction indicated the image noise magnitude and texture. They found that the ML-EM algorithm had more local noise pattern, while the FBP algorithm had more distributed noise pattern. A unified recursive covariance expression was also proposed for a general iterative algorithm, not limited to the emission ML-EM algorithm [12]. For emission data, the OS-EM (ordered subset expectation maximization) algorithm and the FBP algorithm were also compared [13, 14]. It was observed that the FBP algorithm had anisotropic noise covariance images, while the OS-EM algorithm had isotropic noise covariance images.

In recent years, we developed noise-weighted FBP algorithms, which emulated the iterative Landweber algorithm with the ability of imposing ray-by-ray noise weighting [15-19]. The noise-weighted FBP algorithms had very different noise properties than the conventional FBP algorithm. The noise-weighted FBP algorithms are analytical, consisting of two steps: filtering and backprojection.

The main difference between the noise-weighted FBP and the conventional FBP is in the filter. In a conventional FBP algorithm, the filter is a ramp filter that may have a fixed apodization window function. In a noise-weighted FBP algorithm this apodization window function contains projection ray dependent noise weighting function, an emulated iteration number, and a relaxation parameter. The goal of this paper is to show that the noise-weighted FBP algorithm is able to alter the noise covariance and produce approximately isotropic noise covariance images.

II. METHODS

A. The noise variance image for a shift-variant FBP algorithm

The conventional FBP algorithm is a linear shift-invariant algorithm, in the sense that the pre-backprojection filter is shift invariant. A shift-invariant filter can be implemented as convolution or in the Fourier-domain as multiplication.

In a conventional FBP algorithm, let p(s, θ) be sinogram data at view angle θ and detector bin s. Let h(s) be a shift-invariant convolution kernel for the pre-backprojection filter. Then the filtered sinogram can be expressed as

$$q(\widehat{s},\theta )=\underset{-\infty }{\overset{\infty }{\int }}h(\widehat{s}-s)p(s,\theta )ds.$$ (1)

Here the filtered sinogram $q(\widehat{s},\theta )$ is ready for backprojection and $\widehat{s}$ is calculated as

$$\widehat{s}=-x\sin \theta +y\cos \theta .$$ (2)

The FBP reconstruction f(x, y) is the backprojection of $q(\widehat{s},\theta )$ as

$$f(x,y)=\underset{0}{\overset{\pi }{\int }}q(\widehat{s},\theta )d\theta .$$ (3)

For discrete implementation, we can assume that the noise in the sinogram is zero-mean and independent, with the variance ${\sigma }_p^2(s,\theta )$ Thus the image noise variance can be calculated as [1-4]

$${\sigma }_f^2(x,y)=\underset{0}{\overset{\pi }{\int }}\underset{-\infty }{\overset{\infty }{\int }}{h}^2(\widehat{s}-s){\sigma }_p^2(s,\theta )dsd\theta .$$ (4)

This expression (4) is almost the same as the conventiona FBP algorithm, except that the sinogram p(s, θ) is replaced by it noise variance ${\sigma }_p^2(s,\theta )$ and the convolution kernel is squared.

For a noise-weighted FBP algorithm, the pre-backprojection filter h is shift variant. The filter kernel becomes $h(\widehat{s},s,\theta )$. The derivation of the image noise variance is almost the same as the procedure in deriving (4). It is straightforward to obtain the counterpart of (4 for the shift-variant filter as

$${\sigma }_f^2(x,y)=\underset{0}{\overset{\pi }{\int }}\underset{-\infty }{\overset{\infty }{\int }}{h}^2(\widehat{s},s,\theta ){\sigma }_p^2(s,\theta )dsd\theta ,$$ (5)

where the integration over variable s is no longer a convolution.

For a noise-weighted FBP algorithm, the kernel $h(\widehat{s},s,\theta )$ is defined in the Fourier domain as [17, 18]

$$H_{\widehat{s},\theta }(\omega )=\mid \omega \mid [1-(1-\frac{\alpha \times w_{\widehat{s},\theta }}{\mid \omega \mid }{)}^k],$$ (6)

where $H_{\widehat{s},\theta }(\omega )$ is the one-dimensional Fourier transform of $h(\widehat{s},s,\theta )$ with respect to the spatial-domain variable s, and we define $H_{\widehat{s},\theta }(0)=0$.

In our implementation, the noise-weighted FBP algorithm reconstructs an image from a set of discrete sinogram data X (represented by a vector) by minimizing a weighted least square objective function

$$F=(AX-P{)}^{T}W(AX-P),$$ (7)

where W is a diagonal weighting matrix, matrix A represents the forward projector, and vector P contains the sinogram measurements.

Traditionally, objective function (7) is minimized by an iterative algorithm, for example, an iterative Landweber algorithm

$$X^{(k)}=X^{(k-1)}-\alpha A^{T}W(A{X}^{(k-1)}-P),$$ (8)

where α is the relaxation parameter (also known as the step size) and X^(k) reconstruction result of the kth iteration. This relaxation parameter α corresponds to the parameter α in (6) and the diagonal elements in W correspond to $w_{\widehat{s},\theta }$. The parameter k in (6) is the emulated iteration number corresponding to the iteration number k in (8).

In discrete implementation of (6), ∣ω∣ is the ramp filter, and α is the relaxation parameter satisfying

$$0<\mid 1-\frac{\alpha \times w_{\widehat{s},\theta }}{\mid \omega \mid }\mid <1$$ (9)

for all non-zero discrete values of ω. The noise weighting factor $w_{\widehat{s},\theta }$ is determined by the sinogram value $p(\widehat{s},\theta )$ A typical value of the noise weighting factor $w_{\widehat{s},\theta }$ is the reciprocal of the noise variance in $p(\widehat{s},\theta )$ that is,

$$w_{\widehat{s},\theta }=\frac{1}{{\sigma }_p^2(\widehat{s},\theta )}.$$ (10)

In our discrete implementation, the sinogram values are quantized into 10 discrete values for calculation of the noise weighting factors $w_{\widehat{s},\theta }$, and thus the noise weighting factors are quantized into 10 discrete values.

B. The noise covariance image for a shift-variant FBP algorithm

Noise covariance matrix (also known as, covariance image, or covariance function) can reveal an important character of noise texture. For the conventional shift-invariant FBP algorithm, the image noise covariance at point (x₀, y₀) is given as [9]

$$C_{(x_0,y_0)}(u,v)=\underset{0}{\overset{\pi }{\int }}\underset{-\infty }{\overset{\infty }{\int }}h(\stackrel{⌢}{s}-s)h(\breve{s}-s){\sigma }_p^2(s,\theta )dsd\theta ,$$ (11)

where (x₀, y₀) is the point of interest, (x₀+u, y₀+v)is a point close to (x₀, y₀), and

$$\stackrel{⌢}{s}=-x_0\sin \theta +y_0\cos \theta ,$$ (12)

$$\breve{s}=-(x_0+u)\sin \theta +(y_0+v)\cos \theta .$$ (13)

For a noise-weighted FBP algorithm, the pre-backprojection filter h is shift-variant. The filter kernel becomes $h(\widehat{s},s,\theta )$ and is defined by (6) in the Fourier domain. Using exactly the same derivation procedure as presented in [9] to reach (11), the counterpart of (11) for the shift-variant filter is

$$C_{(x_0,y_0)}(u,v)=\underset{0}{\overset{\pi }{\int }}\underset{-\infty }{\overset{\infty }{\int }}h(\stackrel{⌢}{s},s,\theta )h(\breve{s},s,\theta ){\sigma }_p^2(s,\theta )dsd\theta .$$ (14)

with $\stackrel{⌢}{s},\breve{s}$, u and v defined in (12) and (13). Because the weighting factor in the noise-weighted FBP algorithm is projection data dependent, the noise-weighted FBP algorithm is nonlinear. As a result, the proposed variance (5) and covariance (14) formulas are approximations.

C. Computer simulations

Image noise properties depend on the image reconstruction algorithm and its parameters, depend on the object to be imaged, and also depend on the noise model. Since we have closed-form expressions for the image noise variance (5) and covariance (14), we do not need to perform multiple noise realizations to estimate the ensemble variance and covariance. All we need to do is to assign a weighting factor $w_{\widehat{s},\theta }$ in (6) according to the noise model, form the kernel H, and evaluate the variance (5) and covariance (14). No random noise generator is required for our computer simulations.

The computer generated phantom is shown in Fig. 1, which had an elliptical background, two large high intensity discs, two large low intensity discs, and 14 small high intensity dots. The ellipse had a long semi-axis of 85.5 pixels, and a short semi-axis of 45 pixels. The four large discs had a radius of 14.4 pixels. The small dots had a radius of 2.7 pixels. The image array was 180 pixels × 180 pixels. The detector had 259 detectors. The detector size was the same as the pixel size. The imaging geometry was assumed to be parallel and the detector rotated over 180° with 180 stops. The relaxation parameter α in (6) was 0.0001.

Fig. 1.

Phantom

As a transmission phantom, the ellipse had a linear attenuation coefficient of 0.02 per pixel length. The high intensity discs and dots had a linear attenuation coefficient of 0.04 per pixel length. The low intensity discs had a linear attenuation coefficient of 0.01 per pixel length. The sinogram p(s, θ) was analytically calculated line-integral of the phantom. The transmission sinogram noise variance is

$${\sigma }_p^2(\widehat{s},\theta )=\frac{1}{I_0{e}^{-p(\widehat{s},\theta )}}=\frac{1}{I_0}{e}^{p(\widehat{s},\theta )},$$ (15)

where I₀ is the X-ray flux before entering the patient and is assumed to be a constant. The noise weighting factor for the transmission noise model is

$$w_{\widehat{s},\theta }=\frac{1}{{\sigma }_p^2(\widehat{s},\theta )}=I_0{e}^{-p(\widehat{s},\theta )}\propto e^{-p(\widehat{s},\theta )},$$ (16)

Note that the X-ray flux I₀ does not need to be included in the noise weighting $w_{\widehat{s},\theta }$, because it is a constant and any constant scaling factor of the weighting factor will be absorbed into the relaxation parameter α in (6). Without loss of generality, we always assume I₀ = 1 in (16).

In fact, $w_{\widehat{s},\theta }=1∕{\sigma }_p^2(\widehat{s},\theta )$ is not the only way to assign the weighting function; an alternative way is [20]

$$w_{\widehat{s},\theta }=(\frac{1}{{\sigma }_p^2(\widehat{s},\theta )}{)}^{\gamma },$$ (17)

where γ is a parameter making the noise weighting more aggressive (γ > 1) or less aggressive (γ < 1). When γ = 0, there is no weighting as in the conventional FBP. In the computer simulations, we tested some cases with γ = 1 and some cases with γ = 1.5 and γ = 1.4.

As an emission phantom, sinogram noise is Poisson distributed and its variance is

$${\sigma }_p^2(\widehat{s},\theta )=p(\widehat{s},\theta ).$$ (18)

Thus, the noise weighting factor for the emission noise model is

$$w_{\widehat{s},\theta }=\frac{1}{{\sigma }_p^2(\widehat{s},\theta )}=\frac{1}{p(\widehat{s},\theta )}$$ (19a)

or

$$w_{\widehat{s},\theta }=(\frac{1}{{\sigma }_p^2(\widehat{s},\theta )}{)}^{\gamma }=(\frac{1}{p(\widehat{s},\theta )}{)}^{\gamma }.$$ (19b)

Image noise in this paper is characterized by the relative standard uncertainty, R, which is defined by the image noise standard deviation divided by the reconstructed image value:

$$R(x,y)=\frac{{\sigma }_f(x,y)}{f(x,y)},\text{if}f(x,y)\ne 0.$$ (20)

In our computer simulations, the emulated iteration number k is set to different values to observe its effects on noise covariance. Three points were selected to evaluate the noise covariance image: (90, 90), (120, 90) and (175, 90), where the point (175, 90) was on the edge of the phantom.

D. Real SPECT phantom study

A Hoffman brain phantom was used in a SPECT study. The phantom was injected with Tc-99m and scanned for 20 minutes. Three low-energy high-resolution collimators were used in a three-detector IRIX scanner. The detector pixel size was 2.3 mm and the detected photons were stored in a 256 × 256 array on each detector. Data acquisition was the step-and-shoot mode with 120 views over 360°. One slice of the phantom data was used for image reconstruction, and the total photon counts for that slice were 1.4×10⁶. The reconstructed image was stored in an 88 × 88 array. An arbitrarily selected point (66, 44) was used to evaluate the covariance. No attenuation correction was performed during image reconstruction.

III. Results

A. Transmission noise model, with conventional FBP

Using conventional FBP reconstruction algorithm, the image noise variance is calculated according to (4) and displayed in Fig. 2, which resembles a lowpass filtered version of the phantom image itself. The relative standard uncertainty image, R(x,y), is calculated by (20) and is displayed in Fig. 3. Outside the elliptical phantom, f (x, y) = 0 and (20) does not apply. When f (x, y) < 0.01, we set f (x, y) = 0.01 so that we could use (20). Images shown in Figs. 1, 2, and 3 are 180 × 180 images. Each image is displayed from its own minimum value to its own maximum value.

Fig. 2.

The noise variance image of conventional FBP reconstruction with the transmission noise model.

Fig. 3.

The relative noise standard uncertainty image of conventional FBP reconstruction with the transmission noise model.

Fig. 4 shows a noise covariance image with respect to the point (x₀, y₀) = (90, 90). Fig. 5 shows a noise covariance image with respect to the point (x₀, y₀) = (120, 90). These two noise covariance images are anisotropic, and they are both elongated in the horizontal direction. Images shown in Figs. 4(Top) and 5(Top) are 21 × 21 images.

Fig. 4.

Noise covariance for conventional FBP reconstruction with the transmission noise model. Left: Noise covariance image at point (90, 90). Middle: Central horizontal line profile. Right: Central vertical line profile.

Fig. 5.

Noise covariance for conventional FBP reconstruction with the transmission noise model. Left: Noise covariance image at point (120, 90). Middle: Central horizontal line profile. Right: Central vertical line profile.

B. Transmission noise model, with noise-weighted FBP

Figs. 6, 7, and 8 are the results using three different emulated iteration numbers k = 1.8×10⁵, k = 1.8×10⁶, and k = 1.8×10⁷ in (6) with γ = 1.5. The covariance images in Fig. 8 are for the reference point (90, 90). At k ≈ 1.8×10⁶, the covariance image is approximately isotropic. At larger k values, the covariance image is elongated in the horizontal direction. At smaller k values, the covariance image is elongated in the vertical direction. When γ = 1 (images not shown), the covariance image is approximately isotropic when k ≈ 3×10⁴.

Fig. 6.

The noise variance images of noise-weighted FBP reconstruction with the transmission noise model, using γ = 1.5. From left to right: k = 1.8×10⁶, k = 1.8×10⁷, and k = 1.8×10⁵.

Fig. 7.

The relative noise standard uncertainty images of noise-weighted FBP reconstruction with the transmission noise model, using γ = 1.5. From left to right: k = 1.8×10⁶, k = 1.8×10⁷, and k = 1.8×10⁵.

Fig. 8.

Noise covariance images at point (90, 90) for noise-weighted FBP reconstructions with the transmission noise model, using γ = 1.5. From top to bottom: covariance images, central horizontal line profiles, and central vertical line profiles. From left to right: k = 1.8×10⁶, k = 1.8×10⁷, and k = 1.8×10⁵.

Figs. 9, 10, and 11 are the results using three different emulated iteration numbers k = 1×10³, k = 1×10⁴, and k = 1 × 10⁵ with γ = 1.5. The covariance images in Fig. 11 are for the reference point (120, 90). At k ≈ 1×10⁴, the covariance image is approximately isotropic. At larger k values, the covariance image is elongated in the horizontal direction. At smaller k values, the covariance image is elongated in the vertical direction. When γ = 1 (images not shown), the covariance image is approximately isotropic when k ≈ 1×10³.

Fig. 9.

The noise variance images of noise-weighted FBP reconstruction with the transmission noise model, using γ = 1.5. From left to right: k = 1×10⁴, k = 1×10⁵, and k = 1×10³.

Fig. 10.

The relative noise standard uncertainty images of noise-weighted FBP reconstruction with the transmission noise model, using γ = 1.5. From left to right: k = 1×10⁴, k = 1×10⁵, and k = 1×10³.

Fig. 11.

Noise covariance images at point (120, 90) for noise-weighted FBP reconstructions with the transmission noise model, using γ = 1.5. From top to bottom: covariance images, central horizontal line profiles, and central vertical line profiles. From left to right: k = 1×10⁴, k = 1×10⁵, and k = 1×10³.

The variance images and the relative noise standard uncertainty images are almost the same by visual inspection for all cases. Of course, their image values are different from case to case. The general trend is that as the emulated iteration number gets larger, the variance and the relative noise standard uncertainty become larger. The high intensity regions have higher variance. The low intensity regions have higher relative noise standard uncertainty. The relative noise standard uncertainty is a better indicator for the image error. It can be observed that the noise weighting significantly affects the noise covariance as illustrated by the covariance images. The effects depend on the emulated iteration number k, the weighting aggressive parameter γ, the object, the noise model, and the location of reference point (x₀, y₀).

C. Emission noise model, with conventional FBP

For the emission noise model, the sinogram noise variance is the same as the line integral value. We used the same phantom as in the transmission noise model studies, except that the phantom intensity was scaled up by 100 times. Using the conventional FBP reconstruction algorithm, the image noise variance is calculated according to (4) and displayed in Fig. 12, which resembles a lowpass filtered version of the phantom image itself. The relative standard uncertainty image, R(x,y), is calculated by (20) and is displayed in Fig. 13. Outside the elliptical phantom, f (x, y) = 0 and (18) does not apply. When f (x, y) < 0.5, we set f (x, y) = 100 so that we could use (20). The relative standard uncertainty image, R(x,y), is meaningless outside the phantom.

Fig. 12.

The noise variance image of conventional FBP reconstruction with the emission noise model.

Fig. 13.

The relative noise standard uncertainty image of conventional FBP reconstruction with the emission noise model.

Fig. 14 shows a noise covariance image with respect to the point (x₀, y₀) = (120, 90). This noise covariance image is anisotropic, and is elongated in the horizontal direction.

Fig. 14.

Noise covariance for conventional FBP reconstruction with the emission noise model. Left: Noise covariance image at point (120, 90). Middle: Central horizontal line profile. Right: Central vertical line profile.

D. Emission noise model, with noise-weighted FBP

Figs. 15, 16, and 17 are the results using three different emulated iteration numbers: k = 2×10⁶, k = 1×10⁷, and k = 1×10⁵ with γ = 1.4. The covariance images in Fig. 15 are for the reference point (120, 90). At k ≈ 2×10⁶, the covariance image is approximately isotropic. At larger k values, the covariance image is elongated in the horizontal direction. At smaller k values, the covariance image is elongated in the vertical direction.

Fig. 15.

The noise variance images of noise-weighted FBP reconstruction with the emission noise model, using γ = 1.4. From left to right: k = 2×10⁶, k = 1×10⁷, and k = 1×10⁵.

Fig. 16.

The relative noise standard uncertainty images of noise-weighted FBP reconstruction with the emission noise model, using γ = 1.4. From left to right: k = 2×10⁶, k = 1×10⁷, and k = 1×10⁵.

Fig. 17.

Noise covariance images at point (120, 90) for noise-weighted FBP reconstructions with the emission noise model, using γ = 1.4. From top to bottom: covariance images, central horizontal line profiles, and central vertical line profiles. From left to right: k = 2×10⁶, k = 1×10⁷, and k = 1×10⁵.

Similar to the cases in transmission noise model studies, the variance images and the relative noise standard uncertainty images are almost the same by visual inspection for all cases. The high intensity regions have higher variance. The low intensity regions have higher relative noise standard uncertainty. It can be observed that the noise weighting affects the noise covariance as illustrated by the covariance images.

Fig. 18 shows the covariance at a location (175, 90), which is on the edge of the phantom. Both the conventional FBP algorithm and the noise-weighted FBP (γ = 1, α = 0.0001) are used. Two different k values are used: k = 1×10⁸ and k = 1×10³, and the covariance images for these two k values are quite different. The covariance image for the conventional FBP and that for the noise-weighted FBP with k = 1×10⁸ are almost identical.

Fig. 18.

Noise covariance images at an edge point (175, 90) for conventional FBP and noise-weighted FBP (γ = 1, α = 0.0001) reconstructions with noisy emission data. From top to bottom: covariance images, central horizontal line profiles, and central vertical line profiles. From left to right: conventional FBP, k = 1×10⁸, and k = 1×10³.

For the noise-weighted FBP algorithm with emission noise model, some point spread functions (psf) are shown in Figs. 19 and 20. The diameter of the point source was 3.6 pixels. The psf images are reconstructed by using (1), (3), and (6), and projection sinogram is generated by using a point source at the location of interest. During implementation, the weighting factor $w_{\widehat{s},\theta }$ is quantized into 10 values and corresponding 10 versions of $H_{\widehat{s},\theta }(\omega )$ are obtained. The filtered sinogram (1) is generated by using these 10 versions of the window functions $H_{\widehat{s},\theta }(\omega )$ as described in detail in [16]. Figs. 19 and 20 show the psf images of the emission noise-weighted FBP (γ = 1, α =0.0001) algorithm at there locations: (120, 90), at (173, 90), and at (90,134) with two different k values: k = 1×10³ and k = 1×10⁸, respectively. All these psf images seem almost isotropic.

Fig. 19.

Point spread function (psf) of the noise-weighted FBP (γ = 1, α = 0.0001, k = 1×10³) reconstructions with emission noisy model. From top to bottom: psf images, central horizontal line profiles, and central vertical line profiles. From left to right: at (120, 90), at (173, 90), and at (90,134).

Fig. 20.

Point spread function (psf) of the noise-weighted FBP (γ = 1, α = 0.0001, k =1× 10⁸) reconstructions with emission noisy model. From top to bottom: psf images, central horizontal line profiles, and central vertical line profiles. From left to right: at (120, 90), at (173, 90), and at (90,134).

E. Real SPECT phantom study

Fig. 21 shows the reconstructed images of the Hoffman brain phantom with k = 1×10¹⁴ and k = 1×10⁴, respectively, with the emission noise-weighted FBP algorithm (γ = 1, α = 0.0001). Fig. 22 shows the associated noise covariance images at an arbitrarily chosen location of (66, 44). In this phantom study, we are unable to find a parameter k that can give an isotropic covariance image at the point (66, 44).

Fig. 21.

Real SPECT reconstructions using noise-weighted FBP (γ = 1, α =0.0001). Left: k = 1×10¹⁴; Right: k = 1×10⁴.

Fig. 22.

Noise covariance images at point (66, 44) for noise-weighted FBP (γ = 1, α = 0.0001) reconstructions with noisy emission data. From top to bottom: covariance images, central horizontal line profiles, and central vertical line profiles. From left to right: k = 1×10¹⁴, and k = 1×10⁴.

Our noise-weighted FBP algorithm has three parameters to determine: γ , α and k. The parameter γ is fixed to the default value of 1, which corresponds to the noise model. The two parameters α and k are closely related in such a way that the product αk approximately determines the convergence rate. The parameter α can be arbitrarily chosen (as long as it is small enough to avoid divergence). The user can only change one parameter k, which corresponds to the iteration number in an iterative algorithm. Since the noise weighted algorithms are shift-variant, in general, an isotropic covariance cannot be achieved for the entire image for a fixed parameter k. We can only select a parameter k to obtain an isotropic covariance at certain sub-regions. We do not have a matured guidance to select k other than trial-and-error methods. It may happen that the covariance is never isotropic at certain locations no mater what parameter k is applied.

IV. Conclusions

The conventional FBP algorithm does not model shift-variant sinogram noise, and generally has anisotropic noise covariance as represented by the elongations in noise covariance images. The recently developed FBP algorithm is able to model projection noise in the ray-by-ray manner and also has closed-form expressions for its variance and covariance images [15, 16]. On the other hand, the iterative algorithms do not yet have a closed-form expression for their variance and covariance images for a given iteration number [21]. Since the noise-weighted FBP algorithm is a good approximation of the noise-weighted iterative Landweber algorithm, one can use our closed-form expressions to make indications of the noise behavior of an un-regularized iterative Landweber algorithm. Noise behavior of the noise-weighted FBP algorithm is complex and depends on the emulated iteration number parameter. For the high values of that parameter, the noise covariance resembles the non-weighted FBP algorithm. At low values of that parameter, the noise weighting tends to over-correct the noise covariance. Given a fixed value of the emulated iteration number, image has different noise covariance at different location.

In the past, people drew over-simplified observations, such as “the iterative algorithm can give isotropic noise covariance.” In fact, the noise behavior is more complex than that. The noise covariance depends on the number of iterations. At very high iterations, the noise weighting loses its power, and the noise covariance becomes almost the same as that without noise weighting. At very low iterations, the noise weighting tends to “over-correct” the noise covariance. The noise covariance is shift variant. In other words, given a fixed iteration number, the image has different noise covariance at different locations, because the convergence rate is location dependent. The object itself and the noise model also affect the noise covariance.