Estimation of the Initial Image’s Contributions to the Iterative Landweber Reconstruction

Abstract

This paper develops a closed-form frequency-domain transfer function to predict the initial image’s contributions in an iterative Landweber reconstruction for a given number of iterations. The transfer function shows that the initial image’s high frequency components play an important role when the iterative algorithm is stopped before convergence. The contributions from the initial image diminish as the iteration number increases if the imaging matrix has full rank.

I. Introduction

Initial image (or initial condition) can influence the result of the iterative image reconstruction, especially when the number of iterations is low. The basic principles of iterative image reconstruction are classic and well covered in textbooks [1]–[4].

Two methods are commonly practiced in our medical imaging community for iterative image reconstruction. The first method is to use a constant as the initial image. If the algorithm uses the additive update scheme, the initial image is set as a zero image. If the algorithm is multiplicative, the initial image is chosen as an image with a constant ONE (or any positive constant) for its pixels. The motivation of using a constant initial image is to avoid any undesired structures that are in the initial image and may propagate into the reconstruction.

The other method is based on the strategy of speeding up the iterative reconstruction by using an initial image that is close to the expected reconstruction. The filtered backprojection (FBP) algorithm [5] is fast but suffers from reconstruction noise if the noise is not properly controlled. The FBP image can be used as the initially image in an iterative algorithm for a faster convergence [6]–[8].

If the projections are not sufficiently measured, for example, due to data truncation, special initial condition, or image support are considered [9], [10]. This paper only considers the situation where the projections are sufficiently measured.

To gain more insights of how the initial image influences the reconstruction, this paper derives a frequency-domain transfer function that relates the initial image to the reconstructed image. This transfer function is a high-pass filter in nature and its maximum gain diminishes as the iteration number becomes higher.

The derivation of the transfer function is based on our previous work [11]–[14]. For sake of making this paper self-contained, the entire derivation is given in Section II. Computer simulations are provided.

II. Methods

Let us consider an iterative Landweber algorithm that minimizes the following quadratic objective function:

$$f={(AX-P)}^T(AX-P)+\beta X^{T}X$$ (1)

where X is the image array expressed as a vector, P is the projection array expressed as a vector, A is the imaging matrix, and β is a parameter controlling influence of the minimum-norm regularization term on the objective function f. If β = 0, the minimum-norm regularization is not effective. The Landweber algorithm is a gradient descent algorithm, and the gradient of the objective function (1) is given as [15], [16]

$$\nabla f=2A^{T}AX-2A^{T}P+2\beta X.$$ (2)

The Landweber algorithm that minimizes (1) is given as

$$X^{(k)}=X^{(k-1)}+\alpha \left(A^{T}P-A^{T}A{X}^{(k-1)}-\beta X^{(k-1)}\right)$$ (3)

where X^(k) is the result of X at the kth iteration and the relaxation parameter α controls the step size. For a quadratic objective function (1), the Landweber algorithm will converge if the relaxation parameter α > 0 is small enough. The recursive expression (3) can be transformed to a non-recursive form as follows:

$$\begin{gathered} X^{(k)}=X^{(k-1)}+\alpha A^T\left(P-A{X}^{(k-1)}\right)-\alpha \beta X^{(k-1)} \\ =\alpha A^{T}P+\left(I-\alpha A^{T}A-\alpha \beta I\right)X^{(k-1)} \\ =\alpha A^{T}P+\left(I-\alpha A^{T}A-\alpha \beta I\right)\left[\alpha A^{T}P+\left(I-\alpha A^{T}A-\alpha \beta I\right)\times X^{(k-2)}\right] \\ =\left[\sum _{n=0}^{k-1}{\left(I-\alpha A^{T}A-\alpha \beta I\right)}^n\right]\alpha A^{T}P+{\left(I-\alpha A^{T}A-\alpha \beta I\right)}^k\times X^{(0)}. \end{gathered}$$ (4)

If the square matrix (I – M) is nonsingular, we have the identity

$$\sum _{n=0}^{k-1}{M}^n={(I-M)}^{-1}\left(I-M^k\right)$$ (5)

which can be readily verified by premultiplying (I – M) on both sides. In fact, premultiplying the left-hand-side of (5) by (I − M), we obtain

$$\begin{gathered} (I-M)\sum _{n=0}^{k-1}{M}^n=\sum _{n=0}^{k-1}\left(M^n-M^{n+1}\right) \\ =I-M+M-\dots +M^{k-1}-M^k \\ =I-M^k. \end{gathered}$$ (6)

Thus the nonrecursive form (4) can be further written as a closed form without the Σ sign

$$\begin{gathered} X^{(k)}={\left(\alpha A^{T}A+\alpha \beta I\right)}^{-1}\left[I-{\left(I-\alpha A^{T}A-\alpha \beta I\right)}^k\right]\alpha A^{T}P+{\left(I-\alpha A^{T}A-\alpha \beta I\right)}^k{X}^{(0)} \\ ={\left(A^{T}A+\beta I\right)}^{-1}\left[I-{\left(I-\alpha A^{T}A-\alpha \beta I\right)}^k\right]A^{T}P+{\left(I-\alpha A^{T}A-\alpha \beta I\right)}^k{X}^{(0)}. \end{gathered}$$ (7)

Expression (7) can be treated as a linear system with two separate inputs: A^TP and X⁽⁰⁾. In this paper, we focus on X^(k)’s dependency on X⁽⁰⁾.

In tomography, A^TA is a projection-backprojection operator in the matrix form. When it operates upon an image X, it equivalently convolves the image X by a 2-D 1/r kernel, where r is the distance to the origin [4]. In the Fourier domain, this 1/r convolution kernel corresponds to the function $1/\Vert \overrightarrow{\omega }\Vert$, where $\overrightarrow{\omega }$ is the frequency vector in the 2-D Fourier domain.

We thus obtain the equivalence between the matrix A^TA and the Fourier-domain function $1/\Vert \overrightarrow{\omega }\Vert$. We also notice that the identity matrix I corresponds to the function 1 in the Fourier domain. Multiplication with a matrix k times is equivalent to multiplying the filter function k times in the frequency domain.

The operator of the second term on the right-hand-side of (7) in the matrix form is (I − αA^TA − αβI)^k, and in the frequency domain, it becomes

$$F(\Vert \overrightarrow{\omega }\Vert )={(1-\alpha /\Vert \overrightarrow{\omega }\Vert -\alpha \beta )}^k,\text{ if }\Vert \overrightarrow{\omega }\Vert \ne 0$$ (8)

The transfer function (8) is undefined when ω ≠ 0. If we require that the transfer function be smooth (as seen in Figs. 1 and 2), we must have F(0) = 0. The parameter α is chosen such that negative values of F are not allowed. Thus, we define the transfer function (8) to be 0 when $\Vert \overrightarrow{\omega }\Vert =0$.

Fig. 1.

Transfer function (8) with α = 0.001, β = 0, and 4 different k values.

Fig. 2.

Transfer function (8) with α = 0.001, β = 10, and 4 different k values.

According to [19], the correct way to obtain the discrete frequency-domain transfer function is to take the discrete Fourier transform of its discrete spatial-domain convolution kernel. If the convolution kernel is not available (as in our case), discretization of the continuous frequency-domain transfer function can cause errors at very low frequencies. However, these errors can be made arbitrarily small by decreasing the frequency-domain sampling interval, which is equivalent to pad enough zeros in the spatial domain.

Some examples of this transfer function are shown in Fig. 1 with α = 0.001, β = 0, and some values of parameter k. Similar examples are shown in Fig. 2 for β = 10.

We must point out that the parameter α in the spatial-domain expressions and in the discrete frequency-domain expressions are different. They follow different rules. In the spatial domain, the convergence requires ∥I − αA^TA − αβ∥ < 1, in which α depends on the maximum eigenvalue of A^TA. The matrix A is different for different applications. Usually the value of α is chosen by trial-and-error.

On the other hand, in the discrete frequency domain, we require that $\left|1-\alpha /\Vert \overrightarrow{\omega }\Vert -\alpha \beta \right|<1$, which is the same as $0-\alpha <2/\left(\beta +1/\Vert \overrightarrow{\omega }\Vert \right)$. If one assumes the discrete frequencies are between −0.5 and 0.5 with a frequency sampling interval Δ, the lowest non-zero frequency is 1/Δ and our requirement becomes 0 < α < 2/(β +1/Δ). For example, if the 512-point discrete Fourier transform is used, Δ is 1/512. In this case, parameter α can be chosen anywhere in 0 < α < 2/(β + 512).

In the frequency-domain implementation, the selection of the value of α is independent from the imaging matrix A. The upper-bound of α is approximately 2/(the number of frequency-domain samples). In matrix implementation, α is approximately upper bounded by 2/σ_max(A), where σ_max(A) is the maximum singular value of the matrix A (as known as the L₂ norm of the matrix A). If the projection matrix A and backprojection matrix A^T are normalized, the selection of α in matrix implementation does not depend on matrix A.

Some observations can be obtained from the curves in Figs. 1 and 2, where the parameter k is proportional to the iteration number in an iterative algorithm.

1. When k is very small, almost all frequency components are passed to the reconstruction except for the DC (i.e., $\Vert \overrightarrow{\omega }\Vert =0$ component.

2. As k increases, the low-frequency components are gradually attenuated. The high-frequency components are dominating.

3. The overall contributions from the initial image to the reconstruction are suppressed as an exponential function of k. When k is very large, the contributions from the initial image to the reconstruction gradually diminish.

4. The parameter β makes the contributions from the initial image diminish faster.

5. The contributions of the initial image are independent from the noise in the projection data.

The above observations from the frequency-domain transfer function can be used to predict the effects of the initial image in iterative reconstruction. In the next section, the results are presented from two iterative algorithms: 1) the iterative Landweber algorithm and 2) iterative ML-EM (maximum likelihood expectation maximization) algorithm [17].

III. Results

The computer simulations were carried out using MATLAB (The MathWorks, Inc., Natick, MA, USA). The image size was 180 × 180. The number of view angles was 180 over 180°. The image pixel size was the same as the detector bin size. The computer generated phantom had an elliptical shape, containing two brighter discs and two darker discs. The initial image was a square, which looked very different from the elliptical phantom. In this section, the parameter β is set to zero except for the studies in Fig. 4.

Fig. 4.

Predicted contributions from the (left) square initial image and (right) associated central horizontal line profiles with α = 0.001 and β = 10. Row 1: k = 1. Row 2: k = 10. Row 3: k = 100. Row 4: k = 1000.

A. Prediction by the Frequency-Domain 2-D Filter

First, the frequency-domain transfer function (8) was implemented as a 2-D filter and applied to the initial image. If a uniform square is chosen as the initial image, some resultant images and central horizontal line (at row 90) profiles are shown in Fig. 3 with α = 0.001 and β = 0. The maximum values in the images are 0.8742, 0.7241, 0.5724, and 0.1390 for k = 1, 10, 100, and 1000, respectively.

Fig. 3.

Predicted contributions from the (left) square initial image and (right) associated central horizontal line profiles with α = 0.001 and β = 0. Row 1: k = 1. Row 2: k = 10. Row 3: k = 100. Row 4: k = 1000.

The similar results for β = 10 are shown in Fig. 4. The maximum value in the images are 0.8560, 0.6476, 0.2070, and 5.8601 ×10⁻⁶ for k = 1, 10, 100, and 1000, respectively. The results from Figs. 3 (β = 0) and 4 (β = 10) are similar. In Parts B and C of this section, studies with β = 0 only will be presented.

If the 180 × 180 FBP reconstruction is chosen as the initial image, some resultant images and central horizontal line (at row 90) profiles are shown in Fig. 5. The maximum value in the images are 18.6018, 12.3005, 3.7197, and 0.3874 for k = 1, 10, 100, and 1000, respectively.

Fig. 5.

Predicted contributions from the (left) FBP-reconstructed initial image and (right) associated central horizontal line profiles with α = 0.001 and β = 0. Row 1: k = 1. Row 2: k = 10. Row 3: k = 100. Row 4: k = 1000.

Each image is displayed using the linear gray scale from its own minimum value to its own maximum value.

B. Iterative Landweber Reconstructions

To avoid committing the inverse-problem crime, the projection matrix A in reconstruction algorithms and that in projection data generation were different. In data generation, the image array was up-sampled ten times, and the detection array was also up-sampled ten times. The detection array was then down-sampled ten times, and Poisson noise was incorporated in the down-sampled detection array.

The iterative Landweber algorithm is expressed as

$$X^{(k)}=X^{(k-1)}+\alpha A^T\left(P-A{X}^{(k-1)}\right).$$ (9)

We remind the reader that the parameters α and k in the spatial-domain iterative algorithm are not the same as in the frequency domain, even when they produce similar effects. The following cases with the square initial image are shown in Fig. 6 with α = 0.001: k = 5, 15, 105, and 1005, respectively. As the iteration number k increases, the phantom image gets sharper and the contributions from the initial image gradually diminish. The low frequency components of the initial image disappear first. The high frequency edges of the initial image stay in the reconstruction a little longer.

Fig. 6.

Iterative Landweber reconstructions using the (left) square initial image with α = 0.001 and (right) associated central horizontal line profiles. Row 1: k = 5. Row 2: k = 15. Row 3: k = 105. Row 4: k = 1005.

C. Iterative ML-EM Reconstructions

The iterative ML-EM algorithm is popular in emission tomography image reconstruction. It can be symbolically expressed as

$$X^{(k)}=\left\{A^T\left[P./\left(A{X}^{(k-1)}\right)\right]\right\}./\left[A^{T}1\right]$$ (10)

where 1 is the image (formatted as a vector) with all its pixel values being the constant 1 and “./” is the element-wise division as in MATLAB. There is no parameter α in this algorithm. The ML-EM algorithm has a built-in weighting for the Poisson noise for the projections P. However, we did not put the noise weighting in Part B for the iterative Landweber algorithm. Some computer simulation results are shown in Fig. 7 with the initial images as a square.

Fig. 7.

Iterative ML-EM reconstructions using the (left) square initial image and (right) associated central horizontal line profiles. Row 1: k = 1. Row 2: k = 10. Row 3: k = 100. Row 4: k = 1000.

For the iterative Landweber algorithm, the image value in the square is 1 and the background value is 0. The initial image value cannot be set to 0 for the MLEM algorithm, because the zero-valued image pixels will remain zero forever. For the iterative MLEM algorithm, the image value in the square is 2 and the background value is 1.

The ML-EM algorithm has a shift-variant Poisson noise weighting and a non-negativity constraint. These two facts prevent the ML-EM algorithm from being presented in the frequency domain. In the image regions where the image values are above zero and the Poisson noise variance does not change rapidly, the iterative Landweber algorithm can be a good approximation for the ML-EM when the ML-EM algorithm is written in the additive form [3], [4] and the proposed initial image effect model can be applied.

In our derivation of the frequency-domain transfer function (8), no noise weighing was considered. Even though noise weighting is considered in the ML-EM algorithm, the contributions from the initial image as a function of the iteration number k still follow the general trend as discussed at the end of the Section II.

IV. Discussion and Conclusion

The traditional treatment of the initial condition effects uses the null-space theory and assumes the convergence of the algorithm [18]. If a non-zero image, I₀, belongs to the null-space of matrix A (that is, A I₀ = 0) and image I is a solution of the converged iterative algorithm (that is, A I = P), then I + I₀ is also a solution. If the image I₀ is used as the initial image, then I₀ will remain in final solution. The classic treatment of the initial image effect only considers two extreme cases of the iteration numbers: 1) zero and 2) infinity. On the other hand, we have a closed-form expression for each iteration number.

This paper investigates the initial image effects in a nontraditional way—by using frequency-domain analysis. It shows how the contributions from the initial image to the reconstruction change with the iteration number k. As the parameter k increases, the propagation of the frequency components into the reconstruction gets attenuated more and more. Eventually, the influence of the initial image diminishes. The low frequency components disappear sooner than the high frequency components. This novel frequency-domain analysis perspective gives some new insights about the iteration number dependency of the initial image influence on the reconstruction. Our computer simulations support the following.

The contributions from the initial image to the reconstructed image have a trend that the entire initial image contributes at very low iterations, the lower frequency components then get attenuated, and eventually the influences vanish.

The traditional null-space theory only gives a “binary” conclusion whether the initial image 100% remains in the final image. On the other hand, our theory presents a “relative attenuation” (less than 100%) concept and its dependency on the iteration number.

The reader may ask: Does the frequency-domain model (8) always hold? The answer is no. The foundation of the result (8) is the equivalence of the matrix A^TA and the convolution kernel 1/r. In reality, this relationship is only an approximation due to the finite pixel size, finite image array size, and weighting factors W when A^TA is replaced by A^TWA. This relationship is further damaged if the number of views is not sufficient or the projections are truncated due to a relative small detector and a large object. In the situation of limited-angle tomography, where the view angles do not cover the required 180°, A^TWA cannot be approximated by convolution with the convolution kernel 1/r. The bottom line is that if A^TWA does not deviate from 1/r too much, the model (8) is still useful to indicate the trend of the initial image’s influence on the reconstruction.

In practice, the iterative algorithm is usually stopped early before its convergence. The initial image can provide some information about the sharp edges. The DC values in the initial image are almost lost while propagating into the reconstruction. When using the FBP image as the initial image, the iterative algorithm can converge much faster than using the constant as the initial condition. However, one must be aware that the high-frequency noise in the FBP reconstruction may affect the iterative reconstruction in a negative way. It may be helpful that a denoising filter is applied to the FBP image before it is used as the initial image. It is desirable that this denoising filter is nonlinear and can preserve the edges. It is beyond the scope of this paper to suggest an optimal initial image.