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Are you sure that your lens its not scratched or dirty? post some pictures
That's odd, well you can always stop by a Sony repair center
Try shooting using 8MP only. Most of the options are disabled if you are using other resolution rather than 8MP.
Here are some tips to effectively utilize your Z1 camera: (based on my personal experience)
1. Increasing ISO setting increases camera sensitivity w/c means lesser blur when taking moving subjects. Take note, you must get the right setting for this. ISO algorithm increases sharpness therefore more noise and grainy pictures if tweaked badly. It is best to set ISO setting to "AUTO" mode. This setting increases the chance of getting the right or acceptable ISO setting in Z1.
2. More chances of getting beautiful shots when Z1 recognizes the scene or environment. You will notice some icon under your viewfinder if Z1 successfully recognizes the scene. Some samples of this icons are flower for macro or closeup shots, a candle for low light conditions, mountain for landscapes etc. Do not take pictures when a man moving icon is displayed, this means the camera is not steady. Always enable the "image stabilizer" option for stabilty. Try moving the phone left or right or find a better place or angle to take pictures until it recognizes a scene. Remember Z1 only optimizes the scene when it does recognize it.
3. Do not use superior auto if you already know your scene. Sometimes superior auto fails to suggest the right SCN. For example you go outside of your house on a bright sunlight, you already know that it is sunny. Then superior auto indicates a face icon w/c means soft skin or face detection. This is not right, instead go to manual mode tap SCN and use HDR backlight correction, that setting is more suitable strong backlight scenes. Always use the SCN option in manual mode if the options available there fits your environment or scene. What if the scene you are taking does not included in SCN option in manual mode? Then it is time to use superior auto in these situation. If I'm not mistaken, superior auto uses the BIONZ processing algorithm w/c is optimized for up to 32 different SCN. Some of the optimized SCN included in the BIONZ processing algorithm are the low light SCN symbolizes the candle icon, flower icon for macro shots etc. These settings are not available in the SCN manual mode and can be used only in superior auto mode. Remember superior auto only uses 8MP not 20.7
4. Use superior auto in most low light conditions.
5. If you do not prefer using the manual SCN options. It is best to set the white balance options first before tweaking other options in manual mode. Set the right light condition for the right scene. Select the sun icon when shooting unser the sun, fluorescent icon in inside the house or under a fluorescent lamp, bulb if shooting in a bulb lighted environment etc and you'll be stunned and shocked with the Z1 camera capabilities.
Hope this helps (Y)