These are supplementary materials for the paper 'Additive Tensor Decomposition Considering Structural Data Information'. It includes three videos for better illustration of the simulation/case study results. The codes for simulations studies can be found in Exp1 and Exp2 folder. You can run main.m files in each folder to reproduice the results.
The data for monitoring the crack growing process include consecutive measurement images of size 40×40. These images form a tensor M in R^(30×40×40). We simulate M by summing up two tensors
and
that represent the background and the crack, respectively. Each
is generated by a 2D smooth Gaussian process representing the background. Most values in the image
are zeros, and the non-zero values of
gradually grows when the index i increases from 1 to 30, representing the crack growing on the wall. These values are generated from i.i.d.
random variables to represent the random lighting and shadowing conditions. The first image in the following video illustrates M, the second images illustrate the corresponding images for actual crack which is a continuous line and the third image illustrate the images for the simulated crack under irregularly illuminated conditions.
We then decompose M into two components
and
by solving Problem (2). In the ADMM algorithm, the step size is
and the tuning parameters are λ_1,1=λ_1,2=1,λ_2,1=10, and λ_2,2=0.08. The fifth images of Figures 6 (a), (b) illustrate the estimated crack of the 20th and the 30th image using the ATD-based method. It is shown that the ATD-based method captures the growth of the whole crack accurately.
The tensor M in Example 2 is generated to simulate the consecutive measurements taken from a thermal camera in a heated surface monitoring process. It also contains 30 images of size 40×40, and it is generated by summing up three tensors of the same size that represents the true background, the static hotspot, and the moving hotspot respectively. Among them, each mode 1 slice of the tensor
is generated from
where
is a 40×40 matrix representing the heating effect of the heating process,
is a matrix of the same size representing the cooling effect and
is a
random variable representing a random combination of the two effects.
The images representing the matrices
and
are shown in the following figure.
To simulate the heating effect of a single point heating source at the center of this image, we generate using the value of
, where
is the probability density function of
, where I is a 2×2 identity matrix. Then, we transform all values of
linearly such that the maximum and minimum value of
are 1 and 0, respectively. With this setup, the maximum value within
is 1, located at the center of the image; when the pixel moves farther way from the center, the value of
gradually drops to 0. To simulate the cooling effect, we generate
using a linear function
, where
and
are adjusted so that the maximum and minimum value within the matrix
are 0 and 1 respectively. It represents that the coolant for the surface flows from the upper-left corner to the bottom-right corner of the image.
Each image within the tensor are the same, and the non-zero values in these images are located in a fixed 2×2 rectangle with intensity value 1 in their lower-left corners. The non-zero values in each image of the tensor
are also located in a 2×2 rectangle with intensity value 1. However, this rectangle locates on the upper-left part of the image. When the image index i increases, the rectangle in
moves from the left side to the right side across the images.
We decompose the tensor M into components ,
and
by solving Problem (3). The step size
. Tuning parameters are λ_1,1=λ_1,2=30, λ_1,3= λ_2,1=1, λ_2,2=1.9 and λ_3,1=2.
In the following video, we illustrate the simulated images, with the decomposed background, the static hotspot, and the moving hotspot using the ATD method. The decomposed background and the moving object from the RPCA method are illustrated in the third column. The video illustrating the result of the decomposition is provided in the supplementary material. We can see that the proposed ATD method successfully separates the static background, static hotspot and the moving hotspot. Comparing the result of the ATD, the RPCA can only identify the moving object but fails to separate the true background and the static hotspot. This is because the RPCA does not consider the spatial smoothness of the background.
With tuning parameters λ_1,1=30, λ_2,1=0.7, λ_2,2=0.16 and λ_3,1=1, we solved this problem using the ADMM algorithm. The images in first row of the following video illustrates the non-zero values from the 3rdth-7th images in the solution of , reflecting the AVC regions, that determines the boundary of the 3D-printed prototype of patients’ aortic root anatomies. It shows that the smooth change of the AVC regions across images is captured and it reflects the anatomic reality. The following video illustrates the result of the decomposition.
For comparison, we also used SSD to extract the AVC regions from individual images (the second row of the video). It achieves similar performance in the 3rd-6th images. In the 7th image, however, the SSD method failed to fully extract the anomaly region. The reason is that the intensity of the AVC regions is not so high. Unlike the ATD formulation, SSD does not consider the similarity of the AVC regions across images so that the pixels with low intensity on the AVC regions cannot be preserved. This case study further illustrates the versatility of the ATD framework and demonstrates its ability to solving real-world problems.
