Detecting a curve with certain characteristics in a spectrogram
This answer might sound hard initially, but once understood it is intuitive and certainly deals well with the noise.

One smart way to model such constraints is either through HMM's or through MRFs. The design is similar, but for HMM I would give you some hint:

Prepare your transition matrix in such a way that Viterbi decoding would provide you the resulting curve. For example, the first state will be at 0 with certain variance, since it starts there. Then it is only allowed to increase, within a given tolerance. This tolerance would be obtained from a Gaussian prior. Say, the slope of the line (thus the variance of the Gaussian) would range from 1 to 5. Finally, you should only allow for a decrease or keeping constant. Similarly, for every other constraint you could do this. Here allowance means that the probability of transition from the constant state to an increasing state is 0. Otherwise you have a probability (might be due to any distribution, prior etc.)

You could also make use of Hierarchical HMMs to elaborate your model with such tracking parameters:

http://en.wikipedia.org/wiki/Hierarchical_hidden_Markov_model
http://www.cs.princeton.edu/courses/archive/spr06/cos598C/papers/FineSingerTishby1998.pdf