Experimental Study to Determine the Best Compression Ratio of High-Resolution Images of Small Bodies for the Martian Moons Exploration Mission*

Trans. Japan Soc. Aero. Space Sci. Vol. 63, No. 5, pp. 212–221, 2020 DOI: 10.2322/tjsass.63.212

Experimental Study to Determine the Best Compression Ratio of High-Resolution Images of Small Bodies for the Martian Moons eXploration Mission*

Yuta SHIMIZU,1)† Hiroaki KAMIYOSHIHARA,2) Takafumi NIIHARA,1) and Hideaki MIYAMOTO1)

1)Department of Systems Innovation, School of Engineering, The University of Tokyo, Tokyo 113–8654, Japan 2)Department of Earth and Planetary Science, School of Science, The University of Tokyo, Tokyo 113–8654, Japan

Data transmission rate is one of the biggest limiting factors in space exploration because transmitting images is largely restricted by the bitrate. This issue can be crucial for JAXA’s future sample-return mission, such as the Martian Moons eXploration (MMX) mission. High-resolution images are important in selecting the most scientifically interesting and safest landing/sampling sites, although the strategy of rapidly sending each image (> 114 MB) with a limited bitrate (< 32 Kbps) has to be considered. Lossy compression can significantly reduce the file size; however, the highly compressed images can be scientifically worthless. Therefore, determining the best compression ratio is necessary. Here, we develop a method to analyze the influence of image compression using image quality indices, Structural SIMilarity (SSIM) and image entropy. Moreover, by measuring the Cumulative Size Frequency Distribution (CSFD) of regolith grains in images with different amounts of compression, the loss of scientific value of images can be measured. We then carefully confirm that image quality indices can truly determine the best compression ratio. Using this method, the spacecraft can autonomously find the best compression ratio, compress an image and send it to the ground in a few minutes using a limited transmission rate.

Key Words: Data Transmission Rate, Best Compression Ratio, Martian Moons eXploration (MMX), Space Exploration

Nomenclature restrial bodies in the solar system. Among the most common onboard instruments is a visible imaging camera, which usu- B: constant used in the definition of CSFD ally requires the largest amount of data to be transmitted. C: constant used in the definition of SSIM Camera specifications, such as the focal length, total pixels, D: diameter of a grain particle angle of view, image size, shutter speed, and required resour- H: entropy of an image ces (i.e., size, weight, and power) are determined by various K: constant used in the definition of SSIM factors including scientific/engineering needs, available re- : mean intensity (value) sources of spacecraft, and the cost. However, due to the re- L: bit depth cent technological advances in camera performance, data N: number of image pixels transmission rate has become one of the biggest issues. Ob- p: compression ratio taining high-resolution images is now an easier task than ·: standard deviation (error) transmitting them. : image vector of a raw image Such a situation will have significant impacts on the de- : image vector of a compressed image sign of the future spacecraft missions. This is, in fact, an im- Subscripts minent issue for the Martian Moons eXploration (MMX) min: indicator for the minimum mission, which is scheduled for launch in 2024.1) The mis- i: indicator for the ith value sion is a sample-return mission, and the major objective is j: indicator for the jth value to obtain the most scientifically interesting samples safely x: member of from Phobos, one of the satellites of Mars. Phobos has two y: member of hypotheses for its origin: the capture of a primitive asteroid2) and the giant impact,3) which come from analyzing the data 1. Introduction of previous missions. The MMX mission is expected to col- lect samples that will clarify which formation theory is cor- More than 20 spacecraft are currently exploring extrater- rect. Importantly, the selection of the safest, most scientifically interesting landing/sampling sites depends largely on © 2020 The Japan Society for Aeronautical and Space Sciences detailed analyses of the remote-sensing data4) consisting + Presented at the 32nd International Symposium on Space Technology mostly of visible and infrared images. Thus, much higher- – and Science, 15 21 July 2019, Fukui, Japan. resolution images than those collected on previous missions Received 30 June 2019; final revision received 19 March 2020; accepted for publication 20 April 2020. will be required to determine the critical areas for landing and †Corresponding author, [email protected] sampling.

212 Trans. Japan Soc. Aero. Space Sci., Vol. 63, No. 5, 2020

For this reason, the onboard cameras of the MMX spacecraft, TElescopic Nadir imager for GeOmOrphology (TENGOO) and Optical RadiOmeter composed of CHromatic Imagers (OROCHI), will have higher resolutions than the camera of previous JAXA’s missions such as Hayabusa2;5) the size and the bit depth of TENGOO/ OROCHI are planned to be 3296 2472 pixels and 16 bits, respectively. The file size of a raw image can be up to 114 MB with 7 colors, which can take approximately 8 hours for transmission to Earth even with 32 Kbps bitrate, which is not practical for the limited observational period (i.e., in total, approximately 2 yr in the Phobos proximity orbit). Fur- thermore, during the landing phase, the spacecraft can stay on the surface of Phobos for only approximately 3 hr. During this short time, scientists/spacecraft should somehow determine the exact locations for collecting samples, which does Fig. 1. Simulated image of the surface of Phobos. not allow us to spend more than 1 hr for data transmission. Therefore, it is necessary to develop an adequate strategy for transmitting high-resolution images with important infor- Tagish lake-Based (UTPS-TB)6) was used, in which many mation using a limited transmission rate. kinds of rocks on the Earth were crushed and blended with The efficient transmission of image data can be achieved each other in order to make their reflectance and mechanical using the data compression. Many lossy compression properties match with the Phobos regolith, which were con- schemes have been proposed to significantly reduce the size strained by observations from both prior spacecraft missions of an image; however, highly compressed images can be dis- and ground-based telescopes. The surface of Phobos can be torted and become scientifically useless. Theoretical studies simulated by processing that simulant and preparing appro- based on compressional sensing theory may be required to priate illumination conditions. By taking digital photographs, figure out the best compression scheme for a given image, high-resolution images similar to those expected from MMX though this approach can take a large amount of time to be cameras can be obtained. developed. Our approach is to determine the best compres- The simulated Phobos surface image was obtained using sion ratio by using the compression scheme for a general pur- the following processes. UTPS-TB was spread on a founda- pose, which is already available for space missions. The best tion having a size of approximately 30 30 cm. A camera compression ratio should make the sizes of images as small having the focal length of 100 mm was fixed so that the im- as possible without losing scientific importance. We explore age obtained was not blurred due to camera shake. The cam- a strategy to figure out the best compression ratio for high- era took shots from 30 cm directly above the simulant. The resolution images expected to be obtained during the Phobos exposure time was 0.6 s and the f-number was 32, which proximity phase of the MMX mission. were determined to make the image bright enough, and at the same time avoid any objects in the image from getting 2. Image of the Surface of Phobos blurred due to depth of field. The size of the image was set to be 32962472 pixels, which is the same size as that of 2.1. Simulated image of Phobos the MMX cameras. A floodlight was used as the sunlight If we had high-resolution images of Phobos similar to and the incidence angle (phase angle) was set to 30°, which those expected to be obtained during the MMX mission, is a favorable angle to detect each object on the surface. The we could simply use them to study the best image compres- color is grayscale and the resolution of the image is 10 Lm/ sion ratio. However, no such images are available from pre- pixel. Note that this resolution is sufficient for identifying vious missions, which poses the challenge of determining a several tens of centimeters of surface rocks, which is required compression ratio for images which have not yet been ob- for the landing safety of the MMX lander. The simulated im- tained. Simulating realistic surface images by computer age is shown in Fig. 1. graphics may be possible, but is not that easy because of 2.2. Compression of the image by JPEG2000 many unknown variables, such as the shape of surface rocks, The image was then compressed using JPEG2000. their textures, and their responses to different illumination JPEG2000 is one of the famous and widely used image com- conditions which can significantly impact scientific analyses. pression techniques developed by the Joint Photographic Ex- We thus take an experimental approach, where we develop a perts Group (JPEG) and standardized by ISO/IEC 15444.7,8) simulated Phobos surface and take high-resolution images It is generally used for medical images,9) astronomical im- using similar illumination conditions to those which will ages,10,11) and the High-Resolution Imaging Science Experi- be experienced during the MMX mission. ment (HiRISE), which is a camera instrument on NASA’s As a simulated material of Phobos regolith (often simply Mars Reconnaissance Orbiter (MRO).12) Moreover, our use called “simulant”), the Univ. Tokyo Phobos Simulant, of JPEG2000 is motivated by its planned use as a lossy com-

©2020 JSASS 213 Trans. Japan Soc. Aero. Space Sci., Vol. 63, No. 5, 2020 pression technique for the MMX mission. Although there are an enormous number of image compression techniques such as JPEG, PNG, and TIFF, our study is catered to the MMX mission, and therefore focuses on JPEG2000 compression. Future studies may address the influence of other compression techniques that apply to other camera instruments for future space probes. The image was compressed in different compression ratios, p, which is defined as ! size p ¼ 1 compressed 100 ½%; ð1Þ sizeoriginal where, sizecompressed and sizeoriginal are file sizes of compressed, original images, respectively. Examples of compressed images are shown in Fig. 2. Note that the original file size is 8.1 MB, and the minimum compression ratio image for JPEG2000 is a lossless compression image where pixel values are exactly the same as those in the raw image. For the image used in this study, this lossless compression has a value of p ¼ 70%. 2.3. The influence of the compression Because of the unique compression technique of JPEG2000, which involves Discrete Wavelet Transform (DWT) and Embedded Block Coding with Optimal Trunca- tion (EBCOT), high-frequency components in images are more likely to be lost rather than low-frequency components, which means that the smaller the size of the structure, the ear- lier it is blurred and becomes difficult to be distinguished. The lowest compression ratio image, 70.00% (Fig. 1), has all of the same pixel values as the raw image, and thus, they cannot be distinguished from one another. Finding differences between the raw image and each compressed image in the range from 80.00% to 98.00% is difficult without zooming in, although each file size is shrunk at the expense of some information. However, after the compression ratio reaches 99.00% (Fig. 2-top), fine surface roughness starts to change slightly, and at higher compression ratios, this difference in roughness becomes more noticeable. At p>99:90%, the image is completely altered and becomes blurred (Fig. 2-middle). In addition to this change, the outline of each particle in the image starts to get blurred as the compression ratio rises. This difference becomes prominent and many small particles are not able to be detected at a ratio higher than 99.90%. At the ratio of 99.99%, the image is totally distorted, and noth- Fig. 2. Compressed images in 99.00% (top), 99.90% (middle) and 99.99% ing can be reliably identified (Fig. 2-bottom). In short, as the (bottom). compression ratio increases, initially the surface roughness changes, then the profile of particles gets blurred and small particles become difficult to be identified. Finally, the image for identifying the best compression ratio using a computer is completely distorted and nothing can be seen. algorithm. 3.1. Structural SIMilarity (SSIM) 3. Analysis of the Image Quality There are many objective methods to analyze the quality of images, and Structural SIMilarity (SSIM) is one of the To determine the best compression ratio, we explore ob- most widely used quality indices.13) The basic concept of jective methods involving the calculation of image quality SSIM to evaluate the quality of images consists of three com- indices to analyze the change in image quality caused by im- parisons: luminance, contrast, and structure comparisons. age compression. Because such methods can be completed These comparisons are conducted, mainly focusing on the without the aid of manual analysis, they provide a fast means mean intensity and standard deviation of images. Suppose

is a raw image containing all of the pixel values, while The value of SSIM is in the range from 0.0 to 1.0, which represents a compressed image. xi and yi are the pixel values shows a higher quality when the value is higher. It takes 1.0 of and . The mean intensity of and are defined as if an image is totally the same with the original image. 1 XN 3.2. Image entropy x ¼ xi; ð2Þ In a basic sense, compression of images changes each N i¼1 pixel value from the original image, which leads to the loss and of important information and scientific value. How much 1 XN each value of a pixel changes from the original can be meas- y ¼ yi: ð3Þ ured by subtracting the values of a compressed image from N i¼1 the values in the raw image. We thus obtain difference im- In SSIM, the luminance comparison function l( , ) is writ- ages for every compression ratio by calculating the pixel dif- ten as ferences between Raw and compressed images. 2 þ C Figure 3 shows difference images for three different com- lð ; Þ¼ x y 1 ð Þ 2 2 4 pression ratios: 70.00%, 99.90% and 99.99%. The 70.00% þ þ C1 x y compressed image show any differences compared to the where, C1 is a constant. Then, we can conduct a contrast raw image, so its difference image has no features and has comparison. The standard deviation of and are defined as a value of zero for all pixels (Fig. 3-top). At first, when the "#1 ff XN 2 compression ratio rises, the di erence images remain fea- 1 2 ð Þ x ¼ ðxi xÞ ; 5 tureless. However, when p reaches 99.00% some patterns ap- N 1 i¼1 pear and become prominent at higher ratios such as 99.90% "#1 XN 2 (Fig. 3-middle). The patterns then start to exhibit recogniz- 1 2 ð Þ y ¼ ðyi yÞ : 6 able structures as the ratio approaches 99.99% (Fig. 3- N 1 i¼1 bottom). This trend can also be explained using a histogram The contrast comparison function c( , ) is written as of differences in the pixels, which is shown in Fig. 4. In the 2 þ C difference image of 70.00%, all pixels have the value of zero. cð ; Þ¼ x y 2 ; ð Þ 2 2 7 When the compression ratio increases, each pixel starts to þ þ C2 x y have some differences larger/smaller than zero and the vari- where, C2 is a constant. Structure comparison is not affected ety of differences becomes larger at higher ratios, drawing by luminance and contrast, and thus the structures of the two some patterns and structures in the difference images. images are expressed as ( xÞ=x and ( yÞ=y. The When we can see some patterns or structures in the differ- structural similarity can be measured by correlating the two ence images, this means that the compressed images have vectors, and that correlation is equivalent to the correlation lost some pattern or structure information. Stated differently, coefficient between and . In this case, the structure com- we can measure how much information has been lost in the parison function s( , ) can be written as compression images by analyzing the difference images. We therefore make use of the image entropy, in addition to þ C sð ; Þ¼ xy 3 ; ð Þ SSIM, to analyze the trend in the difference images, where þ 8 xy C3 differences of the pixels start to have a wider variety of val- where, C3 is a constant. Finally, SSIM( , )isdefined (us- ues as the compression ratio becomes higher. ing Eqs. (4), (7), (8)) as Suppose n½k; l is the value of the kth column and the lth row in a difference image. The probability of having the val- SSIMð ; Þ¼½lð ; Þ ½cð ; Þ ½sð ; Þ : ð9Þ ue of n½k; l for each pixel, p½k; l can be calculated as Here, the three parameters of >0;>0;>0 are intro- Nn½k; l duced to adjust the relative importance of each component. p½k; l¼ ; ð12Þ Ntotal For simplicity, in this study ¼ ¼ ¼ 1 and C3 ¼ C2=2 13) stand as in Wang et al. Using these values, the SSIM can where, Nn½k; l is the number of pixels having the value of be rewritten as n½k; l and Ntotal is the total number of pixels in the image, ð2xy þ C1Þð2xy þ C2Þ which is 3296 2472 pixels in this study. With reference SSIMð ; Þ¼ : ð10Þ 14) fi ð2 þ 2 þ C Þð2 þ 2 þ C Þ to Shannon and Weaver, the image entropy H is de ned as x y 1 x y 2 X X ¼ ½ ð ½ Þ 13) H p k; l log2 p k; l : ð Þ Moreover, we define C1 and C2 in reference to Wang et al. 13 k l such that This entropy becomes higher if the degree of randomness in C ¼ðK LÞ2;C¼ðK LÞ2; ð11Þ 1 1 2 2 an image is higher, which means that having a larger variety where, L is the bit depth of the image (i.e., 255 for this study). of differences and showing patterns/structures in a difference K1 and K2 are small constants that have conditions of image increases the entropy value. In other words, an in- K1 1 and K2 1 . In this work, K1 is 0.01 and K2 is crease in entropy shows the loss of information and a change 0.03 as in Wang et al.13) in the image quality.

Fig. 4. Histogram showing the number of pixels as a function of the value of diﬀerence of pixels in 70.00%, 99.90%, and 99.99% compressed images.

Fig. 5. SSIM (top), and image entropy (bottom) as functions of compression ratio.

compression ratio just before the sudden decrease in image Fig. 3. Diﬀerence images for 70.00% (top), 99.90% (middle), and 99.99% quality (i.e., 98.00%) can reasonably be determined to be (bottom). the best compression ratio.

4. Confirmation Using Manual Analysis 3.3. Image quality analysis by SSIM and image entropy Figure 5 shows the result of the analysis of the image SSIM and image entropy can be useful methods to deter- quality using SSIM and image entropy. Overall, SSIM de- mine the best compression ratio; however, for space explora- creases as the compression ratio increases. Particularly, the tion missions that usually require massive funding and have value suddenly starts to decrease when the compression ratio little chance to be conducted again, the reliability of those reaches 99.00%. It took 1.25 min (75 s) to calculate the SSIM methods must be confirmed. We have to evaluate whether for all images. On the other hand, the image entropy contin- or not the compressed images obtained using those methods uously increases as the ratio rises. Similarly, the entropy gets are truly suitable for space exploration missions. By defining dramatically higher after the ratio of 99.00%. The calculation the scientific value of high-resolution images for missions, time for image entropy was 1.20 min (72 s), which was the amount of the scientific value loss caused by image com- slightly faster than calculating SSIM. All of the computations pression can be measured, which enables us to evaluate com- were conducted on a laptop computer equipped with an Intel pressed images. Thus, we define the scientific value using the Core i7 3.5 GHz processor and RAM of 16 GB and using concept of Cumulative Size Frequency Distribution (CSFD) Python scripts. of rock gravel and measure the amount of loss. The two analyses show that, as the compression ratio rises, 4.1. Cumulative Size Frequency Distribution (CSFD) the image quality is steadily diminished, and then it dramat- One of the most attractive objects in images of the surface ically decreases for p>99:00%. Therefore, the value of the of celestial bodies is rock gravel. In geology, particles in rock

Table 1. Classification of rock gravel particles. Minimum size (mm) Name of particle 256 Boulder 64 Cobble 4 Pebble 2 Granule 1 Very-coarse sand grain 1/2 Coarse sand grain 1/4 Fine sand grain 1/8 Very-fine sand grain 1/16 Silt particle 1/256 Clay particle gravel are classified by their size (Table 1).15) Previous studies have reported that those particles (i.e., boulders, cobbles and pebbles) are ubiquitously distributed over the surface of 16–18) celestial bodies. They have some key information to Fig. 6. CSFD for Raw image of the simulated surface of Phobos, which is clarify geological processes and the origin of bodies, because shown in Fig. 1. Cumulative number of particles is fit by power-law mod- they seem to originate from impact events that form cra- el (dashed line). Detailed calculation methods for and its error are in the ters,19–21) or the disruption of the parent body, which forms Section 4.4. the body itself.22,23) Some studies have described granular processes of the surface of bodies such as migration and par- portant, and miscalculation of the value of in an image ticle sorting by observing rock gravel in photographs.24) In would contribute to the loss of its scientific value. this manner, previous studies have made new scientific find- Therefore, this study defined the scientific value of images ings by focusing on gravel in images of the surface of celes- as the detectability of rock gravel in the image. The slope of tial bodies. CSFD therefore provides an index by which to measure the In particular, CSFD of rock gravel has been widely used loss of scientific value. A change in the value of the slope of to explain geological processes of many celestial bodies, CSFD in compressed images can be used to quantify the loss such as Ceres,20) Phobos,19,25) asteroid 433 Eros,26) 25143 of the image’s scientific value. Itokawa,16,23,24,27) 21 Lutetia,18) 4179 Toutatis,28) comet 4.3. Manual measurements of the size of particles 67P/Churumov-Gerasimenko,29–32) and 103P/Hartley 2.33) Using Fiji,36) which is an open image-processing software, In the basic concept of CSFD, a number of particles N is writ- the size of each particle was measured manually. The out- ten as lines of all particles in the images were overlapped with ellipses (Fig. 7). This process exactly determines the size of par- N ¼ BD; ð14Þ ticles, and therefore overlapping was carefully conducted. If where, B is a constant and D is the size of particles. Gener- the ellipses did not properly fit the profile of particles, we ally, a cumulative number is defined as the total number of moved the ellipses or changed the size of the ellipses until particles having a size larger than D, or a cumulative number they best-fitted the outline of particles. Then, the length of per area is used as N. represents an index of power-law the semi-major axis of each ellipse was measured as the size (i.e., slope of CSFD), and this value is important because it of each particle. This method of size measurement by fitting describes the relationship between small and large particles ellipses was used in previous studies, such as Tancredi et (i.e., larger value of reflects enrichment of small particles). al.37) Especially for fragmented objects, primarily takes values In this manner, the size of particles was measured in seven between 2 and 4.34) An example of CSFD is shown in different compression ratio images, which were raw Fig. 6, which shows the CSFD for the raw image of the sim- (70.00%), 90.00%, 95.00%, 98.00%, 99.00%, 99.90% and ulated photograph of the surface of Phobos (Fig. 1). Detailed 99.95% compressed images. Those were selected based on explanations are provided in the Section 4.4. the observation of the Section 2.3, which shows that in the 4.2. Definition of scientific value and measurements of range of 70.00% to 98.00% there are no huge differences be- scientific loss tween the raw image and compressed images, while higher The difference between from different celestial bodies ratio compressed images revealed dramatic changes as com- and areas has enormously contributed to the understanding pared to the raw image. of the characteristics and geological processes of celestial 4.4. Method of fitting power-law distribution bodies. Moreover, one of the most important elements for Using the data of the size of each particle, the slope values the safety of landing a spacecraft is the particle (boulder) dis- of CSFD were calculated by plotting the cumulative number tribution.35) Only a difference of 0.5 in the value of leads to of particles per area and fitting power-law distribution to it. very different conclusions; for example, about the surface of As the fitting method, Clauset et al.38) showed that a method properties of celestial bodies and the geological processes consisting maximum-likelihood fitting method and goodness- shaping them. Thus, calculating precisely is extremely im- of-fit tests based on Kolmogorov-Smirnov static and likeli-

C has the condition of Z 1 Z 1 ð þ Þ n pðxÞdx ¼ Cx 1 dx ¼ ; ð17Þ xmin xmin area where, n is the total number of particles in the image and area is the unit of the surface area in the image. Therefore, C can be defined as n C ¼ x : ð18Þ area min In that case, Eqs. (15) and (16) are written as ! ðþ1Þ n x pðxÞdx ¼ dx; ð19Þ area xmin xmin Fig. 7. Example of manual measurements of the size of particles in a raw ! image. n x CCDFðxÞ¼ : ð20Þ area xmin hood is a statistically correct and reproduceable way to fit power-law distribution to empirical data, rather than other Then, let us calculate of power-law distribution that best famous techniques such as the least-squares method. Some fits the observed data having a size is larger than xmin. can previous studies began using this method and discussed the be calculated using maximum likelihood estimators (MLE). distributions of particles on the surface of celestial The probability that the data can be fitted by the model is bodies,20,37,39) and therefore, this study adopts the method written as ! of Clauset et al.38) ðþ1Þ Yn n ^ 1 x The fitting process, is conducted in the following manner: pðxj^Þ¼ i ; ð21Þ area xmin xmin a) Define xmin, which is the minimum size of particles used i¼1 for fitting where, xi represents the size of particles observed. This prob- b) Find the power-law distribution that best fits the data ability is called likelihood of the data given by the model and observed and calculate the slope of power-law, the model (power-law distribution) best fits the data observed c) Calculate the goodness-of-fit D of the data observed when this likelihood is maximized, which leads to the calcu- and the fitted power-law distribution lation of . In order to maximize the likelihood, the logarithm d) Change the value of xmin and calculate D until D is L of the likelihood is frequently used, and L is maximized at minimized the condition of @L=@ ¼ 0. By solving that condition for , e) Check the feasibility of the fitting by calculating the can be calculated as "# p-value 1 Xn x Suppose x is the quantity for the distribution under discus- ¼ n ln i : ð22Þ sion and x represents the size of particles in this study. Let i¼1 xmin us define xmin as the minimum value of x, which also means Moreover, the standard error of is derived as the minimum size of particles of the distribution following ! 1 power-law. Power-law distribution is written by using the ¼ pffiffiffi þO ; ð23Þ probability density function pðxÞ as n n pðxÞdx ¼ Prðx Xxmin Z 1 where xi is the observed size of each particle and yi is the 0 0 ¼ pðx Þdx cumulative number per area of each particle. This D repre- ð16Þ x sents the goodness-of-fit. The best fitting can be achieved C ¼ x : when D is minimized and the best xmin is obtained under that condition.

Finally, the feasibility of the fitting can be checked using the p-value. To calculate the p-value, synthetic data is made by applying a semi-parametric bootstrap. In a semi- parametric bootstrap, one data component is obtained ran- domly from the data observed having a size smaller than the best xmin and the other data is generated using the best- fit power-law distribution under the condition that each value is larger than the best xmin. The synthetic data is made com- bining those two data elements. Note that the total number of the synthetic data is the same as the number of data observed. Fig. 8. The value of alpha of power-low and number of boulders observed In this study, synthetic data was made 1000 times. For each in compression images as a function of compression ratios. synthetic data, the fitting processes are conducted and the minimum of D is calculated. Then, how many times that D fi is smaller than the D of the first fitting is calculated and Table 2. Results of manual measurement of particles and parameters of t- ted CSFD at different compression ratios. Comp. and Num. mean com- fi the p-value is de ned as the proportion of this number to pression and number, respectively. xmin represents the minimum size of the total number of synthetic data, which is 1000 in this particles that was used for fitting. is the error of slope values. study. If the p-value is larger than 0.1, the fitting is consid- Comp. ratio x Num. of min p-value ered to be feasible.38) (%) (mm) fitted particles 4.5. Result of the manual measurement and fitting Raw (70.00) 1.371 118 0.259 0.759 Figure 8 shows the value calculated for alpha and number 90.00 1.505 91 0.302 0.800 95.00 1.346 113 0.271 0.803 of particles detected as functions of the compression ratio. 98.00 1.412 101 0.290 0.403 The alpha does not change at ratios of 95.00% or less. When 99.00 1.494 101 0.286 0.680 the ratio gets higher, a slight change is detected, and partic- 99.90 1.904 53 0.439 0.271 ularly for ratios larger than 99.00%, the value changes dra- 99.95 1.587 39 0.325 0.115 matically. The number of particles detected also decreases slightly for ratios 98.00% or less, and decreases dramatically for ratios larger than 98.00%. Table 2 shows the parameters ploration missions that do not have enough transmission rate of each fitting. xmin, the number of fitted particles, and to send high-resolution images, such as the MMX mission. change due to the differences in the particle size data in each Note that, in critical phases for the future missions, for ex- compressed image. More importantly, in all images, the ample, when selecting the landing sites for the MMX mis- p-value is larger than 0.1, which shows that all fittings are sion, this result should be carefully interpreted. As Fig. 9 feasible to some extent. However feasibility starts to be lost, shows, for p ¼ 98:00%, particles having a size smaller than especially when the ratio rises above 99.00%. Given these re- 1 mm are unable to be identified completely and the total sults, scientific information seems to be lost at ratios larger number of particles identified is 148 less than the number than 99.00%, which means that the best compression ratio in the raw image. Suppose the resolution of the image is should be 99.00% or less. 65 mm/pixel, which is the resolution planned for TENGOO 4.6. Best compression ratio during the quasi-satellite orbit of Phobos 11 km from the sur- Judging from the results of the manual measurement, com- face.5) Scaling our image up, the particles 1 mm in size are pressed images obtained using the two objective methods equivalent to particles 6.5 m in size, based on our image res- (98.00%) can be regarded as truly suitable for space explora- olution of 10 Lm/pixel for the Phobos simulant. If the reso- tion missions, and the reliability of the two methods is also lution is lower than 65 mm/pixel, it is probable that particles confirmed. This means that the best compression ratio can smaller than 6.5 m cannot be identified, which may not be ap- be determined in approximately 1 min using SSIM and im- propriate for choosing the landing site. Therefore, when se- age entropy. lecting the landing site, images having a compression ratio In order to use the methods, the preparation of several im- lower than 98.00% may be needed for safe landing site selec- ages at different compression ratios is needed beforehand. tion. In order to determine the best compression ratio in every However, the process of compression can be minimized critical phase during future missions, different and appropri- within a few seconds,41) and thus it takes only a few minutes ate methods/criteria are needed, which is a topic for future for preparing all of the compressed images. In this study, the research. file size of the image can be shrunk to 2% (162 KB) of the original (8.1 MB), and thus it takes about 40 s to send it to 5. Conclusion Earth even if the data transmission rate is only 32 Kbps. This means that the process of compressing images, determining We analyzed the change in image quality caused by image the best compression ratio, and transmitting the compressed compression using the two objective methods, SSIM and im- image to the ground can be completed within a few minutes, age entropy, in order to determine the best compression ratio while sending a raw image takes as long as 34 min. There- for high-resolution images of the surface of small bodies and fore, this method can be a powerful tool for future space ex- resolve the problem of limited data transmission rate for

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