(Continued from last entry)

 

The solution to this problem is both simple and unintuitive. It is to make 300,000 measurements which are totally unrelated to the wavelet basis - despite all that I have said above regarding how this is the best basis in which to view and compress images. In fact, the best types of measurements to make are (pseudo-)random measurements - generating, say, 300,000 random “mask” images and measuring the extent to which the actual image resembles each of the masks. Now, these measurements (or “correlations”) between the image and the masks are likely to be all very small, and very random. But - and this is the key point - each one of the 2 million possible wavelets which comprise the image will generate their own distinctive “signature” inside these random measurements, as they will correlate positively against some of the masks, negatively against others, and be uncorrelated with yet more masks. But (with overwhelming probability) each of the 2 million signatures will be distinct; furthermore, it turns out that arbitrary linear combinations of up to a hundred thousand of these signatures will still be distinct from each other (from a linear algebra perspective, this is because two randomly chosen 100,000-dimensional subspaces of a 300,000 dimensional ambient space will be almost certainly disjoint from each other). Because of this, it is possible in principle to recover the image (or at least the 100,000 most important components of the image) from these 300,000 random measurements. In short, we are constructing a linear algebra analogue of a hash function.

There are however two technical problems with this approach. Firstly, there is the issue of noise: an image is not perfectly the sum of 100,000 wavelet coefficients, but also has small contributions from the other 1.9 million coefficients also. These small contributions could conceivably disguise the contribution of the 100,000 wavelet signatures as coming from a completely unrelated set of 100,000 wavelet signatures; this is a type of “aliasing” problem. The second problem is how to use the 300,000 measurements obtained to recover the image.

Let us focus on the latter problem first. If we knew which 100,000 of the 2 million wavelets involved were, then we could use standard linear algebra methods (Gaussian elimination, least squares, etc.) to recover the signal. (Indeed, this is one of the great advantages of linear encodings - they are much easier to invert than nonlinear ones. Most hash functions are practically impossible to invert - which is an advantage in cryptography, but not in signal recovery.) However, as stated before, we don’t know in advance which wavelets are involved. How can we find out? A naive least-squares approach gives horrible results which involve all 2 million coefficients and thus lead to very noisy and grainy images. One could perform a brute-force search instead, applying linear algebra once for each of the possible set of 100,000 key coefficients, but this turns out to take an insanely impractical amount of time (there are roughly 10^{170,000} combinations to consider!) and in any case this type of brute-force search turns out to be NP-complete in general (it contains problems such as subset-sum as a special case). Fortunately, however, there are two much more feasible ways to recover the data:

    * Matching pursuit: locate a wavelet whose signature seems to correlate with the data collected; remove all traces of that signature from the data; and repeat until we have totally “explained” the data collected in terms of wavelet signatures.
    * Basis pursuit (or l^1 minimisation): Out of all the possible combinations of wavelets which would fit the data collected, find the one which is “sparsest” in the sense that the total sum of the magnitudes of all the coefficients is as small as possible. (It turns out that this particular minimisation tends to force most of the coefficients to vanish.) This type of minimisation can be computed in reasonable time via convex optimisation methods such as the simplex method.

Note that these image recovery algorithms do require a non-trivial (though not ridiculous) amount of computer processing power, but this is not a problem for applications such as sensor networks since this recovery is done on the receiver end (which has access to powerful computers) rather than the sensor end (which does not).

There are now rigorous results which show that these approaches can reconstruct the original signals perfectly or almost-perfectly with very high probability of success, given various compressibility or sparsity hypotheses on the original image. The matching pursuit algorithm tends to be somewhat faster, but the basis pursuit algorithm seems to be more robust with respect to noise. Exploring the exact range of applicability of these methods is still a highly active current area of research. (Sadly, there does not seem to be an application to P\neq NP; the type of sparse recovery problems which are NP-complete are the total opposite (as far as the measurement matrix is concerned) with the type of sparse recovery problems which can be treated by the above methods.)

As compressed sensing is still a fairly new field (especially regarding the rigorous mathematical results), it is still a bit premature to expect developments here to appear in actual sensors. However, there are proof-of-concept prototypes already, most notably the single-pixel camera developed at Rice.

Finally, I should remark that compressed sensing, being an abstract mathematical idea rather than a specific concrete recipe, can be applied to many other types of contexts than just imaging. Some examples include:

    * Magnetic resonance imaging (MRI). In medicine, MRI attempts to recover an image (in this case, the water density distribution in a human body) by taking a large but finite number of measurements (basically taking a discretised Radon transform (or x-ray transform) of the body), and then reprocessing the data. Because of the large number of measurements needed, the procedure is lengthy for the patient. Compressed sensing techniques can reduce the number of measurements required significantly, leading to faster imaging (possibly even to real-time imaging, i.e. MRI videos rather than static MRI). Furthermore, one can trade off the number of measurements against the quality of the image, so that by using the same number of measurements as one traditionally does, one may be able to get much finer scales of resolution.
    * Astronomy. Many astronomical phenomena (e.g. pulsars) have various frequency oscillation behaviours which make them very sparse or compressible in the frequency domain. Compressed sensing techniques then allow one to measure these phenomena in the time domain (i.e. by recording telescope data) and being able to reconstruct the original signal accurately even from incomplete and noisy data (e.g. if weather, lack of telescope time, or simply the rotation of the earth prevents a complete time-series of data).
    * Linear coding. Compressed sensing also gives a simple way for multiple transmitters to combine their output in an error-correcting way, so that even if a significant fraction of the output is lost or corrupted, the original transmission can still be recovered. For instance, one can transmit 1000 bits of information by encoding them using a random linear code into a stream of 3000 bits; and then it will turn out that even if, say, 300 of the bits (chosen adversarially) are then corrupted, the original message can be reconstructed perfectly with essentially no chance of error. The relationship with compressed sensing arises by viewing the corruption itself as the sparse signal (it is only concentrated on 300 of the 3000 bits).

Many of these applications are still only theoretical, but nevertheless the potential of these algorithms to impact so many types of measurement and signal processing is rather exciting. From a personal viewpoint, it is particularly satisfying to see work arising from pure mathematics (e.g. estimates on the determinant or singular values of Fourier minors) end up having potential application to the real world.

(the end)

Brains could be considered as sensor networks, right? 

I’ve had a number of people ask me (especially in light of some recent publicity) exactly what “compressed sensing” means, and how a “single pixel camera” could possibly work (and how it might be advantageous over traditional cameras in certain circumstances). There is a large literature on the subject, but as the field is relatively recent, there does not yet appear to be a good non-technical introduction to the subject. So here’s my stab at the topic, which should hopefully be accessible to a non-mathematical audience.

 

For sake of concreteness I’ll primarily discuss the camera application, although compressed sensing is a more general measurement paradigm which is applicable to other contexts than imaging (e.g. astronomy, MRI, statistical selection, etc.), as I’ll briefly remark upon at the end of this post.

The purpose of a camera is, of course, to record images. To simplify the discussion, let us think of an image as a rectangular array, e.g. a 1024 x 2048 array of pixels (thus there are 2 megapixels in all). To ignore the (minor) issue of colour, let us assume that we are just taking a black-and-white picture, so that each pixel is measured in grayscale as an integer (e.g. an 8-bit integer from 0 to 255, or a 16-bit integer from 0 to 65535) which signifies the intensity of each pixel.

Now, to oversimplify things quite a bit, a traditional digital camera would take one measurement of intensity for each of its pixels (so, about 2 million measurements in the above example), resulting in a relatively large image file (2MB if one uses 8-bit grayscale, or 4MB if one uses 16-bit grayscale). Mathematically, this file can be represented by a very high-dimensional vector of numbers (in this example, the dimension is about 2 million).

Before I get to the new story of “compressed sensing”, I have to first quickly review the somewhat older story of plain old “compression”. (Those who already know how image compression works can skip forward a few paragraphs.)

The images described above can take up a lot of disk space on the camera (or on some computer where the images are later uploaded), and also take a non-trivial amount of time (and energy) to transfer from one medium to another. So, it is common practice to get the camera to compress the image, from an initial large size (e.g. 2MB) to a much smaller size (e.g. 200KB, which is 10% of the size). The thing is that while the space of all images has 2MB worth of “degrees of freedom” or “entropy”, the space of all interesting images is much smaller, and can be stored using much less space, especially if one is willing to throw away some of the quality of the image. (Indeed, if one generates an image at random, one will almost certainly not get an interesting image; instead, one will just get random noise looking much like the static one can get on TV screens.)

How can one compress an image? There are many ways, some of which are rather technical, but let me try to give a non-technical (and slightly inaccurate) sketch of how it is done. It is quite typical for an image to have a large featureless component - for instance, in a landscape, up to half of the picture might be taken up by a monochromatic sky background. Suppose for instance that we locate a large square, say 100 \times 100 pixels, which are all exactly the same colour - e.g. all white. Without compression, this square would take 10,000 bytes to store (using 8-bit grayscale); however, instead, one can simply record the dimensions and location of the square, and note a single colour with which to paint the entire square; this will require only four or five bytes in all to record, leading to a massive space saving. Now in practice, we don’t get such an impressive gain in compression, because even apparently featureless regions have some small colour variation between them. So, given a featureless square, what one can do is record the average colour of that square, and then subtract that average off from the image, leaving a small residual error. One can then locate more squares where the average colour is significant, and subtract those off as well. If one does this a couple times, eventually the only stuff left will be very small in magnitude (intensity), and not noticeable to the human eye. So we can throw away the rest of the image and record only the size, location, and intensity of the “significant” squares of the image. We can then reverse this process later and reconstruct a slightly lower-quality replica of the original image, which uses much less space.

Now, the above algorithm is not all that effective in practice, as it does not cope well with sharp transitions from one colour to another. It turns out to be better to work not with average colours in squares, but rather with average colour imbalances in squares - the extent to which the intensity on (say) the right half of the square is higher on average than the intensity on the left. One can formalise this by using the (two-dimensional) Haar wavelet system. It then turns out that one can work with “smoother” wavelet systems which are less susceptible to artefacts, but this is a technicality which we will not discuss here. But all of these systems lead to similar schemes: one represents the original image as a linear superposition of various “wavelets” (the analogues of the coloured squares in the preceding paragraph), stores all the significant (large magnitude) wavelet coefficients, and throws away (or “thresholds”) all the rest. This type of “hard wavelet coefficient thresholding” compression algorithm is not nearly as sophisticated as the ones actually used in practice (for instance in the JPEG 2000 standard) but it is somewhat illustrative of the general principles in compression.

To summarise (and to oversimplify somewhat), the original 1024 \times 2048 image may have two million degrees of freedom, and in particular if one wants to express this image in terms of wavelets then one would need thus need two million different wavelets in order to reconstruct all images perfectly. However, the typical interesting image is very sparse or compressible in the wavelet basis: perhaps only a hundred thousand of the wavelets already capture all the notable features of the image, with the remaining 1.9 million wavelets only contributing a very small amount of “random noise” which is largely invisible to most observers. (This is not always the case: heavily textured images - e.g. images containing hair, fur, etc. - are not particularly compressible in the wavelet basis, and pose a challenge for image compression algorithms. But that is another story.)

Now, if we (or the camera) knew in advance which hundred thousand of the 2 million wavelet coefficients are going to be the important ones, then the camera could just measure those coefficients and not even bother trying to measure the rest. (It is possible to measure a single coefficient by applying a suitable “filter” or “mask” to the image, and making a single intensity measurement to what comes out.) However, the camera does not know which of the coefficients are going to be the key ones, so it must instead measure all 2 million pixels, convert the image to a wavelet basis, locate the hundred thousand dominant wavelet coefficients to keep, and throw away the rest. (This is of course only a caricature of how the image compression algorithm really works, but we will use it for sake of discussion.)

Now, of course, modern digital cameras work pretty well, and why should we try to improve on something which isn’t obviously broken? Indeed, the above algorithm, in which one collects an enormous amount of data but only saves a fraction of it, works just fine for consumer photography. Furthermore, with data storage becoming quite cheap, it is now often feasible to use modern cameras to take many images with no compression whatsoever. Also, the computing power required to perform the compression is manageable, even if it does contribute to the notoriously battery-draining energy consumption level of these cameras. However, there are non-consumer imaging applications in which this type of data collection paradigm is infeasible, most notably in sensor networks. If one wants to collect data using thousands of sensors, which each need to stay in situ for long periods of time such as months, then it becomes necessary to make the sensors as cheap and as low-power as possible - which in particular rules out the use of devices which require heavy computer processing power at the sensor end (although - and this is important - we are still allowed the luxury of all the computer power that modern technology affords us at the receiver end, where all the data is collected and processed). For these types of applications, one needs a data collection paradigm which is as “dumb” as possible (and which is also robust with respect to, say, the loss of 10% of the sensors, or with respect to various types of noise or data corruption).

This is where compressed sensing comes in. The main philosophy is this: if one only needs a 100,000 components to recover most of the image, why not just take a 100,000 measurements instead of 2 million? (In practice, we would allow a safety margin, e.g. taking 300,000 measurements, to allow for all sorts of issues, ranging from noise to aliasing to breakdown of the recovery algorithm.) In principle, this could lead to a power consumption saving of up to an order of magnitude, which may not mean much for consumer photography but can be of real importance in sensor networks.

But, as I said before, the camera does not know in advance which hundred thousand of the two million wavelet coefficients are the important ones that one needs to save. What if the camera selects a completely different set of 100,000 (or 300,000) wavelets, and thus loses all the interesting information in the image?

(to be continued)

一不小心熬过了点，没什么睡意，不如记一笔昨晚看的雷人话剧。

 

当时豆瓣上收到邮件说回复报名可以免费看一场根据易卜生绝笔剧本改编的话剧《当我们死者醒来》，代价只是向贫困儿童捐两本书。各么我就回复了，一来我还没看过除阿加莎以外的话剧，二来手头闲书不少。

 

也许我贪了便宜还在背后说人坏话是很不厚道的。。。可是我忍不住阿，整剧用苏州话里的四个字来形容实在再合适不过了————假泥模模

 

先讲讲剧情吧，我是空白一片去看的。根据表演，剧情大致是：

 

某搞雕塑的艺术中年塑了一尊名为《复活日》的少女像，很美很纯洁，带来荣誉金钱地位。该艺术男作为生物自然为模特少女心动，但他强压冲动拒绝把模特少女当人看，于是模特少女非常鄙视该艺术男，愤而出走。后来艺术男又遇上一位少女，还许下不少老男人骗脑残少女的承诺，结果几年平淡生活过去，少女发现该艺术男啥也给不了她，正好又出现了一位强壮肌肉青年，于是跟青年跑了。艺术男也正好在疗养院又遇到了貌似已成死人的模特女，于是在模特女的质问与喝斥下，艺术男回顾了对艺术的追求和理解，承认了为追求艺术而压制人性是不人道的做法。两人重归于好，在池塘边嬉戏一番。这时天气变糟，艺术男非要开辟新道路下山，模特女表示紧紧跟随。结果，迷路，一番群魔乱舞后，少女倒地死亡，艺术男好像死了好像疯了。。。

 

当然，我回来后查了下易老师到底写了哪一出。有兴趣的可以查查《当我们死而复生时》。

关于改编的程度，对于易老师原作的理解等等，我当时其实也没怎么觉得雷，只想着自己到底不是文艺青年看不大懂其中的深意么总归是水平问题；而且偶尔也有几句台词听上去颇耐人寻味的。但是舞美服装道具真的把我惊到：太抽象了！太需要想象力了！！！

 

一进去但见舞台上只有一桌二椅，以及塑料绿叶在地板上圈出的一个圆。圆上吊着一段树枝。

 

开场，艺术男和后来跟肌肉男跑掉的少女坐在桌边，艺术男在看上海晨报！少女捧着十字绣用无影针在上下穿梭！接着，少女端起两个空酒杯邀艺术男喝酒，两人就着空杯喝得啧啧有声！————你们，你们在门外花两块钱买瓶纯净水倒上吧~~~

 

大概由于布景太简陋，连灯光也很不好意思，只打在人物身上，周围一片漆黑。两把椅子上很诡异地吊着红绳子，少女又总是把头伸在两根红绳中间讲话。我一遍遍提醒自己，这不是谋杀剧，不是谋杀剧。

 

接着一口上海普通话的肌肉男登场。他拿着一根木头，各角均为直角的原生态长木条。恕我想象力贫乏，直到肌肉男端起木头驾在肩膀上嘴里发出“砰~”的声音我终于悟出那其实是一把猎枪！

 

少女躺在地板上一直在擦地板上的灰，后来我明白了，人家是躺在草地上拔草玩。。。

 

模特女登场。插一句，说真的介个mm很美（很对得起观众）。可是剧组你们能不能不要这么省钱，唉，不用花钱买票的后果就是让人家美女只穿着一身棉毛衫裤套着一块黑布光着脚上台~~~ 把人家当大理石像也得找身比棉毛衫裤更专业一点的服装吧，我想起了小时候和表妹披着床单当斗篷的艰难而美好的时光。。。

 

说良心话，艺术男和模特女的几段戏虽然场景简陋，背景音乐切换粗糙，但是论表演还是挺引人入胜的。除了艺术男因为悔恨懊恼而啜泣的声音总让我怀疑他是不是因为感冒而流下清涕又没法擦只好不停地吸。。。我坐在第一排，真的看见他的鼻涕了。。。

 

那一圈绿叶围成的圆先是充当了雕塑的台基；后来，后来又成为两人嬉戏的池塘。。。艺术男捧着皇帝的新船，模特女逗弄着皇帝的飞鸟与游鱼，玩得不亦乐乎。

 

最神奇的是最后一段，艺术男和模特女在山间遭遇风暴。肌肉男和戴了面具的少女也出场。两个女孩就手握无形匕首，一边痛苦地扭动一边往身上扎去，两个男子背朝观众也做同样动作。无奈少女2号整部剧都手足僵硬身体语言患上“失语症”，以至于这段自杀式扭动怎么看怎么像广播体操。

 

雷人的高潮来了：当模特女倒下，艺术男背朝观众静止。舞台四角分别出现了四个人。分别是肌肉男和戴面具的少女，以及两位全身黑色棉毛衫裤戴面具的路人甲路人乙。猜得到他们在干嘛吗？他们每人拿着一小瓶公园门口卖给小朋友玩的吹肥皂泡的东西在努力地吹肥皂泡！他们一边沿对角线走，一边吹着肥皂泡！吹阿吹，吹阿吹，吹阿吹。。。然后下台了。

 

音乐嘎然而止。沉默。沉默。沉默。终于又有背景音乐出来，艺术男立马仰头哈了一声，一甩头（他怎么不再跺下脚呢。。），下台了。留下了匍匐在地的穿着棉毛衫裤的美女。。。

 

然后，没了。

 

大概这就是传说中的莫装B装B遭雷劈吧。

 

 

以下全部为摘抄。。。

 

十二律吕：

童鞋们，盆友们！我，活着回来了！

 

本人于2008年11月30日参加了人生第一个半程马拉松（东丽杯上海国际马拉松赛），顺利完成赛前三大目标：活着回来；跑到终点；跑进2小时半小时（呃，貌似我对自己要求不高啊）

 

净成绩：2小时07分；320名。。。（豆瓣上报成绩的都是2小时以内，搞得我都不好意思跟人打招呼orz）

 

－－－－－－－－－－关于准备工作的分割线－－－－－－－－－－－

虽然很多人在听说我报名马拉松（尽管是半程）后表示了惊诧和怀疑，尤其是听说我从前未曾跑过那么长距离（之前最长距离大概就是初中冬季长跑吧，1.5km还是3km来着:p），不过凡事总有第一次么，本人自诩体质向来不错，再加上一点长久没有跑步不知长跑之艰难而带来的盲目乐观（报名前有印象的近期跑步经历就是赶着去火车站），从一开始就对跑下半马就蛮有信心的说:p

 

大概是9月2x号，练了第一次：当时真是没概念啊，穿着一般的白球鞋，也没有做准备活动，在马路上跑了十几分钟，累得上气不接下气。当天就开始肌肉酸痛。而且一起报名的师姐表示退出。打击啊。。

 

几天后，换了适合慢跑的鞋去徐家汇公园。徐家汇公园其实不是一个适合练习跑步的地方，倒不是说路面比塑胶跑道或跑步机对膝盖伤害大，而是路实在曲里拐弯太多；好处则是我一开始就适应了公路跑:) 因为对公园地形不熟，绕的圈子不大，跑了将近二十分钟，除了气喘吁吁且腿脚酸痛外，绕得也有些晕头转向。

 

接着到了十一假期，出去玩了几天，也就没有练习。期间除打了回乒乓球，其实也不耗体力的说，没有运动量可言。

 

10月9日晚，第二次徐家汇公园。发现有志同道合者，便跟在人家屁股后头跑，基本上沿着公园最外围的路，3－4圈，用时半小时。还是气喘。

 

10月16日晚，第三次徐家汇公园。虽然我有一个梦：每周至少跑两三次，但是，到最后它还只是个梦=.=" 在豆瓣上看了不少关于马拉松的理论知识，开始注意简单的拉伸热身。再加上对地形和路线都已熟悉，这一次进步甚大，是上一次练习的翻倍：7圈，用时50分钟。后来根据网上测距显示，大约1.2km每圈，也就是跑了约8km。5km左右非常困难，但熬过去之后脚步反而感觉很轻快。公园有段小小的下坡，中途摔了一跤，蹭破了左膝。事实证明今天到最后最痛的还是左膝。

 

10月22日晚，第四次，徐家汇公园。周末回家参加朋友婚礼，于是离上一回又隔了又将近一礼拜。下雨，公园人很少，边听广播边跑了一个多钟头，10km左右。也是练习期间唯一一次10km。也许是空气清新之故？总之感觉非常好，仍然在5km左右出现困境。另外发现听广播对保持稳定的节奏并不有利，所以后来都养成了不带耳机边跑边想八卦的习惯，呵呵。尽管注意了跑后的拉伸，第二天恢复到底还有些困难。

 

10月28日晚，第五次，徐家汇公园。周末去普陀山玩，竟然还是腐败游，完全不消耗体力。由于打算加大周训练量，这次只跑了2km。我错了，应该每次都跑多一点的。。。至此第一个月的练习都是每周一次（现在回看练得实在太少了！）。

 

11月4日早，第六次，徐家汇公园。中间身体缘故还是没能完成加大周训练量的梦想。因为上马当天是一早7点半开跑，试试调整时间。半小时，将近4圈，节奏很稳定，每圈都是8分钟。早上空气很不错的说。不过早起真是艰难。

 

11月8日晚，第七次，徐家汇公园。终于在一周内跑了两次。保持稳定节奏，6圈，即6－7km。

 

11月11日早，第八次，徐家汇公园。仍然是4圈，略微提速，气温降低明显，于是疲劳感和呼吸不顺畅也变得明显。遇到帅哥搭讪，非常激动，心活念念以后每次都早上来跑，可惜后来天冷下来早上再没爬起来过>.<

 

11月19日下午，第九次，上医操场。吃了三次闭门羹后终于去成一次操场。在跑道上跑果然轻松很多，10圈，前五圈2'30"每圈，后四圈2'15"每圈，最后一圈全力冲刺。报名全程的ZW鼓励我说跑完半程应该没问题，但要在2小时内跑完速度得加快。但离比赛还有十天，练习速度已经没希望了。。。

 

最后十天，除了吃得比较多，没有任何准备工作-______-|||另外这十天中，最担心的就是身体问题，巧不巧地比赛日就在mc期间，虽然也有过mc期间参加校运会800m的经验，可是毕竟这个得跑2个多小时啊。。。好在还算没遇到最严重的时期。然后，就到了正式比赛啦。。。（晕，原来练了十次都不到。。。太无知无畏了orz）

 

－－－－－－－－－－准备与参赛的分割线－－－－－－－－－－

 

经验从来都是重要的：比如，我不知道只提前半小时到起点是来不及做热身练习的。。。组委会安排了存物车，安排得那叫一个乱啊，我在浙江中路、南京东路附近跑了两圈都没找到自己的包该存到哪辆车。要不是有辆车的负责人帅哥可怜我起跑时间将到还在没头苍蝇一样找车遂收留了我的包，估计我就得背着我的羽绒服跑了。。。赶到起跑处，人挤人的情况下（貌似有1万多人参加全程＋半程马拉松）只能活动下手腕脚腕和膝盖，以及站着牵拉下大腿部位的肌肉。拉伸没做开灰常失败。

 

起跑的枪声完全没有听到，只有队伍前面的人群发出的欢呼声。。。同去的同伴也早消失在人海中，随便跟着人潮往前走（对，是走，一开始哪里跑得开。。。），地上不时有别人扔下的一次性雨衣和外套。人很多，以很慢的速度往前边跑边观赏四周的跑步者和路边的观众，大家都很high，最初的五千米见到无数装备非常专业的鬼子、穿着老虎装的大叔（衣服上有条大尾巴。。。）、顶着五彩假发的大叔、戴着斗笠屁股上印着日本膏药的日本大叔等等，可惜手机很土拍不了照。当然还有上海各高校的长跑社团与外地组团来的长跑爱好者，相比之下觉得自己跑得有点孤单呢，以后若是所里或者院里的童鞋搞个logo或者旗子之类一起跑感觉会更好些吧。

 

到8km处几乎跑了1个钟头，比平时速度慢，和心理预期差不多。这一段是整个路程中跑得最好的一段，节奏和练习时一致，脚步很轻快，还能边跑边注意诸如大腿带动、足中着地等传说中的理论要点。路上隔一段就有一群以中年阿姨为主力的啦啦队，敲锣打鼓地喊着“运动员加油”，每当遇到这些笑得像花儿一般灿烂的观众我那小小的人来疯就有点激动，跑得比较带劲//shy 跑步者中男子比女子明显多很多，跑的时候我在想这个活动主要是男的跑女的喊嘛。。。

 

10km处第一次喝到水。这时候已经有些疲劳感出现，但趁着喝水的当口走了十数步的小歇息十分有效，再次跑起来似乎力量有所恢复。跑步的时候还有享受阳光的心情。

 

11km处的标牌让我产生又欣慰又绝望的心情：终于跑过一半路程，可是，还有10km呐！到14km之间的一段路非常疲劳，每一千米都觉得似乎跑了很久很久才看到标牌，不时地有停下来走走的念头，甚至还想这一定是我的大脑中白介素－6开始发挥作用，告诉我的肌肉我跑不了了。中途有座桥，许多人从上桥开始慢下脚步开始步行。真感谢路边一位喊加油的阿姨，非常及时地对我喊了一声“小姑娘加油”，我终于克制住步行的欲望。14km刚过边上一位帅哥掏出手机，非常轻松非常欢快地打起电话来：嗨～我在跑马拉松啊～大概还有半个小时到啊～ 我很汗颜。

 

15km左右是全程和半程的分叉点。别着绿色号码布的半程者开始往终点跑去，而别着白色号码布的全程者却往南面绕大圈子去了。那个时候我是多么地同情跑全程的人啊：我终于捱到看见希望的时候了，你们居然还没跑过一半，娃哈哈～ 刚过分叉点，就能感受到路边志愿者的热情了，个个在大呼“快到终点了哈”，手上还抓满了香蕉和糖果，跑步者伸手即得。可是，最后的四五公里更加漫长。脚步沉重地似乎每一步都从脚底直震到骨盆！大腿完全用不出力，似乎能感觉到里面的碳水化合物消耗殆尽。我后来看着自己那么粗的腿哀叹：里面都是脂肪哇～ 跑过其中一个交通管制的路口，听到有人向交警抱怨：怎么还不放我们走啦，人家第一第二名老早跑到终点勒～ 我好羞愧呀。。。

 

将近终点处有个折返点，还没到折返点就看到一些跑完半程的人领着纪念品往回走了。。。@@ 得赶紧呀，再不跑到终点，纪念品都被领光了哇。。。当时我的动力就是这样猥琐。。。

 

最后的五十米，居然还有力气冲刺，^_^ 而在终点等候多时的阿伏居然还一眼看到了我，还给我留下了帅照！不像某人，在终点看得大概都快睡着了，我跑过终点他都没看到=.="

 

然后，就结束啦:) 我还活着，也没用到发给跑步者人手一瓶的麝香保心丸:)

－－－－－－－－－－我是结束的分割线－－－－－－－－－－－

下午睡了一觉。累死了呜呜呜。好想坐在轮椅上。。。。

 

以此流水帐纪念我的第一个半马:p

 

//tea all